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a/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/2301.00276v1.pdf.txt b/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/2301.00276v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e66a8205be67556d1328a802c99e031cf4b4213 --- /dev/null +++ b/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/2301.00276v1.pdf.txt @@ -0,0 +1,4010 @@ +arXiv:2301.00276v1 [cs.IT] 31 Dec 2022 +1 +Impact of Phase-Shift Error on the Secrecy +Performance of Uplink RIS Communication +Systems +Abdelhamid Salem, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, and Chan-Byoung Chae, +Fellow, IEEE +Abstract +Reconfigurable intelligent surface (RIS) has been recognized as a promising technique for the sixth gen- +eration (6G) of mobile communication networks. The key feature of RIS is to reconfigure the propagation +environment via smart signal reflections. In addition, active RIS schemes have been recently proposed to +overcome the deep path loss attenuation inherent in the RIS-aided communication systems. Accordingly, this +paper considers the secrecy performance of up-link RIS-aided multiple users multiple-input single-output (MU- +MISO) communication systems, in the presence of multiple passive eavesdroppers. In contrast to the existing +works, we investigate the impact of the RIS phase shift errors on the secrecy performance. Taking into account +the complex environment, where a general Rician channel model is adopted for all the communication links, +closed-form approximate expressions for the ergodic secrecy rate are derived for three RIS configurations, +namely, i) passive RIS, ii) active RIS, iii) active RIS with energy harvesting (EH RIS). Then, based on the +derived expressions, we optimize the phase shifts at the RIS to enhance the system performance. In addition, +the best RIS configuration selection is considered for a given target secrecy rate and amount of the power +available at the users. Finally, Monte-Carlo simulations are provided to verify the accuracy of the analysis, +and the impact of different system parameters on the secrecy performance is investigated. The results in this +Abdelhamid Salem is with the department of Electronic and Electrical Engineering, University College London, London, UK, (emails: +a.salem@ucl.ac.uk). +Kai-Kit Wong is with the department of Electronic and Electrical Engineering, University College London, London, UK, Kai-Kit +Wong is also affiliated with Yonsei University, Seoul, Korea (email: kai-kit.wong@ucl.ac.uk). +Chan-Byoung Chae is with Yonsei University, Seoul, Korea (e-mail: cbchae@yonsei.ac.kr). +The work is supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/V052942/1. For the +purpose of open access, the authors will apply a Creative Commons Attribution (CCBY) licence to any Author Accepted Manuscript +version arising. + +2 +paper show that, an active RIS scheme can be implemented to enhance the secrecy performance of RIS-aided +communication systems with phase shift errors, especially when the users have limited transmission power. +Index Terms +Reconfigurable intelligent surface, Physical layer security, MU-MISO, MRC. +I. INTRODUCTION +Reconfigurable intelligent surface (RIS), also known as intelligent reflecting surface (IRS), has been +proposed recently as a promising technique to extend the coverage and improve the spectral efficiency +of wireless communication networks [1], [2]. Specifically, RIS is composed of reflecting elements, each +of which independently imposes a phase shift on the incident signals. By tuning the phase shifts of the +reflecting elements, RIS can convert the propagation environments into smart ones and thus enhance +the received signals quality [1], [2]. Due to these advantages, RIS techniques have been extensively +considered in the literature. For instance, in [3], the fundamental capacity limit of RIS-aided multiple- +input multipleoutput (MIMO) communication systems has been considered. The achievable ergodic rate +of a RIS-assisted MIMO system which comprises links of a Rician channel was derived in [4]. In [5], a +closed-form asymptotic ergodic sum rate of a RIS-assisted MIMO communication system was derived +under the assumption that the number of base station (BS) antennas tends to infinity. In [6], the up-link +achievable rate in RIS-aided massive MIMO systems has been analyzed and optimized. The authors +in [7], [8] analyzed the achievable rate of RIS-assisted multiple users (MU) up-link massive MIMO +system under Rician fading channels. In [9], [10], a closed-form expression of ergodic achievable rate +for RIS-aided massive MIMO systems with zero forcing (ZF) detector has been derived. In addition, a +closed-form analytical expression for the symbol error probability and the upper bound on the channel +capacity of a RIS communication system have been derived in [11]. The work in [12] considered the +impact of hardware impairments on a general RIS MU-MISO system with Rayleigh fading channels. +The ergodic capacity of RIS MIMO networks over Rayleigh-Rician channels was considered in [13]. +However, the practical implementation of passive RIS-aided communication systems may face several +challenges. For instance, the transmitted signal propagates through the RIS experiences a double-fading +attenuation, e.g, source-RIS and RIS-destination links. This issue has been tackled in the literature by +increasing the number of passive RIS elements [14]. However, this solution leads to an increase in + +3 +the size of the RIS module, which is impractical in some scenarios. To tackle this issue the authors +in [15] proposed RIS with active elements. The main idea of active RIS is to adjust the phase shifts +and also amplify the reflected signal attenuated from the first link with extra power consumption. +Theoretical comparison between the active RIS-assisted system and the passive RIS-aided system has +been presented in [16]. The results in [16] show that the active RIS has better performance than +passive RIS. The use of active RIS elements to overcome the double-fading problem has been also +investigated in [17], where the results illustrated that using active elements results in a severe reduction +in the physical size of RIS to achieve a certain performance. To reduce the power consumption of +active RIS, a sub-connected architecture has been proposed in [18]. The energy efficiency in an active +RIS-aided MU-MISO down-link system has been investigated in [19]. +Although fixed embedded batteries can be used to power the RIS, these batteries cannot be relied on +for long time and uninterrupted operations. In addition, wired charging might not be possible to use if +the RIS is deployed in inaccessible places. Therefore, equipping RIS elements with energy harvesting +(EH) modules can solve these issues. Accordingly, a self-sustainable RIS approach was proposed and +studied in the resent researches on RIS. In this regard, in [20] time switching (TS) and power splitting +(PS) EH protocols for the RIS to harvest sufficient amount of energy from an access point have +been proposed and investigated. The work in [21] considered a self-sustainable RIS-aided MU-MISO +communication systems, in which the RIS collected energy from the radio frequency (RF) transmitter +using the PS protocol. In [22], a novel transmission policy for a communication network assisted by +self-sustainable RIS has been proposed, where the RIS harvests energy from an energy transmitter to +support its operation. In [23], self-sustainable RIS with the PS protocol to assist broadcasting network +was studied. In [24], self-sustainable RIS-aided communication between a gateway and a device was +studied, in which the RIS harvested energy prior communication. +Moreover, due to the broadcast nature of wireless channels, confidential messages are vulnerable +to eavesdropping attacks. For the provision of secure transmission, physical layer security (PHYSec) +has been proposed from the information theory perspective [25], [26]. PHYSec exploits the nature of +wireless channels to enhance the system security [25], [26]. PHYSec of RIS systems has also been +studied in the literature. In [27], the secrecy throughput maximization problem has been formulated +and solved to enhance the secrecy performance of the RIS-assisted MIMO systems. In [28], a novel +active RIS design to enhance the security of wireless transmission was proposed. PHYSec of RIS- + +4 +aided wireless networks has been considered in [29] to achieve secure transmission between a source +and a legitimate user in the presence of a malicious eavesdropper. In [30], RIS has been used to +perform secure transmission from a multiple antennas transmitter to a multiple antennas legitimate +receiver. Further work in [31] considered the secrecy transmission in a RIS-aided multiple antennas +communication, where the secrecy rate was improved by optimizing the RIS location. In [32], an active +RIS-aided multiple antennas PHYSec transmission scheme was considered, where the active RIS was +designed to amplify the signal actively. +Accordingly, this paper investigates the impact of phase shift error on the secrecy performance of +up-link RIS-aided MU-MISO systems in the presence of multiple eavesdroppers. The BS receives +the users messages only through the RIS, while eavesdroppers can receive the signals from both the +direct and reflected links. Under Rician fading channels and phase shift errors, the ergodic secrecy +rate is analyzed for three RIS configurations, namely, 1) passive RIS, 2) active RIS, and 3) EH RIS. +Based on the derived rate expressions, the phase shifts at the RIS are optimized to enhance the system +performance. Then, the best RIS configuration selection is considered based on the target secrecy rate +and amount of power available at the users. For clarity we list the main contributions of this work as +follows: +1) We investigate the impact of RIS phase shift error on the secrecy performance of up-link MU- +MIMO systems in the presence of multiple passive eavesdroppers. +2) New closed-form explicit analytical expressions for the ergodic secrecy rate are derived for +the RIS-assisted MU-MIMO systems, when the RIS is passive, active and EH node under Rician +fading channels. This channel model is more general but also very challenging to be considered +mathematically. The derived secrecy rate expressions are simple, explicit and in closed form, and +provide several important practical design insights. +3) Based on the derived expressions, a genetic algorithm (GA)-based approach is used to obtain the +optimal phase shifts. Also, a simple suboptimal technique is proposed to enhance the secrecy rate for +a legitimate user. +4) Given a target secrecy rate, we calculate the required user power, and we present steps to select +best RIS configuration which depend mainly on the available power at the users. +5) Finally, Monte-Carlo simulations are performed to validate the analytical expressions. Then, the +impact of several system parameters on the secrecy performance are investigated. + +5 +The results in this work show that active RIS is an efficient scheme to achieve secure communication +in the presence of phase shift errors at the RIS, especially when there is no sufficient amount of power +at the users. +Next, Section II presents the RIS-aided uplink MU-MISO system model. In Section III, we derive +the ergodic secrecy rate of the passive RIS model. Section IV presents the ergodic secrecy rate of the +active RIS scheme. Section V derives the ergodic secrecy rate of the EH RIS scheme. Section VII +depicts our numerical results. Our main conclusions are summarized in Section VIII. +II. SYSTEM MODEL +Consider a typical up-link RIS-aided MU-MISO communication system consisting of a multiple +antennas BS, an RIS and K single-antenna users in the presence of J single antenna passive eaves- +droppers. The BS is equipped with N antennas, and the RIS is equipped with M reflecting elements, +as shown in Fig. 1. +UE 1 +UE k +UE K +. +. +. +. +Eave J +. +. +Eave 1 +BS +RIS +Figure 1: An RIS-aided uplink MU-MISO system with N BS antennas, M RIS elements, K users and +J eavesdroppers. +The BS and RIS are connected to control and adjust the phase shifts of the the RIS elements. It is +assumed that the eavesdroppers can hear the signals from the direct and reflected links, and trying to +eavesdrop a specific confidential message in the system. On the other side, the direct links between the +users and BS are assumed to be blocked, which justifies the use of the RIS. It is known that, the RIS is +most likely to be installed on the buildings, and thus it can create channels dominated by line-of-sight +(LoS) path along with scatters. Accordingly, a Rician fading model is considered for the RIS channels. +The channel matrix between the RIS and the BS is denoted by G ∈ CN×M , and the channel vector + +6 +between user k and the RIS is presented by hr,k ∈ CM×1. The mathematical expressions of the channel +matrix G and the channel vector hr,k can be expressed, respectively, as +G = +�� +ρb +ρb + 1 +¯G + +� +1 +ρb + 1 +˜G +� +, +hr,k = +�� +ρk +ρk + 1 +¯hr,k + +� +1 +ρk + 1 +˜hr,k +� +(1) +where ρb and ρk are the Rician factors, ¯G and ¯hr,k are the LoS components and ˜G and ˜hr,k are the +NLoS components, in which +¯G = aN (φa +r, φe +r) aH +M (φa +t , φe +t) , +¯hr,k = aM (φa +kr, φe +kr) +(2) +where φa +kr, φe +kr denote the azimuth and elevation angles of arrival (AoA) from user k to the RIS , +respectively, φa +t , φe +t are the azimuth and elevation angles of departure (AoD) at the BS from the RIS, +respectively, φa +r, φe +r are the azimuth and elevation AoA from the RIS to the BS, respectively. The kth +element of the vector aX can be written as [aX (φ1, φ2)]k = ej2π d +λ (xk sin φ1 sin φ2+yk cos φ2), where λ is the +wavelength, d is the elements/antennas spacing, and xk = (k − 1) mod√x, yk = k−1 +√ +X . On the other +hand, the channel vector between the RIS and eavesdropper j is presented by hej,r ∈ C1×M , and +the channel from user k to eavesdropper j is hej,k ∈ C1×1. The direct channel fading is assumed to +be Rayleigh fading due to extensive scatterers, while for the RIS-related channels, is assumed to be +Rician fading. Thus the expression of hej,ris given by +hej,r = +�� +ρej,r +ρej,r + 1 +¯hej,r + +� +1 +ρej,r + 1 +˜hej,r +� +(3) +where ρej,r is the Rician factor, ¯hej,r and ˜hej,r are the LoS of NLoS components, respectively. +The channel state information (CSI) of the eavesdroppers is assumed to be unknown at the BS/RIS +(only statistical information can be known), and the eavesdroppers are non-colluding. Therefore, the +ergodic secrecy rate can be calculated by [33] +ˆRs = +� +ˆRbk − ˆRej,k +�+ +(4) +where [l]+= max (0, l), ˆRbk = E {Rbk}, Rbk is the up-link rate of user k, and ˆRej,k = max E +� +Rej,k +� +, +Rej,k is the rate at eavesdropper j. + +7 +In the following sections, we consider the secrecy performance of the three RIS configurations. +III. PASSIVE RIS +As we have mentioned earlier, passive RIS reflects the users messages constructively to the BS with +passive elements. Thus, the received signal at the BS can be expressed as +yb = +K +� +k=1 +� +pk Luk,bG˜Θhr,kxk + nb +(5) +where Luk,b = d−αr +uk,rd−αb +r,b +is the large scale fading, duk,r is the distance between user k and RIS, dr,b +is the distance between RIS and the BS, αr and αb are the path-loss exponents, nb is the additive +wight Gaussian noise (AWGN) at the BS, nb ∼ CN (0, σ2 +bI), ˜Θ = ¯ΘΘ where Θ = diag (θ), and +θ = [θ1, ......, θM]Tis the RIS reflection coefficients with θm = ejϕm, where ϕm∈[0, 2π) is the phase +shift of element m. However, in practical systems, phase shift errors can exist due to imperfect channel +knowledge and finite precision in phase adjustment. Thus, we define ¯Θ = [ej ¯ +ϕ1, ...., ej ¯ +ϕM] as the phase- +shift errors at the RIS. The phase-error is modeled according to Von-Mises (VM) distribution with +zero-mean and a characteristic function (CF) E [ej ¯ +ϕm] = I1(κ) +I0(κ) = ρ (κ), where κ is the concentration +parameter and Ii is the modified Bessel function of the first kind and order i. By applying the receive +beamforming vector wk at the BS, the received signal of user k is +yb,k = +� +pk Luk,bwkG¯ΘΘhr,kxk+ +K +� +i=1 +i̸=k +� +pi Lui,bwkG¯ΘΘhr,ixi + wknb. +(6) +On the other hand, the received signal at eavesdropper j to detect user k signal is +yej,k = √pkxk +�� +d−αe +ej,k hej,k + +� +Luk,ejhej,r ¯ΘΘhr,k +� ++ +K +� +i=1 +i̸=k +√pixi + + + +� +d−αe +ej,i hej,i+ +K +� +i=1 +i̸=k +� +Lui,ejhej,r ¯ΘΘhr,i + + + + nej +(7) +where d−αe +ej,k is the distance between user k and eavesdropper j, αe is the path-loss exponent, Luk,B = +d−αr +uk,rd−e +ej,r and d−e +ej,r denotes the distance between the RIS and eavesdropper j. + +8 +To calculate the ergodic secrecy rate, the ergodic up-link rate for user k and ergodic rate at the +eavesdropper j should be derived, which will be considered in the following sub-sections. +A. Ergodic Up-link rate of user k +To calculate the ergodic user rate, maximum ratio combining (MRC) is adopted at the BS. The +beamforming matrix is given by W = (GΘH)H, and thus wk = hH +r,kΘHGH. The signal to interference +plus noise ratio (SINR) at the BS to decode user k signal can be written as +γbk = +pk Luk,b +��hH +r,kΘHGHGΘ¯Θhr,k +��2 +K� +i=1 +i̸=k +pi Lui,b +��hH +r,kΘHGHGΘ¯Θhr,i +��2 + +��hH +r,kΘHGH��2 σ2 +b +. +(8) +Lemma 1. The ergodic up-link rate of user k in passive RIS-aided MU-MISO systems under Rician +fading channels and with phase shift error can be calculated by +E {Rbk} ≈ log2 + + + + + + + +1 + +pk Luk,bξk +K� +i=1 +i̸=k +pi Lui,bςi + υkσ2 +b + + + + + + + +(9) +where +ξk = E +���hH +r,kΘHGHGΘ¯Θhr,k +��2� += +1 +(ρb + 1)2 (ρk + 1)2 +� +a1N2 + a2NM2 + a3NM + a4N +� +a1 = +� +ρ (κ)2 (ρk + ρb + 1)2 + +� +1 − ρ (κ)2� +ρkρ2 +b + ρ2 +b +� +M2 ++ +�� +(2ρk + 3ρb + 2 − ρkρb) ρ (κ)2 + (1 + ρk) ρb +� +ρbρk |fk|2 + (ρk + ρb + 2)2 +− ρ (κ)2 (ρk + ρb + 1)2 − 2ρ (κ)2 ρkρb − 2 +� +M ++ρ (κ)2 ρ2 +bρ2 +k |fk|4 + 2 +�� +1 − ρ (κ)2� +(ρk + ρb) + 2 +� +ρbρk |fk|2, +a2 = +� +−ρ (κ)2 ρkρb (1 + ρk) M2� ++ (ρk + ρb + 1) ρbρr + (ρk + ρb + 1)2 − (ρk + 1) ρ2 +b, +a3 = +�� +(ρk + 1) ρ (κ)2� ++ (ρk + 1) +� +ρbρk |fk|2 − 2ρbρkρ (κ)2 + 2ρbρk + 2ρk + 2ρb − 1, +a4 = 2ρbρr |fk|2 � +1 + ρ (κ)2� +and + +9 +ςi = E +���hH +r,kΘHGHGΘ¯Θhr,i +��2� += +1 +(ρb + 1)2 (ρk + 1) (ρi + 1) +� +b1N2 + b2NM2 + b3NM +� +b1 = +� +ρi + 1 − ρ (κ)2 ρi +� +M2ρ2 +b ++M +�� +ρi + 1 − ρ (κ)2 ρi +� +ρ2 +bρk |fk|2 + ρ (κ)2 ρ2 +bρi |fi|2 + (ρk + 2ρb + 1) +� +ρi + 1 − ρ (κ)2 ρi +� ++ ρ (κ)2 ρi +� ++ +� +2ρb |fi|2 + ρk +��¯hH +k ¯hi +��2 + 2ρbρkRe +� +f ∗ +kfi¯hH +i ¯hk +�� +ρ (κ)2 ρi ++ +� +ρ (κ)2 ρbρi |fi|2 + 2ρi +� +1 − ρ (κ)2� ++ 2 +� +ρbρk |fk|2 +b2 = +� +(ρb + 1) ρk + (ρb + 1)2 − ρ2 +b +� +(ρi + 1) − (ρb + 1) ρbρiρ (κ)2 − 1 +b3 = (ρi + 1) ρbρk |fk|2 + (ρk + 1) ρ (κ)2 ρbρi |fi|2 +and +υk = E +���hH +r,kΘHGH��2� += +Luk,b +(ρb + 1) (ρk + 1) +� +ρbρk |fk|2 + (ρb + ρk + 1) M +� +Proof: The proof is provided in Appendix A. +B. Ergodic Rate at Eavesdropper j +The SINR at eavesdropper j to decode user k signal can be expressed as +γej,k = +pk +���d +− αr +2 +uk,r d +− αe +2 +ej,r hej,rΘ¯Θhr,k + d +− αe +2 +ej,k hej,k +��� +2 +K� +i=1 +i̸=k +pi +���d +− αr +2 +ui,r d +− αe +2 +ej,r hej,rΘ¯Θhr,i + d +− αe +2 +ej,i hej,i +��� +2 ++ σ2ej +. +(10) +Lemma 2. The ergodic rate at eavesdropper j in up-link passive RIS-aided MU-MISO systems under +Rician fading channels and with phase shift error can be calculated by +E +� +Rej,k +� += log2 + + + + + + + +1 + +pk xk +K� +i=1 +i̸=k +pi yi + σ2 +ej + + + + + + + +(11) +where +xk = +� +d−αr +uk,rd−αe +ej,r +� +ρej +ρej +1 +ρk +ρk+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρk+1M + +ρk +ρk+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρk+1M +� ++ d−αe +ej,r +� +, +and + +10 +yi = d−αr +ui,r d−αe +ej,r +� +ρej +ρej +1 +ρi +ρi+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρi+1M + +ρi +ρi+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρi+1M + d−αe +ej,i +� +. +Proof: The proof is provided in Appendix B. +Finally, the ergodic secrecy rate in passive RIS scheme is presented in the next theorem. +Theorem 1. The ergodic secrecy rate in passive RIS-aided MU-MISO systems under Rician fading +channels and with phase shift error can be calculated by +ˆRs = + + +log2 + + + + + + + +1 + +pk Luk,bξk +K� +i=1 +i̸=k +pi Lui,bςi + υkσ2 +b + + + + + + + +− log2 + + + + + + + +1 + +pk xk +K� +i=1 +i̸=k +pi yi + σ2ej + + + + + + + + + +.+ +(12) +IV. ACTIVE RIS +As we mentioned earlier, active RIS can adjust the phase shifts and also amplify the reflected signal +to compensate the attenuation from the first link with extra power consumption. The signal reflected +by the active IRS can be written as +yr = ˜Θ +K +� +k=1 +� +pk d−αr +uk,rhr,ixi + ˜Θnr +(13) +where nr is the noise at RIS elements nr ∼ CN (0, σ2 +rI). In this case ˜Θ = ¯ΘΘ where Θ = diag (θ), +and θ = [θ1, ......, θM]T with θm = ̺mejϕm, ̺m > 1 and ϕm∈[0, 2π) represents the amplification factor +and phase shift coefficient, respectively, at element m. For simplicity, we assume that ̺m = ̺ and then +define Θ = ̺diag {ejϕ1, ...., ejϕM}. The active RIS amplification power can be expressed as +Pr = +� K +� +k=1 +pk +dαr +uk,r +E +����˜Θhr,i +��� +2� ++ E +����˜Θnr +��� +2�� += +� K +� +k=1 +pk +dαr +uk,r +M̺2 + M̺2σ2 +r +� +(14) +where E +���˜Θnr +��� +2 += M̺2σ2 +r, and E +���˜Θhr,i +��� +2 += +̺2 +ρi+1 +� +ρiE +�¯hH +r,i¯hr,i +� ++ E +� +˜hH +r,i˜hr,i +�� += +̺2 +ρi+1 (ρiM + M) = +M̺2. Thus, the amplification factor for each element on the active RIS is given by +̺ = +� +� +� +� +� +Pr +M +� K +� +k=1 +pk +dαr +uk,r + σ2 +r +�. +(15) + +11 +By applying the receive beamforming vector wk at the BS, the received signal of user k is +yb,k = +� +pk Luk,bwkG¯ΘΘhr,kxk+ +K +� +i=1 +i̸=k +� +pi Lui,bwkG¯ΘΘhr,ixi + +� +d−αr +r,b wkG¯ΘΘnr + wknb. +(16) +On the other hand, the received signal at eavesdropper j to detect user k signal is +yej,k = √pkxk +�� +d−αe +ej,k hej,k + +� +Luk,ejhej,r ¯ΘΘhr,k +� ++ +K +� +i=1 +i̸=k +√pixi + + + +� +d−αe +ej,i hej,i+ +K +� +i=1 +i̸=k +� +Lui,ejhej,r ¯ΘΘhr,i + + + + +� +d−αe +ej,r hej,r ¯ΘΘnr + nej. +(17) +A. Ergodic Up-link rate of user k +Applying MRC beamforming at the BS, the SINRs at the BS to decode user k signal can be expressed +as +γbk = +pk Luk,b +��hH +r,kΘHGHGΘ¯Θhr,k +��2 +K� +i=1 +i̸=k +pi Lui,b +��hH +r,kΘHGHGΘ¯Θhr,i +��2 + d−αr +r,b +��hH +r,kΘHGHG¯ΘΘ +��2 σ2 +r + +��hH +r,kΘHGH��2 σ2 +b +. +(18) +Lemma 3. The ergodic up-link rate of user k in active RIS-aided MU-MISO systems under Rician +fading channels and with phase shift error can be calculated by +E {Rbk} ≈ log2 + + + + + + + +1 + +pk Luk,bξk̺4 +K� +i=1 +i̸=k +pi Lui,bςi̺4 + ̺4d−αr +r,b σ2 +rνk + ̺2υkσ2 +b + + + + + + + +(19) +where +νk = E +��hH +r,kΘHGHG¯ΘΘ +��2 = +1 +(ρb + 1) +� +(ρk + 1) +(X1 + X2) +(20) +and X1 = E +� +|∆1,1|2� ++ E +� +|∆1,2|2� ++ E +� +|∆1,3|2� ++ E +� +|∆1,4|2� ++ E +� +∆1,1∆∗ +1,4 +� + +12 +E +� +|∆1,1|2� += ρ2 +bρk +��� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +���2 × M, +E +� +|∆1,2|2� += ρbρk +��aH +M (φa +kr, φe +kr) ΘHaM (φa +r, φe +r) +��2 NM, +E +� +|∆1,3|2� += ρbρkMN +� +ρ (κ)2 M + +� +1 − ρ (κ)2� +M +� +, E +� +|∆1,4|2� += ρk +� +N2M + NM2� +, +E +� +∆1,1∆∗ +1,4 +� += ρbρk +� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� +× (aM (φa +r, φe +r) Θ) ρkaH +M (φa +kr, φe +kr) ΘHNΘ, +and X2 = E +� +|∆2,1|2� ++ E +� +|∆2,2|2� ++ E +� +|∆2,3|2� ++ E +� +|∆2,4|2� ++ E +� +∆2,1∆∗ +2,4 +� +E +� +|∆2,1|2� += ρ2 +b +��ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) aM (φa +r, φe +r) Θ +��2 +F , +E +� +|∆2,2|2� += ρb +��ΘHaM (φa +r, φe +r) +��2 NM, +E +� +|∆2,3|2� += ρbMN +���aH +M (φa +r, φe +r) Θ¯Θ +��2� += ρbM2N, +E +� +|∆2,4|2� += ρ2 +k +� +N2M + NM2� +, +E +� +∆2,1∆∗ +2,4 +� += ρk +� +ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� +(aM (φa +r, φe +r) Θ) ρkΘNΘH. +Proof: The proof is provided in Appendix C. +B. Ergodic Rate at Eavesdropper j +The SINR at eavesdropper j to decode user k signal in this scenario can be written as + +13 +γej,k = +pk +���d +− αr +2 +uk,r d +− αe +2 +ej,r hej,rΘ¯Θhr,k + d +− αe +2 +ej,k hej,k +��� +2 +K� +i=1 +i̸=k +pi +���d +− αr +2 +ui,r d +− αe +2 +ej,r hej,rΘ¯Θhr,i + d +− αe +2 +ej,i hej,i +��� +2 ++ d−αe +ej,r +��hej,r ¯ΘΘ +��2 σ2r + σ2ej +. +(21) +Lemma 4. The ergodic rate at eavesdropper j in up-link active RIS-aided MU-MISO systems under +Rician fading channels and with phase shift error can be calculated by +E +� +Rej,k +� += log2 + + + + + + + +1 + +pk xj +K� +i=1 +i̸=k +piyi + zjσ2 +r + σ2 +ej + + + + + + + +(22) +where +xj = +� +d−αr +uk,rd−αe +ej,r ̺2 � +ρej +ρej +1 +ρk +ρk+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρk+1M + +ρk +ρk+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρk+1M +� ++ d−αe +ej,r +� +, +yk = d−αr +ui,r d−αe +ej,r ̺2 � +ρej +ρej +1 +ρi +ρi+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρi+1M + +ρi +ρi+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρi+1M + d−αe +ej,i +� +which have been derived in Appemdix B, and +zj = d−αe +ej,r E +���hej,r ˜Θ +��� +2 += d−αe +ej,r +̺2 +ρej,r+1 +� +ρej,rE +� +¯hH +ej,r¯hej,r +� ++ E +� +˜hH +ej,r˜hej,r +�� += d−αe +ej,r +̺2 +ρej,r+1 +� +ρej,rM + M +� += d−αe +ej,r M̺2. +The ergodic secrecy rate in active RIS scheme is presented in the following Theorem. +Theorem 2. The ergodic secrecy rate in active RIS-aided MU-MISO systems under Rician fading +channels and with phase shift error can be calculated by +ˆRs = + + +log2 + + + + + + + +1 + +pk Luk,bξk̺4 +K� +i=1 +i̸=k +pi Lui,bςi̺4 + ̺4d−αr +r,b σ2rνk + ̺2υkσ2 +b + + + + + + + +− log2 + + + + + + + +1 + +pk xj +K� +i=1 +i̸=k +piyi + zjσ2r + σ2ej + + + + + + + + + ++ +. +(23) +V. EH RIS +Following the recent works in [20], [21], [22], [23], [24], in this section, the RIS is an energy +constrained node and it can harvest RF energy to support its operation. Thus, in this scenario the + +14 +whole operation time block, T, is split into two time periods, the energy transfer (ET) slot and the +information transfer (IT) slot. During the ET slot, the BS transmits energy signals to the RIS to support +its operation. During the IT slot, the users deliver their messages to the BS through the RIS. We denote +τT as the time duration for the ET, and (1 − τ) T as the time duration for IT. The received signals at +the RIS in the first sub-slot is expressed as +yr = +� +PbGpWpxp + nr +(24) +where Pb is the BS power, Gp = +�� +ρp +ρp+1 ¯Gp + +� +1 +ρp+1 ˜Gp +� +is the BS-RIS channel in the ET slot, Wp +is the precoding matrix and xp is the energy signals vector. Using the maximum ratio transmission +(MRT) scheme, the harvested power at the RIS can be expressed as Pr = ηeff τPb∥Gp∥2 +F +1−τ +, which can +be written as Pr = +ηeff τPbTr(GbGH +b ) +1−τ +where ηeff is the efficiency of EH. Since GbGH +b +has Wishart +distribution, the average harvested power can be written as +Pr = ηeffτPbE +� +Tr +� +GbGH +b +�� +1 − τ += ηeffτPbNM +1 − τ +. +(25) +By substituting (25) into (15), the amplification factor for each element on the RIS in this case is given +by +ˆ̺ = +� +� +� +� +� +ηeffτPbNM +M (1 − τ) +� K� +k=1 +pk +dαr +uk,r + σ2 +r +�. +(26) +A. Ergodic Up-link rate of user k +Applying MRC beamforming at the BS, the SINR at the BS to decode user k signal can be expressed +as +γbk = +pk Luk,b +��hH +r,kΘHGHGΘ¯Θhr,k +��2 +K� +i=1 +i̸=k +pi Lui,b +��hH +r,kΘHGHGΘ¯Θhr,i +��2 + d−αr +r,b +��hH +r,kΘHGHG¯ΘΘ +��2 σ2r + +��hH +r,kΘHGH��2 σ2 +b +. +(27) +Lemma 5. The ergodic up-link rate of user k in EH RIS-aided MU-MISO systems under Rician fading +channels and with phase shift error can be calculated by + +15 +E {Rbk} ≈ (1 − τ) log2 + + + + + + + +1 + +pk Luk,bξkˆ̺4 +K� +i=1 +i̸=k +pi Lui,bςiˆ̺4 + ˆ̺4d−αr +r,b σ2rνk + ˆ̺2υkσ2 +b + + + + + + + +. +(28) +Proof: This expression can be obtained by following same derivation in Appendix C. +B. Ergodic Rate at Eavesdropper j +The SINR at eavesdropper j to decode user k signal is given by +γej,k = +pk +���d +− αr +2 +uk,r d +− αe +2 +ej,r hej,rΘ¯Θhr,k + d +− αe +2 +ej,k hej,k +��� +2 +K� +i=1 +i̸=k +pi +���d +− αr +2 +ui,r d +− αe +2 +ej,r hej,rΘ¯Θhr,i + d +− αe +2 +ej,i hej,i +��� +2 ++ d−αe +ej,r +��hej,r ¯ΘΘ +��2 σ2 +r + σ2 +ej +. +(29) +Lemma 6. The ergodic rate at eavesdropper j in up-link EH RIS-aided MU-MISO systems under +Rician fading channels and with phase shift error can be calculated by +E +� +Rej,k +� += (1 − τ) log2 + + + + + + + +1 + +pk ˆxj +K� +i=1 +i̸=k +piˆyi + ˆzjσ2r + σ2ej + + + + + + + +(30) +where +ˆxj = +� +d−αr +uk,rd−αe +ej,r ˆ̺2 � +ρej +ρej +1 +ρk +ρk+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρk+1M + +ρk +ρk+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρk+1M +� ++ d−αe +ej,r +� +, +ˆyi = d−αr +ui,r d−αe +ej,r ˆ̺2 � +ρej +ρej +1 +ρi +ρi+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρi+1M + +ρi +ρi+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρi+1M + d−αe +ej,i +� +ˆzj = d−αe +ej,r M ˆ̺2, +which have been derived in the previous section. +Finally, the ergodic secrecy rate in EH RIS scheme is presented in the next Theorem. +Theorem 3. The ergodic secrecy rate of user k in EH active RIS-aided MU-MISO systems under +Rician fading channels and with phase shift error can be calculated by + +16 +ˆRs += + + +(1 − τ) log2 + + + + + + + +1 + +pk Luk,bξk ˆ̺4 +K� +i=1 +i̸=k +pi Lui,bςiˆ̺4 + ˆ̺4d−αr +r,b σ2 +rνk + ˆ̺2υkσ2 +b + + + + + + + +− (1 − τ) log2 + + + + + + + +1 + +pk ˆxj +K� +i=1 +i̸=k +piˆyi + ˆzjσ2r + σ2ej + + + + + + + + + ++ +. +(31) +VI. SYSTEM DESIGN +In this section, based on the derived analytical expressions, we first design the phase shifts of the RIS +configurations considered in this work. Then, the best RIS configuration selection scheme is presented. +A. Phase Shift Optimization +The secrecy rate expressions presented in Theorems, 1, 2 and 3, show that the secrecy performance +relies on the phase shifts of the RIS elements. In this work, it is assumed that the CSI of the +eavesdroppers is unknown at the BS/RIS (only channel distribution known). Therefore, to enhance +the system performance, the RIS phase shifts can be optimized by maximizing the achievable ergodic +sum rate. Since the phase shift at each unit of the RIS lies in the range of [0; 2π), the phase shift +optimization problem can be formulated as +max +Θ +K� +i=1 +ˆRbi +s.t +θm ∈ [0, 2π) , +∨m. +(32) +Due to the complicated formula of the ergodic sum rate, it is difficult to optimize (32) based on +the conventional techniques. However, GA-based methods can be employed to solve this optimization +problem. Due to the page limitation, we refer readers to [6] for more details about the GA methods. + +17 +As an efficient suboptimal solution, the RIS phase shifts can be aligned to user k, who transmits +the confidential message. This presents a simple sub-optimal solution for enhancing the secrecy rate +[6]. Accordingly, the phase shifts should be +θm = −2π d +λ (xmtk + ymlk) , tk = sin φa +kr sin φe +kr − sin φa +t sin φe +t, lk = cos φe +kr − cos φe +t. +(33) +B. RIS Configuration Selection Scheme +Based on the required secrecy rate (rs) and amount of the power available at user k, and the RIS, +we can decide which system configuration, i.e., passive RIS, active RIS or EH RIS, should be selected. +A) If user k has sufficient amount of power to achieve the target secrecy rate, in this case passive +RIS can be implemented. Based on the secrecy rate expression provided in Theorem 1, the required +user k power, pk, to achieve the target secrecy rate, rs, can be obtained by solving +rs = log2 + + + + + + + +1 + +pk Luk,bξk +K� +i=1 +i̸=k +pi Lui,bςi + υkσ2 +b + + + + + + + +− log2 + + + + + + + +1 + +pk xk +K� +i=1 +i̸=k +pi yi + σ2 +ej + + + + + + + +(34) +which can be found as +pk = p1 − p2 +p3 − p4 +(35) +where p1 = +K +� +i=1 +i̸=k +pi Lui,bςi+υkσ2 +b +K +� +i=1 +i̸=k +pi Lui,bςi+υkσ2 +b +, p2 = +2rs +K +� +i=1 +i̸=k +pi yi+2rsσ2 +ej +K +� +i=1 +i̸=k +pi yi+σ2ej +, p3 = +2rs xk +K +� +i=1 +i̸=k +pi yi+σ2ej +and p4 = +Luk,bξk +K +� +i=1 +i̸=k +pi Lui,bςi+υkσ2 +b +. +B) If user k has limited amount of power, e.g., the user power, pk, is less than the power required +in (35). In this case active RIS can be implemented to provide the target secrecy rate. Based on the +secrecy rate expression provided in Theorem 2, the required RIS power, ̺ or Pr ,to achieve the target +secrecy rate, rs, can be obtained by solving + +18 +rs = log2 + + + + + + + +1 + +pk Luk,bξk̺2 +K� +i=1 +i̸=k +pi Lui,bςi̺2 + ̺2d−αr +r,b σ2rνk + υkσ2 +b + + + + + + + +− log2 + + + + + + + +1 + +pk̺2x1 + pkx2 +K� +i=1 +i̸=k +pi̺2y1i+ +K +� +i=1 +i̸=k +piy2i + z1̺2σ2r + σ2ej + + + + + + + +(36) +where x1 = d−αr +uk,rd−αe +ej,r +� +ρej +ρej +1 +ρk +ρk+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρk+1M + +ρk +ρk+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρk+1M +� +, x2 = +d−αe +ej,r , +y1i = d−αr +ui,r d−αe +ej,r +� +ρej +ρej +1 +ρi +ρi+1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej +1 +1 +ρi+1M + +ρi +ρi+1 +1 +ρej +1M + +1 +ρej +1 +1 +ρi+1M +� +, y2i = d−αe +ej,i , +and z1 = d−αe +ej,r M. After some simplifications, the last equation can be expressed as +̺4 (q1 − q3) + ̺2 (q2 − q4 − q5 + q7) + (q8 − q6) = 0 +(37) +where +q1 = +K� +i=1 +i̸=k +pi Lui,bςi +K +� +i=1 +i̸=k +piy1i+d−αr +r,b σ2 +rνk +K� +i=1 +i̸=k +piy1i+ +K +� +i=1 +i̸=k +pi Lui,bςiz1σ2 +r+d−αr +r,b σ2 +rνkz1σ2 +r+ +K� +i=1 +i̸=k +pi Lui,bςipkx1+ +d−αr +r,b σ2 +rνkpkx1, +q2 = + +υkσ2 +b +K +� +i=1 +i̸=k +piy1i + υkσ2 +bz1σ2 +r + υkσ2 +bpkx1 + + , +q3 = +K� +i=1 +i̸=k +piy1ipk Luk,bξk+z1σ2 +rpk Luk,bξk+ +K +� +i=1 +i̸=k +piy1i +K� +i=1 +i̸=k +pi Lui,bςi+z1σ2 +r +K +� +i=1 +i̸=k +pi Lui,bςi+ +K +� +i=1 +i̸=k +piy1id−αr +r,b σ2 +rνk+ +z1σ2 +rd−αr +r,b σ2 +rνk, +q4 = +K� +i=1 +i̸=k +piy2ipk Luk,bξk+σ2 +ejpk Luk,bξk+ +K +� +i=1 +i̸=k +piy2i +K +� +i=1 +i̸=k +pi Lui,bςi+σ2 +ej +K� +i=1 +i̸=k +pi Lui,bςi+ +K� +i=1 +i̸=k +piy2id−αr +r,b σ2 +rνk+ +σ2 +ejd−αr +r,b σ2 +rνk, +q5 = + + +K� +i=1 +i̸=k +piy1iυkσ2 +b + z1σ2 +rυkσ2 +b + +, q6 = +K� +i=1 +i̸=k +piy2iυkσ2 +b + σ2 +ejυkσ2 +b, +q7 = +K� +i=1 +i̸=k +pi Lui,bςi̺22rspkx2+̺2d−αr +r,b σ2 +rνk2rspkx2+ +K +� +i=1 +i̸=k +pi Lui,bςi2rsσ2 +ej+d−αr +r,b σ2 +rνk2rsσ2 +ej+ +K +� +i=1 +i̸=k +pi Lui,bςi2rs +K� +i=1 +i̸=k +piy2i + d−αr +r,b σ2 +rνk2rs +K� +i=1 +i̸=k +piy2i, + +19 +q8 = υkσ2 +b2rspkx2 + υkσ2 +b2rsσ2 +ej + υkσ2 +b2rs +K +� +i=1 +i̸=k +piy2i. +Thus, from (15), the RIS power should be higher than or equal to +Pr = M + +− (q2 − q4 − q5 + q7) ± +� +(q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6) +2 (q1 − q3) + + +� K +� +k=1 +pk +dαr +uk,r ++ σ2 +r +� +. +(38) +C) If user k and the RIS have limited amount of power, e,g., user k power, pk, is less than the +required power in (35) and the RIS power, Pr, is less than the required power in (38). In this case EH +RIS can be implemented to provide the target secrecy rate. Based on (25) and (38), the required BS +power, Pb, to charge the RIS and achieve the target secrecy rate, rs, can be obtained by +Pb = M (1 − τ) +ηeffτNM + +− (q2 − q4 − q5 + q7) ± +� +(q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6) +2 (q1 − q3) + + +× +� K +� +k=1 +pk +dαr +uk,r ++ σ2 +r +� +. +(39) +VII. NUMERICAL RESULTS +In this section, we present simulation and numerical results to assess the accuracy of the derived +expressions and the secrecy performance of the RIS schemes considered in this paper. Monte-Carlo +simulations with 105 independent trials are excuted. The locations of the BS and the RIS are (0 m, 0 +m), (20 m, 20 m), respectively, while the users are scattered on the corners of a square. Specifically, +the coordinates for the users square are (30 m, 5 m), (35 m, 5 m), (30 m,−5 m), and (35 m,−5 m), +respectively, while the eavesdroppers are distributed in a circle centered at (20 m, 0 m) with radius +of 10 m. Unless otherwise specified, the simulation settings are assumed as follows: K = J = 4, +N = 10, M = 5, the users power pi = 2W, the active RIS power Pr = 7W , the BS power in EH +RIS scenario Pb = 50W, and the nodes have same noise variance, σ2 = −70 dBm. In addition, the +path-loss exponent is 2.7, the Rician factors ρ = 0.5. The values of the AoA and AoD of the BS and +the RIS are uni-formally distributed in (0, 2π), and the concentration parameter of RIS phase error +κ = 2. + +20 +Firstly, in Fig. 2, we illustrate the ergodic secrecy rate versus the transmission user power, pk, for +the three considered RIS schemes. Fig. 2a shows the secrecy rate with phase shift errors and Fig. 2b, +presents the secrecy rate for the ideal scenario, when there is no phase error at RIS. It is clear from +this figure that the analytical results are in good agreement with the simulated results, which confirms +the validity of the analysis presented in this paper. It is also evident that for the given parameters +values, the secrecy rate loss due to the imperfect phase shift at the RIS is about 0.75 bits/s/Hz. In +addition, passive RIS achieves the lowest secrecy rate, but with small amount of power consumption. +The secrecy rate gain of active RIS above passive RIS is about 0.8 bits/s/Hz for a given user power. +Furthermore, high secrecy rates can be achieved and controlled by implementing EH RIS. However, +in this case the BS should transmit high power in the EH phase to provide sufficient amount of energy +at the RIS to achieve higher secrecy rates. +0 +10 +20 +30 +40 +50 +60 +pk (W) +0 +0.5 +1 +1.5 +2 +2.5 +Secrecy Rate (bits\sec\Hz) +Active RIS +Passive RIS +Analytical +EH RIS +(a) Secrecy rate versus user, k , power with phase shift error. +0 +10 +20 +30 +40 +50 +60 +pk (W) +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Secrecy Rate (bits\sec\Hz) +Active RIS +Passive RIS +Analytical +EH RIS +(b) Secrecy rate versus user, k , power with no phase shift error. +Figure 2: Secrecy rate versus user, k , power with and without phase shift error. +To explain the impact of the phase errors at the RIS on the secrecy performance, in Fig. 3, we plot +the secrecy rate versus the concentration parameter of the phase error, κ. Additionally, the results of +ideal RIS are also presented in this figure. It can be observed from these results that the secrecy rate +enhances as the concentration parameter, κ, increases. In addition, at high concentration parameter +values, κ −→ ∞, the secrecy rate achieved by imperfect RIS saturates to that achieved by ideal RIS. +This can be explained by the fact that the phase error at the RIS is assumed to follow a Von Mises +distribution, thus high concentration parameter values make the error fluctuate in a smaller range, and + +21 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +Secrecy Rate (bits\sec\Hz) +Ideal RIS without Error + RIS with Error + RIS with Error + RIS with Error +Ideal RIS without Error +Active RIS +Passive RIS +EH RIS +Ideal RIS without Error +Figure 3: Secrecy rate versus concentration parameter, κ, of RIS phase error. +when κ −→ ∞, the error at the RIS tends to zero. Accordingly, the secrecy rate of imperfect RIS +converges to the ideal RIS case as κ −→ ∞, as explained in Fig. 3. +Furthermore, Fig. 4 shows the secrecy rate versus the number of BS antennas N for the all RIS +schemes. It is evident and as expected, increasing the number of BS antennas N enhances the secrecy +performance for the all RIS schemes. It should be pointed out that the number of BS antennas, N, has +impact only on the received signal at the BS, thus increasing N results in enhancing the rate of the +legitimate users. However N dose not have any impact on the rate at the eavesdroppers. Having said +that in EH RIS, increasing N also increases the amount of the harvested energy at the RIS. Thus, in +EH RIS, N has impact on both achievable rates at the BS and the eavesdroppers. +In Fig. 5, we depict the secrecy rate versus the number of RIS elements, M, for the all considered RIS +schemes. To obtain clear insights and results, in this figure the noise variance at the nodes is assumed +to be σ2 = −20 dBm. Notably and as expected, increasing M results in enhancing the secrecy rate +for the all considered scenarios. In addition, as we can notice from the analytical expressions of the +secrecy rate presented in this paper, the number of RIS elements M has impact on both the achievable +rate at the BS and the eavesdroppers, e.g., adding more RIS elements increases the rate at the BS and +the eavesdroppers. However this improvement in the rate is essential at the BS, because the RIS phase +shifts are designed to be toward the BS direction. Furthermore, in the EH RIS scheme, increasing the + +22 +5 +10 +15 +20 +N +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +Secrecy Rate (bits\sec\Hz) +Passive RIS +Analytical +Active RIS +EH RIS +Figure 4: Secrecy rate versus number of BS antennas, N, with phase shift error. +20 +40 +60 +80 +100 +120 +140 +M +0 +1 +2 +3 +4 +5 +6 +Secrecy Rate (bits\Sec\Hz) +Active RIS +Passive RIS +EH RIS +Analytical +Figure 5: Secrecy rate versus number of RIS elements, M, with phase shift error. +number of the RIS elements, M, leads to an increase in the amount of the harvested energy at the RIS +and thus Pr will be high when the number of elements M is very large. +In order to illustrate the RIS configuration selection scheme, in Fig. 6 we plot the user power +versus the target secrecy rate for different values of the concentration parameter of RIS phase error, +κ = 2 and 8. Firstly, in Figs. 6a and 6b, we consider two examples, when the target secrecy rate is +assumed to be rs = 0.75 (bits/s/Hz) and rs = 1.2 (bits/s/Hz) for κ = 2 and 8. As we can see from + +23 +the results in Fig. 6a, when rs = 0.75 (bits/s/Hz), passive RIS can achieve the target secrecy rate +with total transmission power is PT = pk = 50W, (neglecting the small amount of power consuming +at passive RIS elements), and in the active RIS scheme the user transmission power can be reduced +to around pk = 7W and thus the total transmission power is PT = pk + Pr = 14W, while EH +RIS scheme can achieve the target secrecy rate with the smallest amount of the user power which is +about pk = 2.95W, but with the highest total transmission power PT = pk + Pb = 52.95W. Similar +observations can be noticed from the second scenario when rs = 1.2 (bits/s/Hz), passive RIS achieves +the target secrecy rate with the highest user power, while EH RIS achieves, rs, with the smallest user +power but with very high total consumption power, and the active RIS scheme works between these +two regions. In addition, the concentration parameter of RIS phase error, κ, has essential impact on +the required user power. By comparing Figs 6a and 6b, one can notice that as κ increases the required +user power to achieve the target secrecy rate decreases. For instance when the target secrecy rate is +rs = 0.75 (bits/s/Hz), the required user power in the passive RIS scheme is about 50W when κ = 2, +and 20W when κ = 8. This is due to the fact explained in Fig. 3. +Then, in Figs. 6c and 6d, we present the RIS configuration selection scheme when the available +user power is pk = 20W for κ = 2 and 8. In the first case when κ = 2, if the target secrecy rate +is rs ≤ 0.45 (bits/s/Hz), passive RIS can be selected, and active RIS can be implemented if the +target secrecy rate is rs ≤ 1.17 (bits/s/Hz), while EH RIS can be selected if rs ≤ 1.87 (bits/s/Hz). +These secrecy rate regions of the RIS schemes become wider as the concentration parameter of RIS +phase error, κ, increases. In Fig. 6d when κ = 8, passive RIS can be selected to achieve secrecy +rates up to rs ≤ 0.77 (bits/s/Hz), and active RIS can be selected to perform secrecy rates less +than or equal to rs ≤ 1.635 (bits/s/Hz), whilst EH RIS can be used to achieve secrecy rates up to +rs ≤ 2.48 (bits/s/Hz). + +24 +0 +0.5 +1 +1.5 +2 +2.5 +Target Secrecy Rate (bits/s/Hz) +0 +10 +20 +30 +40 +50 +60 +User Power (W) +Passive RIS +Active RIS, Pr=7W +EH RIS, Pb=50W + Target Secrecy Rate +rs=1.2 (bits/s/Hz) + Target Secrecy Rate +rs=0.75 (bits/s/Hz) +(a) The user power versus target secrecy rate when κ = 2 . +0 +0.5 +1 +1.5 +2 +2.5 +3 +Target Secrecy Rate (bits/s/Hz) +0 +10 +20 +30 +40 +50 +60 +User Power (W) +Passive RIS +Active RIS, Pr=7W +EH RIS, Pb=50W + Target Secrecy Rate +rs=1.2 (bits/s/Hz) + Target Secrecy Rate +rs=0.75 (bits/s/Hz) +(b) The user power versus target secrecy rate when κ = 8. +0 +0.5 +1 +1.5 +2 +2.5 + Target Secrecy Rate (bits/s/Hz) +0 +10 +20 +30 +40 +50 +60 +User Power (W) +Passive RIS +Active RIS, Pr=7W +EH RIS, Pb=50W + + Only EH RIS +can be selected + + Active RIS +can be selected + Passive RIS +can be selected +Available + power pk +(c) RIS configuration selection scheme when pk = 20W and κ = 2. +0 +0.5 +1 +1.5 +2 +2.5 +3 +Target Secrecy Rate (bits/s/Hz) +0 +10 +20 +30 +40 +50 +60 + User Power (W) +Passive RIS +Active RIS, Pr=7W +EH RIS, Pb=50W +Available + power pk + Passive RIS +can be selected + + Active RIS +can be selected + + Only EH RIS +can be selected +(d) RIS configuration selection scheme when pk = 20W and, κ = 8. +Figure 6: The user power versus target secrecy rate for different values of the concentration parameter +of RIS phase error, κ. +VIII. CONCLUSIONS +In this paper the impact of phase shift error on the secrecy performance of up-link RIS-aided MU- +MISO systems was considered. Under Rician fading channels and phase shift errors the ergodic secrecy +rate for, passive RIS, active RIS, and EH RIS have been analyzed. Then, the phase shifts at the RIS +have been optimized based on the derived rate expressions. In addition, according to the target secrecy +rate and amount of power available at the users, the best RIS configuration selection scheme has been +considered. The results presented in this work demonstrated that an active RIS scheme can enhance +the secrecy performance of imperfect RIS elements, especially when the users have limited amount of + +25 +power. Furthermore, increasing the number of BS antennas, the concentration parameter of RIS phase +error, and the number of RIS elements lead to the enhancement of the secrecy performance. +APPENDIX A +By using Jensen inequality, the ergodic rate can be expressed as +E {Rbk} ≈ log2 + + + + + + + +1 + E + + + + + + + + + + + + + +pk Luk,b +��hH +r,kΘHGHGΘ¯Θhr,k +��2 +K� +i=1 +i̸=k +pi Lui,b +��hH +r,kΘHGHGΘ¯Θhr,i +��2 + +��hH +r,kΘHGH��2 σ2 +b + + + + + + + + + + + + + + + + + + + + +. +(40) +Due to the paper length limitation, in this Appendix we will explain how to calculate the average +of the first term, similarly and by following similar steps we can find the average of the other terms. +The first term is +E +� +Pk Luk,b +��hH +r,kΘHGHGΘ¯Θhr,k +��2� += Pk Luk,bE +���hH +r,kΘHGHGΘ¯Θhr,k +��2� +(41) +where +hH +r,kΘHGHGΘ¯Θhr,k = hH +r,kΘH +� +ρb +ρb + 1 +¯GH ¯G + +√ρb +ρb + 1 +¯GH ˜G + +√ρb +ρb + 1 +˜GH ¯G + +1 +ρb + 1 +˜GH ˜G +� +Θ¯Θhr,k += +1 +ρb + 1hH +r,kΘH � +ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G +� +Θ¯Θhr,k = +1 +ρb + 1hH +r,kA¯Θhr,k +(42) +where A = ΘH � +ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G +� +Θ. Now (42) can be expressed as +hH +r,kΘHGHGΘ¯Θhr,k = +1 +(ρb + 1) (ρk + 1) +�√ρk¯hH +r,k + ˜hH +r,k +� +A¯Θ +�√ρk¯hr,k + ˜hr,k +� += +1 +(ρb + 1) (ρk + 1) + + +ρk¯hH +r,kA¯Θ¯hr,k +� +�� +� +∆1 ++ √ρk¯hH +r,kA¯Θ˜hr,k +� +�� +� +∆2 ++ √ρk˜hH +r,kA¯Θ¯hr,k +� +�� +� +∆3 ++ ˜hH +r,kA¯Θ˜hr,k +� +�� +� +∆4 + + + +(43) + +26 +The channels are independent and have zero mean. Thus by removing the zero expectation terms, +we can get +E +���hH +r,kΘHGHGΘ¯Θhr,k +��2� += +1 +(ρb + 1)2 (ρk + 1)2E + + + +����� +4 +� +i=1 +∆i +����� +2 + + += +1 +(ρb + 1)2 (ρk + 1)2 +� +4 +� +i=1 +E +� +|∆i|2� ++ 2E {∆1∆∗ +4} +� +(44) +Now the first term +∆1 = ρk¯hH +r,kΘH � +ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G +� +Θ¯Θ¯hr,k += + + +ρbρk¯hH +r,kΘH ¯GH ¯GΘ¯Θ¯hr,k +� +�� +� +∆1,1 ++ √ρbρk¯hH +r,kΘH ¯GH ˜GΘ¯Θ¯hr,k +� +�� +� +∆1,2 ++√ρbρk¯hH +r,kΘH ˜GH ¯GΘ¯Θ¯hr,k +� +�� +� +∆1,3 ++ ρk¯hH +r,kΘH ˜GH ˜GΘ¯Θ¯hr,k +� +�� +� +∆1,4 + + + +(45) +The average of the first term +E +� +|∆1|2� += E +� +|∆1,1|2� ++ E +� +|∆1,2|2� ++ E +� +|∆1,3|2� ++ E +� +|∆1,4|2� ++ 2E +� +∆1,1∆H +1,4 +� +(46) +where ∆1,1 = ρbρk¯hH +r,kΘH ¯GH ¯GΘ¯Θ¯hr,k, which can be written as +∆1,1 = ρbρkaH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aM (φa +r, φe +r) Θ¯ΘaM (φa +kr, φe +kr) , +∆1,1 = ρbρk +� M +� +m=1 +aH +M,m (φa +kr, φe +kr) e−jϕmaH +M,m (φa +r, φe +r) +� � M +� +m=1 +aM,m (φa +kr, φe +kr) ejϕmej ¯ +ϕmaM,m (φa +r, φe +r) +� +. +(47) +The average can now be written as + +27 +E +� +|∆1,1|2� += ρ2 +bρ2 +k +� M +� +m=1 +aH +M,m (φa +kr, φe +kr) e−jϕmaH +M,m (φa +r, φe +r) +�2 +× + +M + ρ (κ)2 +����� +M +� +m1=1 +M +� +m2̸=m1 +� +aM,m1 (φa +kr, φe +kr) ejϕm1aM,m1 (φa +r, φe +r) +� � +aM,m2 (φa +kr, φe +kr) ejϕm2aM,m2 (φa +r, φe +r) +�H +����� +2 + +(48) +E +� +|∆1,1|2� += ρ2 +bρ2 +k |fk|2 �� +1 − ρ (κ)2� +M + ρ (κ)2 |fk|2� +(49) +where fk = +M +� +m=1 +fk,m, fk,m = aH +M.m (φa +r, φe +r) ejϕmaM,m (φa +kr, φe +kr). The second term, +∆1,2 = √ρbρkaH +M (φa +kr, φe +kr) ΘHaM (φa +r, φe +r) aH +N (φa +b, φe +b) ˜GΘ¯ΘaM (φa +kr, φe +kr) += √ρbρkf ∗ +k +M +� +m=1 +N +� +n=1 +aH +N,n (φa +b, φe +b) ˜gnmejϕmej ¯ +ϕmaM,m (φa +kr, φe +kr) , +(50) +E +� +|∆1,2|2� += ρbρ2 +kNM |fk|2 . +(51) +The third term +∆1,3 = √ρbρkaH +M (φa +kr, φe +kr) ΘH ˜GHaN (φa +b, φe +b) aH +M (φa +r, φe +r) Θ¯ΘaM (φa +kr, φe +kr) += √ρbρk +M +� +m=1 +N +� +n=1 +aH +N,n (φa +b, φe +b) ˜gH +nme−jϕmaM,m (φa +kr, φe +kr) +M +� +m=1 +ej ¯ +ϕmfk,m, +(52) +E +� +|∆1,3|2� += ρbρ2 +k +� +NMρ (κ)2 |fk|2 + +� +1 − ρ (κ)2� +NM2� +. +(53) +The forth term +∆1,4 = ρkaH +M (φa +kr, φe +kr) ΘH ˜GH ˜GΘ¯ΘaM (φa +kr, φe +kr) + +28 += ρk +M +� +m1=1 +aH +M,m1 (φa +kr, φe +kr) e−jϕm˜gH +nm1 +M +� +m2=1 +˜gnm2ejϕmej ¯ +ϕmaM,m2 (φa +kr, φe +kr) , +(54) +E +� +|∆1,4|2� += ρkNM +� +Mρ (κ)2 + 1 − ρ (κ)2� ++ NM2. +(55) +The last term +E +� +∆1,1∆∗ +1,4 +� += N |fk|2 � +Mρ (κ)2 + 1 − ρ (κ)2� +. +(56) +Similarly, following the same way we can find the average of the other terms. +APPENDIX B +Using Jensen inequality, the ergodic rate can be written as +E +� +Rej,k +� +≈ log2 + + + + + + + +1 + E + + + + + + + + + + + + + +pk +���d +− αr +2 +uk,r d +− αe +2 +ej,r hej,rΘ¯Θhr,k + d +− αe +2 +ej,k hej,k +��� +2 +K� +i=1 +i̸=k +pi +���d +− αr +2 +ui,r d +− αe +2 +ej,r hej,rΘ¯Θhr,i + d +− αe +2 +ej,i hej,i +��� +2 ++ σ2 +ej + + + + + + + + + + + + + + + + + + + + +(57) +The average of the first term, after removing the zero expectation terms can be calculated by, +E +����d +− αr +2 +uk,r d +− αe +2 +ej,r hej,rΘ¯Θhr,k + d +− αe +2 +ej,k hej,k +��� +2� += d−αr +uk,rd−αe +ej,r E +���hej,rΘ¯Θhr,k +��2� ++ d−αe +ej,r +(58) +where +E +���hej,rΘ¯Θhr,k +��2� += E +����� +�� +ρej +ρej + 1 +� +ρk +ρk + 1 +¯hejΘ¯Θ¯hr,k + +� +ρej +ρej + 1 +� +1 +ρk + 1 +¯hejΘ¯Θ˜hr,k ++ +� +ρk +ρk + 1 +� +1 +ρej + 1 +˜hejΘ¯Θ¯hr,k + +� +1 +ρej + 1 +� +1 +ρk + 1 +˜hejΘ¯Θ˜hr,k +������ +2 + + +(59) +E +���hej,rΘ¯Θhr,k +��2� += +ρej +ρej + 1 +ρk +ρk + 1E +��¯hejΘ¯Θ¯hr,k +��2 + +ρej +ρej + 1 +1 +ρk + 1E +���¯hejΘ¯Θ˜hr,k +��� +2 + +29 ++ +ρk +ρk + 1 +1 +ρej + 1E +���˜hejΘ¯Θ¯hr,k +��� +2 ++ +1 +ρej + 1 +1 +ρk + 1E +���˜hejΘ¯Θ˜hr,k +��� +2 +(60) +Now +¯hejΘ¯Θ¯hr,k = +� M +� +m=1 +aM,m (φa +kr, φe +kr) ejϕmej ¯ +ϕmaM,m +� +φa +ejr, φe +ejr +�� +(61) +E +��¯hejΘ¯Θ¯hr,k +��2 = E +����� +M +� +m=1 +aM,m (φa +kr, φe +kr) ejϕmej ¯ +ϕmaM,m +� +φa +ejr, φe +ejr +������ +2 += M+ρ (κ)2 +M +� +m1=1 +M +� +m2̸=m1 +� +aM,m1 (φa +kr, φe +kr) ejϕm1aM,m1 (φa +r, φe +r) +� � +aM,m2 (φa +kr, φe +kr) ejϕm2aM,m2 (φa +r, φe +r) +�H += M + ρ (κ)2 ξ +(62) +where ξ = +M +� +m1=1 +M +� +m2̸=m1 +(aM,m1 (φa +kr, φe +kr) ejϕm1aM,m1 (φa +r, φe +r)) (aM,m2 (φa +kr, φe +kr) ejϕm2aM,m2 (φa +r, φe +r)) +H. +Similarly, the second term, +¯hejΘ¯Θ˜hr,k = aM +� +φa +ejr, φe +ejr +� +Θ¯Θ˜hr,k = +M +� +m=1 +aMm +� +φa +ejr, φe +ejr +� +ejϕmej ¯ +ϕm � +˜hr,k +� +m +(63) +E +���¯hejΘ¯Θ˜hr,k +��� +2 += E +����� +M +� +m=1 +aMm +� +φa +ejr, φe +ejr +� +ejϕmej ¯ +ϕm � +˜hr,k +� +m +����� +2 +E +���¯hejΘ¯Θ˜hr,k +��� +2 += M+ +E +� +M +� +m1=1 +M +� +m2̸=m1 +� +aMm1 +� +φa +ejr, φe +ejr +� +ejϕm1ej +¯ +ϕm1 � +˜hr,k +� +m1 +� � +aMm2 +� +φa +ejr, φe +ejr +� +ejϕm2ej +¯ +ϕm2 � +˜hr,k +� +m2 +�H� += M +(64) +other terms, + +30 +˜hejΘ¯Θ¯hr,k = +M +� +m=1 +˜hej,mejϕmej ¯ +ϕmaM,m (φa +kr, φe +kr) +(65) +E +���˜hejΘ¯Θ¯hr,k +��� +2 += M+ +E +� +M +� +m1=1 +M +� +m2̸=m1 +�� +˜hej +� +m1 ejϕm1ej +¯ +ϕm1aMm1 (φa +kr, φe +kr) +� �� +˜hej +� +m2 ejϕm2ej +¯ +ϕm2aMm2 (φa +kr, φe +kr) +�H� += M +(66) +and +˜hejΘ¯Θ˜hr,k = +M +� +m=1 +� +˜hej +� +m ejϕmej ¯ +ϕm � +˜hr,k +� +m +(67) +E +���˜hejΘ¯Θ˜hr,k +��� +2 += E +����� +M +� +m=1 +� +˜hej +� +m ejϕmej ¯ +ϕm � +˜hr,k +� +m +����� +2 += M +(68) +Now, we are ready to write the average of the first term, +E +���hej,rΘ¯Θhr,k +��2� += +ρej +ρej + 1 +ρk +ρk + 1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej + 1 +1 +ρk + 1M ++ +ρk +ρk + 1 +1 +ρej + 1M + +1 +ρej + 1 +1 +ρk + 1M +(69) +Similarly we can find the average of the second term as, +E +����d +− αr +2 +ui,r d +− αe +2 +ej,r hej,rΘ¯Θhr,i + d +− αe +2 +ej,i hej,i +��� +2� += d−αr +ui,r d−αe +ej,r E +���hej,rΘ¯Θhr,i +��2� ++ d−αe +ej,i +(70) +E +���hej,rΘ¯Θhr,i +��2� += +ρej +ρej + 1 +ρi +ρi + 1 +� +M + ρ (κ)2 ξ +� ++ +ρej +ρej + 1 +1 +ρi + 1M ++ +ρi +ρi + 1 +1 +ρej + 1M + +1 +ρej + 1 +1 +ρi + 1M +(71) + +31 +APPENDIX C +Using Jensen inequality, the ergodic rate can be written as +E {Rbk} ≈ log2 (1 + E {γbk}) +(72) +We will follow similar steps as in Appendix A, +hH +r,kΘHGHG¯ΘΘ = +1 +ρb + 1hH +r,kA¯Θ +(73) +where A = ΘH � +ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G +� +Θ. Last expression can be written as +hH +r,kΘHGHG¯ΘΘ = +1 +(ρb + 1) +� +(ρk + 1) +�√ρk¯hH +r,k + ˜hH +r,k +� +A¯Θ += +1 +(ρb + 1) +� +(ρk + 1) + + +√ρk¯hH +r,kA¯Θ +� +�� +� +∆1 ++ ˜hH +r,kA¯Θ +� �� � +∆2 + + + +(74) +The average can be written as, +E +���hH +r,kΘHGHG¯ΘΘ +��2� += +1 +(ρb + 1)2 (ρk + 1) +E + + + + + +ρr +������� +¯hH +r,kA¯Θ +� �� � +∆1 +������� +2 ++ +������� +˜hH +r,kA¯Θ +� �� � +∆2 +������� +2 + + + + +(75) +∆1 = √ρk¯hH +r,kΘH � +ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G +� +Θ¯Θ += + + +ρb +√ρk¯hH +r,kΘH ¯GH ¯GΘ¯Θ +� +�� +� +∆1,1 ++ √ρb +√ρk¯hH +r,kΘH ¯GH ˜GΘ¯Θ +� +�� +� +∆1,2 ++√ρb +√ρk¯hH +r,kΘH ˜GH ¯GΘ¯Θ +� +�� +� +∆1,3 ++ √ρk¯hH +r,kΘH ˜GH ˜GΘ¯Θ +� +�� +� +∆1,4 + + + +(76) +E +� +|∆1|2� += E +� +|∆1,1|2� ++ E +� +|∆1,2|2� ++ E +� +|∆1,3|2� ++ E +� +|∆1,4|2� ++ 2E +� +∆1,1∆H +1,4 +� +(77) + +32 +where +∆1,1 = ρb +√ρk +� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� � +aM (φa +r, φe +r) Θ¯Θ +� +(78) +E +� +|∆1,1|2� += ρ2 +bρk +��� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +���2 +× E +���aM (φa +r, φe +r) Θ¯Θ +��2� +(79) +E +� +|∆1,1|2� += ρ2 +bρk +��� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +���2 × M +(80) +The second term can be expressed as, +∆1,2 = √ρb +√ρkaH +M (φa +kr, φe +kr) ΘHaM (φa +r, φe +r) +M +� +m=1 +N +� +n=1 +aH +N,n (φa +b, φe +b) ˜gnmejϕmej ¯ +ϕm +(81) +E +� +|∆1,2|2� += ρbρk +��aH +M (φa +kr, φe +kr) ΘHaM (φa +r, φe +r) +��2 NM +(82) +The third term can be written as +∆1,3 = √ρb +√ρk +M +� +m=1 +N +� +n=1 +aH +M,m (φa +kr, φe +kr) ˜gH +nme−jϕmaN,n (φa +b, φe +b) +M +� +m=1 +aH +M,m (φa +r, φe +r) ej ¯ +ϕmejϕm +(83) +E +� +|∆1,3|2� += ρbρkMN + +E +����� +M +� +m=1 +aH +M,m (φa +r, φe +r) ej ¯ +ϕmejϕm +����� +2 + +(84) +E +� +|∆1,3|2� += ρbρkMN +� +M + ρ (κ)2 +M +� +m1=1 +M +� +m2̸=m1 +aH +M,m1 (φa +r, φe +r) ejϕm1aM,m2 (φa +r, φe +r) e−jϕm2 +� +(85) +E +� +|∆1,3|2� += ρbρkMN +� +ρ (κ)2 M + +� +1 − ρ (κ)2� +M +� +(86) + +33 +The forth term can be represented as, +∆1,4 = √ρk +N +� +n=1 +M +� +m1=1 +aH +M,m1 (φa +kr, φe +kr) e−jϕm˜gH +nm1 +M +� +m2=1 +˜gnm2ejϕmej ¯ +ϕm +(87) +E +� +|∆1,4|2� += ρk +� +N2M + NM2� +(88) +Now the last term can be written as +E +� +∆1,1∆∗ +1,4 +� += ρbρk +� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� +× +� +aM (φa +r, φe +r) Θ¯Θ +� +ρk¯hH +r,kΘH ˜GH ˜GΘ¯Θ +(89) +E +� +∆1,1∆∗ +1,4 +� += ρbρk +� +aH +M (φa +kr, φe +kr) ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� +(aM (φa +r, φe +r) Θ) ρk¯hH +r,kΘHNΘ +(90) +We will repeat similar steps for ∆2, +∆2 = + + +ρb˜hH +r,kΘH ¯GH ¯GΘ¯Θ +� +�� +� +∆2,1 ++ √ρb˜hH +r,kΘH ¯GH ˜GΘ¯Θ +� +�� +� +∆2,2 ++√ρb˜hH +r,kΘH ˜GH ¯GΘ¯Θ +� +�� +� +∆2,3 ++ ˜hH +r,kΘH ˜GH ˜GΘ¯Θ +� +�� +� +∆2,4 + + + +(91) +E +� +|∆2|2� += E +� +|∆2,1|2� ++ E +� +|∆2,2|2� ++ E +� +|∆2,3|2� ++ E +� +|∆2,4|2� ++ 2E +� +∆2,1∆H +2,4 +� +(92) +where +∆2,1 = ρb˜hH +r,kΘH ¯GH ¯GΘ¯Θ +(93) + +34 +E +� +|∆2,1|2� += ρ2 +b +��ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) aM (φa +r, φe +r) Θ +��2 +F +(94) +and +∆2,2 = √ρb˜hH +r,kΘHaM (φa +r, φe +r) +M +� +m=1 +N +� +n=1 +aH +N,n (φa +b, φe +b) ˜gnmejϕmej ¯ +ϕm +(95) +E +� +|∆2,2|2� += ρb +���˜hH +r,kΘHaM (φa +r, φe +r) +��� +2 +NM +(96) +while +∆2,3 = √ρb +M +� +m=1 +N +� +n=1 +˜hH +r,k,nm˜gH +nme−jϕmaN,n (φa +b, φe +b) +M +� +m=1 +aH +M,m (φa +r, φe +r) ej ¯ +ϕmejϕm +(97) +E +� +|∆2,3|2� += ρbMN + +E +����� +M +� +m=1 +aH +M,m (φa +r, φe +r) ej ¯ +ϕmejϕm +����� +2 + = ρbMN +� +ρ (κ)2 M + +� +1 − ρ (κ)2� +M +� +(98) +Finally, +∆2,4 = ρk +N +� +n=1 +M +� +m1=1 +˜hH +r,k,nm1e−jϕm˜gH +nm1 +M +� +m2=1 +˜gnm2ejϕmej ¯ +ϕm +(99) +E +� +|∆2,4|2� += ρ2 +k +� +N2M + NM2� +(100) +and +E +� +∆2,1∆∗ +2,4 +� += E +� +ρbρk +� +˜hH +r,kΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� � +aM (φa +r, φe +r) Θ¯Θ +� +ρk˜hH +r,kΘH ˜GH ˜GΘ¯Θ +� += ρk +� +ΘHaH +M (φa +r, φe +r) aH +N (φa +b, φe +b) aN (φa +b, φe +b) +� +(aM (φa +r, φe +r) Θ) ρkΘNΘH +(101) + +35 +REFERENCES +[1] M. 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Hanzo, “Joint reconfigurable intelligent surface location and passive beamforming +optimization for maximizing the secrecy-rate,” IEEE Transactions on Vehicular Technology, pp. 1–13, 2022. +[30] J. Luo, F. Wang, S. Wang, H. Wang, and D. Wang, “Reconfigurable intelligent surface: Reflection design against passive +eavesdropping,” IEEE Transactions on Wireless Communications, vol. 20, no. 5, pp. 3350–3364, 2021. +[31] J. Bai, H.-M. Wang, and P. Liu, “Robust irs-aided secrecy transmission with location optimization,” IEEE Transactions on +Communications, vol. 70, no. 9, pp. 6149–6163, 2022. +[32] W. Lv, J. Bai, Q. Yan, and H.-M. Wang, “Ris-assisted green secure communications: Active ris or passive ris?” IEEE Wireless +Communications Letters, pp. 1–1, 2022. +[33] P. Xu, G. Chen, G. Pan, and M. D. Renzo, “Ergodic secrecy rate of ris-assisted communication systems in the presence of discrete +phase shifts and multiple eavesdroppers,” IEEE Wireless Communications Letters, vol. 10, no. 3, pp. 629–633, 2021. + diff --git a/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/load_file.txt b/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..00a0aca590a4770fcf89f4ab26d7ee3f89709f8a --- /dev/null +++ b/3tAyT4oBgHgl3EQfb_f6/content/tmp_files/load_file.txt @@ -0,0 +1,1244 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf,len=1243 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='00276v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='IT] 31 Dec 2022 1 Impact of Phase-Shift Error on the Secrecy Performance of Uplink RIS Communication Systems Abdelhamid Salem, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE Abstract Reconfigurable intelligent surface (RIS) has been recognized as a promising technique for the sixth gen- eration (6G) of mobile communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The key feature of RIS is to reconfigure the propagation environment via smart signal reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, active RIS schemes have been recently proposed to overcome the deep path loss attenuation inherent in the RIS-aided communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, this paper considers the secrecy performance of up-link RIS-aided multiple users multiple-input single-output (MU- MISO) communication systems, in the presence of multiple passive eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In contrast to the existing works, we investigate the impact of the RIS phase shift errors on the secrecy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Taking into account the complex environment, where a general Rician channel model is adopted for all the communication links, closed-form approximate expressions for the ergodic secrecy rate are derived for three RIS configurations, namely, i) passive RIS, ii) active RIS, iii) active RIS with energy harvesting (EH RIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, based on the derived expressions, we optimize the phase shifts at the RIS to enhance the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, the best RIS configuration selection is considered for a given target secrecy rate and amount of the power available at the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Finally, Monte-Carlo simulations are provided to verify the accuracy of the analysis, and the impact of different system parameters on the secrecy performance is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The results in this Abdelhamid Salem is with the department of Electronic and Electrical Engineering, University College London, London, UK, (emails: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='salem@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Kai-Kit Wong is with the department of Electronic and Electrical Engineering, University College London, London, UK, Kai-Kit Wong is also affiliated with Yonsei University, Seoul, Korea (email: kai-kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='wong@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Chan-Byoung Chae is with Yonsei University, Seoul, Korea (e-mail: cbchae@yonsei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The work is supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant EP/V052942/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For the purpose of open access, the authors will apply a Creative Commons Attribution (CCBY) licence to any Author Accepted Manuscript version arising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2 paper show that, an active RIS scheme can be implemented to enhance the secrecy performance of RIS-aided communication systems with phase shift errors, especially when the users have limited transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Index Terms Reconfigurable intelligent surface, Physical layer security, MU-MISO, MRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' INTRODUCTION Reconfigurable intelligent surface (RIS), also known as intelligent reflecting surface (IRS), has been proposed recently as a promising technique to extend the coverage and improve the spectral efficiency of wireless communication networks [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Specifically, RIS is composed of reflecting elements, each of which independently imposes a phase shift on the incident signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' By tuning the phase shifts of the reflecting elements, RIS can convert the propagation environments into smart ones and thus enhance the received signals quality [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Due to these advantages, RIS techniques have been extensively considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For instance, in [3], the fundamental capacity limit of RIS-aided multiple- input multipleoutput (MIMO) communication systems has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The achievable ergodic rate of a RIS-assisted MIMO system which comprises links of a Rician channel was derived in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [5], a closed-form asymptotic ergodic sum rate of a RIS-assisted MIMO communication system was derived under the assumption that the number of base station (BS) antennas tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [6], the up-link achievable rate in RIS-aided massive MIMO systems has been analyzed and optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The authors in [7], [8] analyzed the achievable rate of RIS-assisted multiple users (MU) up-link massive MIMO system under Rician fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [9], [10], a closed-form expression of ergodic achievable rate for RIS-aided massive MIMO systems with zero forcing (ZF) detector has been derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, a closed-form analytical expression for the symbol error probability and the upper bound on the channel capacity of a RIS communication system have been derived in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The work in [12] considered the impact of hardware impairments on a general RIS MU-MISO system with Rayleigh fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic capacity of RIS MIMO networks over Rayleigh-Rician channels was considered in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However, the practical implementation of passive RIS-aided communication systems may face several challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For instance, the transmitted signal propagates through the RIS experiences a double-fading attenuation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='g, source-RIS and RIS-destination links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' This issue has been tackled in the literature by increasing the number of passive RIS elements [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However, this solution leads to an increase in 3 the size of the RIS module, which is impractical in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' To tackle this issue the authors in [15] proposed RIS with active elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The main idea of active RIS is to adjust the phase shifts and also amplify the reflected signal attenuated from the first link with extra power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Theoretical comparison between the active RIS-assisted system and the passive RIS-aided system has been presented in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The results in [16] show that the active RIS has better performance than passive RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The use of active RIS elements to overcome the double-fading problem has been also investigated in [17], where the results illustrated that using active elements results in a severe reduction in the physical size of RIS to achieve a certain performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' To reduce the power consumption of active RIS, a sub-connected architecture has been proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The energy efficiency in an active RIS-aided MU-MISO down-link system has been investigated in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Although fixed embedded batteries can be used to power the RIS, these batteries cannot be relied on for long time and uninterrupted operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, wired charging might not be possible to use if the RIS is deployed in inaccessible places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Therefore, equipping RIS elements with energy harvesting (EH) modules can solve these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, a self-sustainable RIS approach was proposed and studied in the resent researches on RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In this regard, in [20] time switching (TS) and power splitting (PS) EH protocols for the RIS to harvest sufficient amount of energy from an access point have been proposed and investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The work in [21] considered a self-sustainable RIS-aided MU-MISO communication systems, in which the RIS collected energy from the radio frequency (RF) transmitter using the PS protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [22], a novel transmission policy for a communication network assisted by self-sustainable RIS has been proposed, where the RIS harvests energy from an energy transmitter to support its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [23], self-sustainable RIS with the PS protocol to assist broadcasting network was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [24], self-sustainable RIS-aided communication between a gateway and a device was studied, in which the RIS harvested energy prior communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Moreover, due to the broadcast nature of wireless channels, confidential messages are vulnerable to eavesdropping attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For the provision of secure transmission, physical layer security (PHYSec) has been proposed from the information theory perspective [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' PHYSec exploits the nature of wireless channels to enhance the system security [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' PHYSec of RIS systems has also been studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [27], the secrecy throughput maximization problem has been formulated and solved to enhance the secrecy performance of the RIS-assisted MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [28], a novel active RIS design to enhance the security of wireless transmission was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' PHYSec of RIS- 4 aided wireless networks has been considered in [29] to achieve secure transmission between a source and a legitimate user in the presence of a malicious eavesdropper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [30], RIS has been used to perform secure transmission from a multiple antennas transmitter to a multiple antennas legitimate receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Further work in [31] considered the secrecy transmission in a RIS-aided multiple antennas communication, where the secrecy rate was improved by optimizing the RIS location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In [32], an active RIS-aided multiple antennas PHYSec transmission scheme was considered, where the active RIS was designed to amplify the signal actively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, this paper investigates the impact of phase shift error on the secrecy performance of up-link RIS-aided MU-MISO systems in the presence of multiple eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The BS receives the users messages only through the RIS, while eavesdroppers can receive the signals from both the direct and reflected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Under Rician fading channels and phase shift errors, the ergodic secrecy rate is analyzed for three RIS configurations, namely, 1) passive RIS, 2) active RIS, and 3) EH RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Based on the derived rate expressions, the phase shifts at the RIS are optimized to enhance the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, the best RIS configuration selection is considered based on the target secrecy rate and amount of power available at the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For clarity we list the main contributions of this work as follows: 1) We investigate the impact of RIS phase shift error on the secrecy performance of up-link MU- MIMO systems in the presence of multiple passive eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2) New closed-form explicit analytical expressions for the ergodic secrecy rate are derived for the RIS-assisted MU-MIMO systems, when the RIS is passive, active and EH node under Rician fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' This channel model is more general but also very challenging to be considered mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The derived secrecy rate expressions are simple, explicit and in closed form, and provide several important practical design insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 3) Based on the derived expressions, a genetic algorithm (GA)-based approach is used to obtain the optimal phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Also, a simple suboptimal technique is proposed to enhance the secrecy rate for a legitimate user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 4) Given a target secrecy rate, we calculate the required user power, and we present steps to select best RIS configuration which depend mainly on the available power at the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 5) Finally, Monte-Carlo simulations are performed to validate the analytical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, the impact of several system parameters on the secrecy performance are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 5 The results in this work show that active RIS is an efficient scheme to achieve secure communication in the presence of phase shift errors at the RIS, especially when there is no sufficient amount of power at the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Next, Section II presents the RIS-aided uplink MU-MISO system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In Section III, we derive the ergodic secrecy rate of the passive RIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Section IV presents the ergodic secrecy rate of the active RIS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Section V derives the ergodic secrecy rate of the EH RIS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Section VII depicts our numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Our main conclusions are summarized in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' SYSTEM MODEL Consider a typical up-link RIS-aided MU-MISO communication system consisting of a multiple antennas BS, an RIS and K single-antenna users in the presence of J single antenna passive eaves- droppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The BS is equipped with N antennas, and the RIS is equipped with M reflecting elements, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' UE 1 UE k UE K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Eave J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Eave 1 BS RIS Figure 1: An RIS-aided uplink MU-MISO system with N BS antennas, M RIS elements, K users and J eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The BS and RIS are connected to control and adjust the phase shifts of the the RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It is assumed that the eavesdroppers can hear the signals from the direct and reflected links, and trying to eavesdrop a specific confidential message in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' On the other side, the direct links between the users and BS are assumed to be blocked, which justifies the use of the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It is known that, the RIS is most likely to be installed on the buildings, and thus it can create channels dominated by line-of-sight (LoS) path along with scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, a Rician fading model is considered for the RIS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The channel matrix between the RIS and the BS is denoted by G ∈ CN×M , and the channel vector 6 between user k and the RIS is presented by hr,k ∈ CM×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The mathematical expressions of the channel matrix G and the channel vector hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k can be expressed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' as G = �� ρb ρb + 1 ¯G + � 1 ρb + 1 ˜G � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = �� ρk ρk + 1 ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + � 1 ρk + 1 ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � (1) where ρb and ρk are the Rician factors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ¯G and ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k are the LoS components and ˜G and ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k are the NLoS components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' in which ¯G = aN (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH M (φa t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = aM (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) (2) where φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr denote the azimuth and elevation angles of arrival (AoA) from user k to the RIS ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φa t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe t are the azimuth and elevation angles of departure (AoD) at the BS from the RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r are the azimuth and elevation AoA from the RIS to the BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The kth element of the vector aX can be written as [aX (φ1, φ2)]k = ej2π d λ (xk sin φ1 sin φ2+yk cos φ2), where λ is the wavelength, d is the elements/antennas spacing, and xk = (k − 1) mod√x, yk = k−1 √ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' On the other hand, the channel vector between the RIS and eavesdropper j is presented by hej,r ∈ C1×M , and the channel from user k to eavesdropper j is hej,k ∈ C1×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The direct channel fading is assumed to be Rayleigh fading due to extensive scatterers, while for the RIS-related channels, is assumed to be Rician fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus the expression of hej,ris given by hej,r = �� ρej,r ρej,r + 1 ¯hej,r + � 1 ρej,r + 1 ˜hej,r � (3) where ρej,r is the Rician factor, ¯hej,r and ˜hej,r are the LoS of NLoS components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The channel state information (CSI) of the eavesdroppers is assumed to be unknown at the BS/RIS (only statistical information can be known), and the eavesdroppers are non-colluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Therefore, the ergodic secrecy rate can be calculated by [33] ˆRs = � ˆRbk − ˆRej,k �+ (4) where [l]+= max (0, l), ˆRbk = E {Rbk}, Rbk is the up-link rate of user k, and ˆRej,k = max E � Rej,k � , Rej,k is the rate at eavesdropper j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 7 In the following sections, we consider the secrecy performance of the three RIS configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' PASSIVE RIS As we have mentioned earlier, passive RIS reflects the users messages constructively to the BS with passive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, the received signal at the BS can be expressed as yb = K � k=1 � pk Luk,bG˜Θhr,kxk + nb (5) where Luk,b = d−αr uk,rd−αb r,b is the large scale fading, duk,r is the distance between user k and RIS, dr,b is the distance between RIS and the BS, αr and αb are the path-loss exponents, nb is the additive wight Gaussian noise (AWGN) at the BS, nb ∼ CN (0, σ2 bI), ˜Θ = ¯ΘΘ where Θ = diag (θ), and θ = [θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='., θM]Tis the RIS reflection coefficients with θm = ejϕm, where ϕm∈[0, 2π) is the phase shift of element m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However, in practical systems, phase shift errors can exist due to imperfect channel knowledge and finite precision in phase adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, we define ¯Θ = [ej ¯ ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='., ej ¯ ϕM] as the phase- shift errors at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The phase-error is modeled according to Von-Mises (VM) distribution with zero-mean and a characteristic function (CF) E [ej ¯ ϕm] = I1(κ) I0(κ) = ρ (κ), where κ is the concentration parameter and Ii is the modified Bessel function of the first kind and order i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' By applying the receive beamforming vector wk at the BS, the received signal of user k is yb,k = � pk Luk,bwkG¯ΘΘhr,kxk+ K � i=1 i̸=k � pi Lui,bwkG¯ΘΘhr,ixi + wknb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (6) On the other hand, the received signal at eavesdropper j to detect user k signal is yej,k = √pkxk �� d−αe ej,k hej,k + � Luk,ejhej,r ¯ΘΘhr,k � + K � i=1 i̸=k √pixi \uf8eb \uf8ec \uf8ed � d−αe ej,i hej,i+ K � i=1 i̸=k � Lui,ejhej,r ¯ΘΘhr,i \uf8f6 \uf8f7 \uf8f8 + nej (7) where d−αe ej,k is the distance between user k and eavesdropper j, αe is the path-loss exponent, Luk,B = d−αr uk,rd−e ej,r and d−e ej,r denotes the distance between the RIS and eavesdropper j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 8 To calculate the ergodic secrecy rate, the ergodic up-link rate for user k and ergodic rate at the eavesdropper j should be derived, which will be considered in the following sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Up-link rate of user k To calculate the ergodic user rate, maximum ratio combining (MRC) is adopted at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The beamforming matrix is given by W = (GΘH)H, and thus wk = hH r,kΘHGH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The signal to interference plus noise ratio (SINR) at the BS to decode user k signal can be written as γbk = pk Luk,b ��hH r,kΘHGHGΘ¯Θhr,k ��2 K� i=1 i̸=k pi Lui,b ��hH r,kΘHGHGΘ¯Θhr,i ��2 + ��hH r,kΘHGH��2 σ2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (8) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic up-link rate of user k in passive RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by E {Rbk} ≈ log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi + υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (9) where ξk = E ���hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHGΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� = 1 (ρb + 1)2 (ρk + 1)2 � a1N2 + a2NM2 + a3NM + a4N � a1 = � ρ (κ)2 (ρk + ρb + 1)2 + � 1 − ρ (κ)2� ρkρ2 b + ρ2 b � M2 + �� (2ρk + 3ρb + 2 − ρkρb) ρ (κ)2 + (1 + ρk) ρb � ρbρk |fk|2 + (ρk + ρb + 2)2 − ρ (κ)2 (ρk + ρb + 1)2 − 2ρ (κ)2 ρkρb − 2 � M +ρ (κ)2 ρ2 bρ2 k |fk|4 + 2 �� 1 − ρ (κ)2� (ρk + ρb) + 2 � ρbρk |fk|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' a2 = � −ρ (κ)2 ρkρb (1 + ρk) M2� + (ρk + ρb + 1) ρbρr + (ρk + ρb + 1)2 − (ρk + 1) ρ2 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' a3 = �� (ρk + 1) ρ (κ)2� + (ρk + 1) � ρbρk |fk|2 − 2ρbρkρ (κ)2 + 2ρbρk + 2ρk + 2ρb − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' a4 = 2ρbρr |fk|2 � 1 + ρ (κ)2� and 9 ςi = E ���hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHGΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='��2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='(ρb + 1)2 (ρk + 1) (ρi + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b1N2 + b2NM2 + b3NM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρi + 1 − ρ (κ)2 ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='M2ρ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρi + 1 − ρ (κ)2 ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bρk |fk|2 + ρ (κ)2 ρ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bρi |fi|2 + (ρk + 2ρb + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρi + 1 − ρ (κ)2 ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+ ρ (κ)2 ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2ρb |fi|2 + ρk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='��¯hH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ¯hi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='��2 + 2ρbρkRe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='f ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kfi¯hH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ¯hk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρ (κ)2 ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρ (κ)2 ρbρi |fi|2 + 2ρi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 − ρ (κ)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='ρbρk |fk|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='(ρb + 1) ρk + (ρb + 1)2 − ρ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='(ρi + 1) − (ρb + 1) ρbρiρ (κ)2 − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b3 = (ρi + 1) ρbρk |fk|2 + (ρk + 1) ρ (κ)2 ρbρi |fi|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='υk = E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='���hH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGH��2� = Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b (ρb + 1) (ρk + 1) � ρbρk |fk|2 + (ρb + ρk + 1) M � Proof: The proof is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Rate at Eavesdropper j The SINR at eavesdropper j to decode user k signal can be expressed as γej,k = pk ���d − αr 2 uk,r d − αe 2 ej,r hej,rΘ¯Θhr,k + d − αe 2 ej,k hej,k ��� 2 K� i=1 i̸=k pi ���d − αr 2 ui,r d − αe 2 ej,r hej,rΘ¯Θhr,i + d − αe 2 ej,i hej,i ��� 2 + σ2ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (10) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic rate at eavesdropper j in up-link passive RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by E � Rej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � = log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk xk K� i=1 i̸=k pi yi + σ2 ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (11) where xk = � d−αr uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rd−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ρej ρej +1 ρk ρk+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρk+1M + ρk ρk+1 1 ρej +1M + 1 ρej +1 1 ρk+1M � + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' and 10 yi = d−αr ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ρej ρej +1 ρi ρi+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρi+1M + ρi ρi+1 1 ρej +1M + 1 ρej +1 1 ρi+1M + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Proof: The proof is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Finally, the ergodic secrecy rate in passive RIS scheme is presented in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic secrecy rate in passive RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by ˆRs = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,bξk K� i=1 i̸=k pi Lui,bςi + υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk xk K� i=1 i̸=k pi yi + σ2ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='+ (12) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ACTIVE RIS As we mentioned earlier, active RIS can adjust the phase shifts and also amplify the reflected signal to compensate the attenuation from the first link with extra power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The signal reflected by the active IRS can be written as yr = ˜Θ K � k=1 � pk d−αr uk,rhr,ixi + ˜Θnr (13) where nr is the noise at RIS elements nr ∼ CN (0, σ2 rI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In this case ˜Θ = ¯ΘΘ where Θ = diag (θ), and θ = [θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='., θM]T with θm = ̺mejϕm, ̺m > 1 and ϕm∈[0, 2π) represents the amplification factor and phase shift coefficient, respectively, at element m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For simplicity, we assume that ̺m = ̺ and then define Θ = ̺diag {ejϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='., ejϕM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The active RIS amplification power can be expressed as Pr = � K � k=1 pk dαr uk,r E ����˜Θhr,i ��� 2� + E ����˜Θnr ��� 2�� = � K � k=1 pk dαr uk,r M̺2 + M̺2σ2 r � (14) where E ���˜Θnr ��� 2 = M̺2σ2 r, and E ���˜Θhr,i ��� 2 = ̺2 ρi+1 � ρiE �¯hH r,i¯hr,i � + E � ˜hH r,i˜hr,i �� = ̺2 ρi+1 (ρiM + M) = M̺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, the amplification factor for each element on the active RIS is given by ̺ = � � � � � Pr M � K � k=1 pk dαr uk,r + σ2 r �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (15) 11 By applying the receive beamforming vector wk at the BS, the received signal of user k is yb,k = � pk Luk,bwkG¯ΘΘhr,kxk+ K � i=1 i̸=k � pi Lui,bwkG¯ΘΘhr,ixi + � d−αr r,b wkG¯ΘΘnr + wknb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (16) On the other hand, the received signal at eavesdropper j to detect user k signal is yej,k = √pkxk �� d−αe ej,k hej,k + � Luk,ejhej,r ¯ΘΘhr,k � + K � i=1 i̸=k √pixi \uf8eb \uf8ec \uf8ed � d−αe ej,i hej,i+ K � i=1 i̸=k � Lui,ejhej,r ¯ΘΘhr,i \uf8f6 \uf8f7 \uf8f8 + � d−αe ej,r hej,r ¯ΘΘnr + nej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (17) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Up-link rate of user k Applying MRC beamforming at the BS, the SINRs at the BS to decode user k signal can be expressed as γbk = pk Luk,b ��hH r,kΘHGHGΘ¯Θhr,k ��2 K� i=1 i̸=k pi Lui,b ��hH r,kΘHGHGΘ¯Θhr,i ��2 + d−αr r,b ��hH r,kΘHGHG¯ΘΘ ��2 σ2 r + ��hH r,kΘHGH��2 σ2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (18) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic up-link rate of user k in active RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by E {Rbk} ≈ log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk̺4 K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi̺4 + ̺4d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk + ̺2υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (19) where νk = E ��hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHG¯ΘΘ ��2 = 1 (ρb + 1) � (ρk + 1) (X1 + X2) (20) and X1 = E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� + E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � 12 E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� = ρ2 bρk ��� aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) ���2 × M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� = ρbρk ��aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ��2 NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbρkMN � ρ (κ)2 M + � 1 − ρ (κ)2� M � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� = ρk � N2M + NM2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � = ρbρk � aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � × (aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ) ρkaH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHNΘ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' and X2 = E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� + E � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� = ρ2 b ��ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ ��2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� = ρb ��ΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ��2 NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbMN ���aH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯Θ ��2� = ρbM2N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� = ρ2 k � N2M + NM2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � = ρk � ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � (aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ) ρkΘNΘH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Proof: The proof is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Rate at Eavesdropper j The SINR at eavesdropper j to decode user k signal in this scenario can be written as 13 γej,k = pk ���d − αr 2 uk,r d − αe 2 ej,r hej,rΘ¯Θhr,k + d − αe 2 ej,k hej,k ��� 2 K� i=1 i̸=k pi ���d − αr 2 ui,r d − αe 2 ej,r hej,rΘ¯Θhr,i + d − αe 2 ej,i hej,i ��� 2 + d−αe ej,r ��hej,r ¯ΘΘ ��2 σ2r + σ2ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (21) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic rate at eavesdropper j in up-link active RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by E � Rej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � = log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk xj K� i=1 i̸=k piyi + zjσ2 r + σ2 ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (22) where xj = � d−αr uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rd−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ̺2 � ρej ρej +1 ρk ρk+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρk+1M + ρk ρk+1 1 ρej +1M + 1 ρej +1 1 ρk+1M � + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' yk = d−αr ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ̺2 � ρej ρej +1 ρi ρi+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρi+1M + ρi ρi+1 1 ρej +1M + 1 ρej +1 1 ρi+1M + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i � which have been derived in Appemdix B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' and zj = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ˜Θ ��� 2 = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ̺2 ρej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r+1 � ρej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rE � ¯hH ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r¯hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � + E � ˜hH ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r˜hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r �� = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ̺2 ρej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r+1 � ρej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rM + M � = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r M̺2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic secrecy rate in active RIS scheme is presented in the following Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic secrecy rate in active RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by ˆRs = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,bξk̺4 K� i=1 i̸=k pi Lui,bςi̺4 + ̺4d−αr r,b σ2rνk + ̺2υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk xj K� i=1 i̸=k piyi + zjσ2r + σ2ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (23) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' EH RIS Following the recent works in [20], [21], [22], [23], [24], in this section, the RIS is an energy constrained node and it can harvest RF energy to support its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, in this scenario the 14 whole operation time block, T, is split into two time periods, the energy transfer (ET) slot and the information transfer (IT) slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' During the ET slot, the BS transmits energy signals to the RIS to support its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' During the IT slot, the users deliver their messages to the BS through the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' We denote τT as the time duration for the ET, and (1 − τ) T as the time duration for IT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The received signals at the RIS in the first sub-slot is expressed as yr = � PbGpWpxp + nr (24) where Pb is the BS power, Gp = �� ρp ρp+1 ¯Gp + � 1 ρp+1 ˜Gp � is the BS-RIS channel in the ET slot, Wp is the precoding matrix and xp is the energy signals vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Using the maximum ratio transmission (MRT) scheme, the harvested power at the RIS can be expressed as Pr = ηeff τPb∥Gp∥2 F 1−τ , which can be written as Pr = ηeff τPbTr(GbGH b ) 1−τ where ηeff is the efficiency of EH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Since GbGH b has Wishart distribution, the average harvested power can be written as Pr = ηeffτPbE � Tr � GbGH b �� 1 − τ = ηeffτPbNM 1 − τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (25) By substituting (25) into (15), the amplification factor for each element on the RIS in this case is given by ˆ̺ = � � � � � ηeffτPbNM M (1 − τ) � K� k=1 pk dαr uk,r + σ2 r �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (26) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Up-link rate of user k Applying MRC beamforming at the BS, the SINR at the BS to decode user k signal can be expressed as γbk = pk Luk,b ��hH r,kΘHGHGΘ¯Θhr,k ��2 K� i=1 i̸=k pi Lui,b ��hH r,kΘHGHGΘ¯Θhr,i ��2 + d−αr r,b ��hH r,kΘHGHG¯ΘΘ ��2 σ2r + ��hH r,kΘHGH��2 σ2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (27) Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic up-link rate of user k in EH RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by 15 E {Rbk} ≈ (1 − τ) log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,bξkˆ̺4 K� i=1 i̸=k pi Lui,bςiˆ̺4 + ˆ̺4d−αr r,b σ2rνk + ˆ̺2υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (28) Proof: This expression can be obtained by following same derivation in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Ergodic Rate at Eavesdropper j The SINR at eavesdropper j to decode user k signal is given by γej,k = pk ���d − αr 2 uk,r d − αe 2 ej,r hej,rΘ¯Θhr,k + d − αe 2 ej,k hej,k ��� 2 K� i=1 i̸=k pi ���d − αr 2 ui,r d − αe 2 ej,r hej,rΘ¯Θhr,i + d − αe 2 ej,i hej,i ��� 2 + d−αe ej,r ��hej,r ¯ΘΘ ��2 σ2 r + σ2 ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (29) Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic rate at eavesdropper j in up-link EH RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by E � Rej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � = (1 − τ) log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk ˆxj K� i=1 i̸=k piˆyi + ˆzjσ2r + σ2ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (30) where ˆxj = � d−αr uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rd−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ˆ̺2 � ρej ρej +1 ρk ρk+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρk+1M + ρk ρk+1 1 ρej +1M + 1 ρej +1 1 ρk+1M � + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ˆyi = d−αr ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ˆ̺2 � ρej ρej +1 ρi ρi+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρi+1M + ρi ρi+1 1 ρej +1M + 1 ρej +1 1 ρi+1M + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i � ˆzj = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r M ˆ̺2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' which have been derived in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Finally, the ergodic secrecy rate in EH RIS scheme is presented in the next Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The ergodic secrecy rate of user k in EH active RIS-aided MU-MISO systems under Rician fading channels and with phase shift error can be calculated by 16 ˆRs = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 (1 − τ) log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,bξk ˆ̺4 K� i=1 i̸=k pi Lui,bςiˆ̺4 + ˆ̺4d−αr r,b σ2 rνk + ˆ̺2υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − (1 − τ) log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk ˆxj K� i=1 i̸=k piˆyi + ˆzjσ2r + σ2ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (31) VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' SYSTEM DESIGN In this section, based on the derived analytical expressions, we first design the phase shifts of the RIS configurations considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, the best RIS configuration selection scheme is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Phase Shift Optimization The secrecy rate expressions presented in Theorems, 1, 2 and 3, show that the secrecy performance relies on the phase shifts of the RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In this work, it is assumed that the CSI of the eavesdroppers is unknown at the BS/RIS (only channel distribution known).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Therefore, to enhance the system performance, the RIS phase shifts can be optimized by maximizing the achievable ergodic sum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Since the phase shift at each unit of the RIS lies in the range of [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2π), the phase shift optimization problem can be formulated as max Θ K� i=1 ˆRbi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='t θm ∈ [0, 2π) , ∨m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (32) Due to the complicated formula of the ergodic sum rate, it is difficult to optimize (32) based on the conventional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However, GA-based methods can be employed to solve this optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Due to the page limitation, we refer readers to [6] for more details about the GA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 17 As an efficient suboptimal solution, the RIS phase shifts can be aligned to user k, who transmits the confidential message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' This presents a simple sub-optimal solution for enhancing the secrecy rate [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, the phase shifts should be θm = −2π d λ (xmtk + ymlk) , tk = sin φa kr sin φe kr − sin φa t sin φe t, lk = cos φe kr − cos φe t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (33) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' RIS Configuration Selection Scheme Based on the required secrecy rate (rs) and amount of the power available at user k, and the RIS, we can decide which system configuration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=', passive RIS, active RIS or EH RIS, should be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' A) If user k has sufficient amount of power to achieve the target secrecy rate, in this case passive RIS can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Based on the secrecy rate expression provided in Theorem 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the required user k power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' to achieve the target secrecy rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' can be obtained by solving rs = log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi + υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk xk K� i=1 i̸=k pi yi + σ2 ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (34) which can be found as pk = p1 − p2 p3 − p4 (35) where p1 = K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+υkσ2 b K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+υkσ2 b ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' p2 = 2rs K � i=1 i̸=k pi yi+2rsσ2 ej K � i=1 i̸=k pi yi+σ2ej ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' p3 = 2rs xk K � i=1 i̸=k pi yi+σ2ej and p4 = Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+υkσ2 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' B) If user k has limited amount of power, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=', the user power, pk, is less than the power required in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In this case active RIS can be implemented to provide the target secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Based on the secrecy rate expression provided in Theorem 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the required RIS power,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ̺ or Pr ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='to achieve the target secrecy rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' can be obtained by solving 18 rs = log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk̺2 K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi̺2 + ̺2d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2rνk + υkσ2 b \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 − log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + pk̺2x1 + pkx2 K� i=1 i̸=k pi̺2y1i+ K � i=1 i̸=k piy2i + z1̺2σ2r + σ2ej \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (36) where x1 = d−αr uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rd−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ρej ρej +1 ρk ρk+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρk+1M + ρk ρk+1 1 ρej +1M + 1 ρej +1 1 ρk+1M � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' x2 = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' y1i = d−αr ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r � ρej ρej +1 ρi ρi+1 � M + ρ (κ)2 ξ � + ρej ρej +1 1 ρi+1M + ρi ρi+1 1 ρej +1M + 1 ρej +1 1 ρi+1M � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' y2i = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' and z1 = d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' After some simplifications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the last equation can be expressed as ̺4 (q1 − q3) + ̺2 (q2 − q4 − q5 + q7) + (q8 − q6) = 0 (37) where q1 = K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi K � i=1 i̸=k piy1i+d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk K� i=1 i̸=k piy1i+ K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςiz1σ2 r+d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνkz1σ2 r+ K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςipkx1+ d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνkpkx1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q2 = \uf8eb \uf8edυkσ2 b K � i=1 i̸=k piy1i + υkσ2 bz1σ2 r + υkσ2 bpkx1 \uf8f6 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q3 = K� i=1 i̸=k piy1ipk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk+z1σ2 rpk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk+ K � i=1 i̸=k piy1i K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+z1σ2 r K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+ K � i=1 i̸=k piy1id−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk+ z1σ2 rd−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q4 = K� i=1 i̸=k piy2ipk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk+σ2 ejpk Luk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bξk+ K � i=1 i̸=k piy2i K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+σ2 ej K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi+ K� i=1 i̸=k piy2id−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk+ σ2 ejd−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q5 = \uf8eb \uf8ed K� i=1 i̸=k piy1iυkσ2 b + z1σ2 rυkσ2 b \uf8f6 \uf8f8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q6 = K� i=1 i̸=k piy2iυkσ2 b + σ2 ejυkσ2 b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' q7 = K� i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi̺22rspkx2+̺2d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk2rspkx2+ K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi2rsσ2 ej+d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk2rsσ2 ej+ K � i=1 i̸=k pi Lui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='bςi2rs K� i=1 i̸=k piy2i + d−αr r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='b σ2 rνk2rs K� i=1 i̸=k piy2i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 19 q8 = υkσ2 b2rspkx2 + υkσ2 b2rsσ2 ej + υkσ2 b2rs K � i=1 i̸=k piy2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, from (15), the RIS power should be higher than or equal to Pr = M \uf8eb \uf8ed− (q2 − q4 − q5 + q7) ± � (q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6) 2 (q1 − q3) \uf8f6 \uf8f8 � K � k=1 pk dαr uk,r + σ2 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (38) C) If user k and the RIS have limited amount of power, e,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=', user k power, pk, is less than the required power in (35) and the RIS power, Pr, is less than the required power in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In this case EH RIS can be implemented to provide the target secrecy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Based on (25) and (38), the required BS power, Pb, to charge the RIS and achieve the target secrecy rate, rs, can be obtained by Pb = M (1 − τ) ηeffτNM \uf8eb \uf8ed− (q2 − q4 − q5 + q7) ± � (q2 − q4 − q5 + q7)2 − 4 (q1 − q3) (q8 − q6) 2 (q1 − q3) \uf8f6 \uf8f8 × � K � k=1 pk dαr uk,r + σ2 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (39) VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we present simulation and numerical results to assess the accuracy of the derived expressions and the secrecy performance of the RIS schemes considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Monte-Carlo simulations with 105 independent trials are excuted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The locations of the BS and the RIS are (0 m, 0 m), (20 m, 20 m), respectively, while the users are scattered on the corners of a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Specifically, the coordinates for the users square are (30 m, 5 m), (35 m, 5 m), (30 m,−5 m), and (35 m,−5 m), respectively, while the eavesdroppers are distributed in a circle centered at (20 m, 0 m) with radius of 10 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Unless otherwise specified, the simulation settings are assumed as follows: K = J = 4, N = 10, M = 5, the users power pi = 2W, the active RIS power Pr = 7W , the BS power in EH RIS scenario Pb = 50W, and the nodes have same noise variance, σ2 = −70 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, the path-loss exponent is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='7, the Rician factors ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The values of the AoA and AoD of the BS and the RIS are uni-formally distributed in (0, 2π), and the concentration parameter of RIS phase error κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 20 Firstly, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2, we illustrate the ergodic secrecy rate versus the transmission user power, pk, for the three considered RIS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2a shows the secrecy rate with phase shift errors and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 2b, presents the secrecy rate for the ideal scenario, when there is no phase error at RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It is clear from this figure that the analytical results are in good agreement with the simulated results, which confirms the validity of the analysis presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It is also evident that for the given parameters values, the secrecy rate loss due to the imperfect phase shift at the RIS is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 bits/s/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, passive RIS achieves the lowest secrecy rate, but with small amount of power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The secrecy rate gain of active RIS above passive RIS is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='8 bits/s/Hz for a given user power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Furthermore, high secrecy rates can be achieved and controlled by implementing EH RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However, in this case the BS should transmit high power in the EH phase to provide sufficient amount of energy at the RIS to achieve higher secrecy rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 0 10 20 30 40 50 60 pk (W) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 Secrecy Rate (bits\\sec\\Hz) Active RIS Passive RIS Analytical EH RIS (a) Secrecy rate versus user, k , power with phase shift error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 0 10 20 30 40 50 60 pk (W) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 Secrecy Rate (bits\\sec\\Hz) Active RIS Passive RIS Analytical EH RIS (b) Secrecy rate versus user, k , power with no phase shift error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Figure 2: Secrecy rate versus user, k , power with and without phase shift error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' To explain the impact of the phase errors at the RIS on the secrecy performance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 3, we plot the secrecy rate versus the concentration parameter of the phase error, κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Additionally, the results of ideal RIS are also presented in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It can be observed from these results that the secrecy rate enhances as the concentration parameter, κ, increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, at high concentration parameter values, κ −→ ∞, the secrecy rate achieved by imperfect RIS saturates to that achieved by ideal RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' This can be explained by the fact that the phase error at the RIS is assumed to follow a Von Mises distribution, thus high concentration parameter values make the error fluctuate in a smaller range, and 21 2 4 6 8 10 12 14 16 18 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='8 2 Secrecy Rate (bits\\sec\\Hz) Ideal RIS without Error RIS with Error RIS with Error RIS with Error Ideal RIS without Error Active RIS Passive RIS EH RIS Ideal RIS without Error Figure 3: Secrecy rate versus concentration parameter, κ, of RIS phase error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' when κ −→ ∞, the error at the RIS tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Accordingly, the secrecy rate of imperfect RIS converges to the ideal RIS case as κ −→ ∞, as explained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 4 shows the secrecy rate versus the number of BS antennas N for the all RIS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It is evident and as expected, increasing the number of BS antennas N enhances the secrecy performance for the all RIS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' It should be pointed out that the number of BS antennas, N, has impact only on the received signal at the BS, thus increasing N results in enhancing the rate of the legitimate users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However N dose not have any impact on the rate at the eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Having said that in EH RIS, increasing N also increases the amount of the harvested energy at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus, in EH RIS, N has impact on both achievable rates at the BS and the eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 5, we depict the secrecy rate versus the number of RIS elements, M, for the all considered RIS schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' To obtain clear insights and results, in this figure the noise variance at the nodes is assumed to be σ2 = −20 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Notably and as expected, increasing M results in enhancing the secrecy rate for the all considered scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, as we can notice from the analytical expressions of the secrecy rate presented in this paper, the number of RIS elements M has impact on both the achievable rate at the BS and the eavesdroppers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=', adding more RIS elements increases the rate at the BS and the eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' However this improvement in the rate is essential at the BS, because the RIS phase shifts are designed to be toward the BS direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Furthermore, in the EH RIS scheme, increasing the 22 5 10 15 20 N 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='8 Secrecy Rate (bits\\sec\\Hz) Passive RIS Analytical Active RIS EH RIS Figure 4: Secrecy rate versus number of BS antennas, N, with phase shift error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 20 40 60 80 100 120 140 M 0 1 2 3 4 5 6 Secrecy Rate (bits\\Sec\\Hz) Active RIS Passive RIS EH RIS Analytical Figure 5: Secrecy rate versus number of RIS elements, M, with phase shift error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' number of the RIS elements, M, leads to an increase in the amount of the harvested energy at the RIS and thus Pr will be high when the number of elements M is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In order to illustrate the RIS configuration selection scheme, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 6 we plot the user power versus the target secrecy rate for different values of the concentration parameter of RIS phase error, κ = 2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Firstly, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 6a and 6b, we consider two examples, when the target secrecy rate is assumed to be rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 (bits/s/Hz) and rs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 (bits/s/Hz) for κ = 2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' As we can see from 23 the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 6a, when rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 (bits/s/Hz), passive RIS can achieve the target secrecy rate with total transmission power is PT = pk = 50W, (neglecting the small amount of power consuming at passive RIS elements), and in the active RIS scheme the user transmission power can be reduced to around pk = 7W and thus the total transmission power is PT = pk + Pr = 14W, while EH RIS scheme can achieve the target secrecy rate with the smallest amount of the user power which is about pk = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='95W, but with the highest total transmission power PT = pk + Pb = 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='95W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Similar observations can be noticed from the second scenario when rs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 (bits/s/Hz), passive RIS achieves the target secrecy rate with the highest user power, while EH RIS achieves, rs, with the smallest user power but with very high total consumption power, and the active RIS scheme works between these two regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, the concentration parameter of RIS phase error, κ, has essential impact on the required user power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' By comparing Figs 6a and 6b, one can notice that as κ increases the required user power to achieve the target secrecy rate decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' For instance when the target secrecy rate is rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 (bits/s/Hz), the required user power in the passive RIS scheme is about 50W when κ = 2, and 20W when κ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' This is due to the fact explained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 6c and 6d, we present the RIS configuration selection scheme when the available user power is pk = 20W for κ = 2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In the first case when κ = 2, if the target secrecy rate is rs ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='45 (bits/s/Hz), passive RIS can be selected, and active RIS can be implemented if the target secrecy rate is rs ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='17 (bits/s/Hz), while EH RIS can be selected if rs ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='87 (bits/s/Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' These secrecy rate regions of the RIS schemes become wider as the concentration parameter of RIS phase error, κ, increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 6d when κ = 8, passive RIS can be selected to achieve secrecy rates up to rs ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='77 (bits/s/Hz), and active RIS can be selected to perform secrecy rates less than or equal to rs ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='635 (bits/s/Hz), whilst EH RIS can be used to achieve secrecy rates up to rs ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='48 (bits/s/Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 24 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 Target Secrecy Rate (bits/s/Hz) 0 10 20 30 40 50 60 User Power (W) Passive RIS Active RIS, Pr=7W EH RIS, Pb=50W Target Secrecy Rate rs=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 (bits/s/Hz) Target Secrecy Rate rs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 (bits/s/Hz) (a) The user power versus target secrecy rate when κ = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 3 Target Secrecy Rate (bits/s/Hz) 0 10 20 30 40 50 60 User Power (W) Passive RIS Active RIS, Pr=7W EH RIS, Pb=50W Target Secrecy Rate rs=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 (bits/s/Hz) Target Secrecy Rate rs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='75 (bits/s/Hz) (b) The user power versus target secrecy rate when κ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 Target Secrecy Rate (bits/s/Hz) 0 10 20 30 40 50 60 User Power (W) Passive RIS Active RIS, Pr=7W EH RIS, Pb=50W Only EH RIS can be selected Active RIS can be selected Passive RIS can be selected Available power pk (c) RIS configuration selection scheme when pk = 20W and κ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='5 3 Target Secrecy Rate (bits/s/Hz) 0 10 20 30 40 50 60 User Power (W) Passive RIS Active RIS, Pr=7W EH RIS, Pb=50W Available power pk Passive RIS can be selected Active RIS can be selected Only EH RIS can be selected (d) RIS configuration selection scheme when pk = 20W and, κ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Figure 6: The user power versus target secrecy rate for different values of the concentration parameter of RIS phase error, κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' CONCLUSIONS In this paper the impact of phase shift error on the secrecy performance of up-link RIS-aided MU- MISO systems was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Under Rician fading channels and phase shift errors the ergodic secrecy rate for, passive RIS, active RIS, and EH RIS have been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Then, the phase shifts at the RIS have been optimized based on the derived rate expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' In addition, according to the target secrecy rate and amount of power available at the users, the best RIS configuration selection scheme has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The results presented in this work demonstrated that an active RIS scheme can enhance the secrecy performance of imperfect RIS elements, especially when the users have limited amount of 25 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Furthermore, increasing the number of BS antennas, the concentration parameter of RIS phase error, and the number of RIS elements lead to the enhancement of the secrecy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' APPENDIX A By using Jensen inequality, the ergodic rate can be expressed as E {Rbk} ≈ log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + E \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 pk Luk,b ��hH r,kΘHGHGΘ¯Θhr,k ��2 K� i=1 i̸=k pi Lui,b ��hH r,kΘHGHGΘ¯Θhr,i ��2 + ��hH r,kΘHGH��2 σ2 b \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (40) Due to the paper length limitation, in this Appendix we will explain how to calculate the average of the first term, similarly and by following similar steps we can find the average of the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The first term is E � Pk Luk,b ��hH r,kΘHGHGΘ¯Θhr,k ��2� = Pk Luk,bE ���hH r,kΘHGHGΘ¯Θhr,k ��2� (41) where hH r,kΘHGHGΘ¯Θhr,k = hH r,kΘH � ρb ρb + 1 ¯GH ¯G + √ρb ρb + 1 ¯GH ˜G + √ρb ρb + 1 ˜GH ¯G + 1 ρb + 1 ˜GH ˜G � Θ¯Θhr,k = 1 ρb + 1hH r,kΘH � ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G � Θ¯Θhr,k = 1 ρb + 1hH r,kA¯Θhr,k (42) where A = ΘH � ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G � Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Now (42) can be expressed as hH r,kΘHGHGΘ¯Θhr,k = 1 (ρb + 1) (ρk + 1) �√ρk¯hH r,k + ˜hH r,k � A¯Θ �√ρk¯hr,k + ˜hr,k � = 1 (ρb + 1) (ρk + 1) \uf8eb \uf8ec \uf8edρk¯hH r,kA¯Θ¯hr,k � �� � ∆1 + √ρk¯hH r,kA¯Θ˜hr,k � �� � ∆2 + √ρk˜hH r,kA¯Θ¯hr,k � �� � ∆3 + ˜hH r,kA¯Θ˜hr,k � �� � ∆4 \uf8f6 \uf8f7 \uf8f8 (43) 26 The channels are independent and have zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Thus by removing the zero expectation terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' we can get E ���hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHGΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� = 1 (ρb + 1)2 (ρk + 1)2E \uf8f1 \uf8f2 \uf8f3 ����� 4 � i=1 ∆i ����� 2\uf8fc \uf8fd \uf8fe = 1 (ρb + 1)2 (ρk + 1)2 � 4 � i=1 E � |∆i|2� + 2E {∆1∆∗ 4} � (44) Now the first term ∆1 = ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH � ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G � Θ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = \uf8eb \uf8ec \uf8edρbρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ¯GΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 + √ρbρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ˜GΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 +√ρbρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ¯GΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3 + ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ˜GΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 \uf8f6 \uf8f7 \uf8f8 (45) The average of the first term E � |∆1|2� = E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� + 2E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆H 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � (46) where ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 = ρbρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ¯GΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' which can be written as ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 = ρbρkaH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯ΘaM (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 = ρbρk � M � m=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) e−jϕmaH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) � � M � m=1 aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕmej ¯ ϕmaM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (47) The average can now be written as 27 E � |∆1,1|2� = ρ2 bρ2 k � M � m=1 aH M,m (φa kr, φe kr) e−jϕmaH M,m (φa r, φe r) �2 × \uf8eb \uf8edM + ρ (κ)2 ����� M � m1=1 M � m2̸=m1 � aM,m1 (φa kr, φe kr) ejϕm1aM,m1 (φa r, φe r) � � aM,m2 (φa kr, φe kr) ejϕm2aM,m2 (φa r, φe r) �H ����� 2\uf8f6 \uf8f8 (48) E � |∆1,1|2� = ρ2 bρ2 k |fk|2 �� 1 − ρ (κ)2� M + ρ (κ)2 |fk|2� (49) where fk = M � m=1 fk,m, fk,m = aH M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r, φe r) ejϕmaM,m (φa kr, φe kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' The second term, ∆1,2 = √ρbρkaH M (φa kr, φe kr) ΘHaM (φa r, φe r) aH N (φa b, φe b) ˜GΘ¯ΘaM (φa kr, φe kr) = √ρbρkf ∗ k M � m=1 N � n=1 aH N,n (φa b, φe b) ˜gnmejϕmej ¯ ϕmaM,m (φa kr, φe kr) , (50) E � |∆1,2|2� = ρbρ2 kNM |fk|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (51) The third term ∆1,3 = √ρbρkaH M (φa kr, φe kr) ΘH ˜GHaN (φa b, φe b) aH M (φa r, φe r) Θ¯ΘaM (φa kr, φe kr) = √ρbρk M � m=1 N � n=1 aH N,n (φa b, φe b) ˜gH nme−jϕmaM,m (φa kr, φe kr) M � m=1 ej ¯ ϕmfk,m, (52) E � |∆1,3|2� = ρbρ2 k � NMρ (κ)2 |fk|2 + � 1 − ρ (κ)2� NM2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (53) The forth term ∆1,4 = ρkaH M (φa kr, φe kr) ΘH ˜GH ˜GΘ¯ΘaM (φa kr, φe kr) 28 = ρk M � m1=1 aH M,m1 (φa kr, φe kr) e−jϕm˜gH nm1 M � m2=1 ˜gnm2ejϕmej ¯ ϕmaM,m2 (φa kr, φe kr) , (54) E � |∆1,4|2� = ρkNM � Mρ (κ)2 + 1 − ρ (κ)2� + NM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (55) The last term E � ∆1,1∆∗ 1,4 � = N |fk|2 � Mρ (κ)2 + 1 − ρ (κ)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' (56) Similarly, following the same way we can find the average of the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' APPENDIX B Using Jensen inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the ergodic rate can be written as E � Rej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � ≈ log2 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 + E \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 pk ���d − αr 2 uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 K� i=1 i̸=k pi ���d − αr 2 ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i + d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ��� 2 + σ2 ej \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (57) The average of the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' after removing the zero expectation terms can be calculated by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E ����d − αr 2 uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2� = d−αr uk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rd−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r (58) where E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� = E ����� �� ρej ρej + 1 � ρk ρk + 1 ¯hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + � ρej ρej + 1 � 1 ρk + 1 ¯hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + � ρk ρk + 1 � 1 ρej + 1 ˜hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + � 1 ρej + 1 � 1 ρk + 1 ˜hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ������ 2\uf8fc \uf8fd \uf8fe (59) E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� = ρej ρej + 1 ρk ρk + 1E ��¯hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2 + ρej ρej + 1 1 ρk + 1E ���¯hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 29 + ρk ρk + 1 1 ρej + 1E ���˜hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 + 1 ρej + 1 1 ρk + 1E ���˜hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 (60) Now ¯hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = � M � m=1 aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕmej ¯ ϕmaM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr �� (61) E ��¯hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2 = E ����� M � m=1 aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕmej ¯ ϕmaM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr ������ 2 = M+ρ (κ)2 M � m1=1 M � m2̸=m1 � aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕm1aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) � � aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m2 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕm2aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m2 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) �H = M + ρ (κ)2 ξ (62) where ξ = M � m1=1 M � m2̸=m1 (aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕm1aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r)) (aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m2 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ejϕm2aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m2 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r)) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ¯hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = aM � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr � Θ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = M � m=1 aMm � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr � ejϕmej ¯ ϕm � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m (63) E ���¯hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 = E ����� M � m=1 aMm � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr � ejϕmej ¯ ϕm � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m ����� 2 E ���¯hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 = M+ E � M � m1=1 M � m2̸=m1 � aMm1 � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr � ejϕm1ej ¯ ϕm1 � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m1 � � aMm2 � φa ejr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe ejr � ejϕm2ej ¯ ϕm2 � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m2 �H� = M (64) other terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' 30 ˜hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = M � m=1 ˜hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='mejϕmej ¯ ϕmaM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) (65) E ���˜hejΘ¯Θ¯hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 = M+ E � M � m1=1 M � m2̸=m1 �� ˜hej � m1 ejϕm1ej ¯ ϕm1aMm1 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) � �� ˜hej � m2 ejϕm2ej ¯ ϕm2aMm2 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) �H� = M (66) and ˜hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k = M � m=1 � ˜hej � m ejϕmej ¯ ϕm � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m (67) E ���˜hejΘ¯Θ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��� 2 = E ����� M � m=1 � ˜hej � m ejϕmej ¯ ϕm � ˜hr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � m ����� 2 = M (68) Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' we are ready to write the average of the first term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k ��2� = ρej ρej + 1 ρk ρk + 1 � M + ρ (κ)2 ξ � + ρej ρej + 1 1 ρk + 1M + ρk ρk + 1 1 ρej + 1M + 1 ρej + 1 1 ρk + 1M (69) Similarly we can find the average of the second term as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E ����d − αr 2 ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i + d − αe 2 ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ��� 2� = d−αr ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='r E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ��2� + d−αe ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i (70) E ���hej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='rΘ¯Θhr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='i ��2� = ρej ρej + 1 ρi ρi + 1 � M + ρ (κ)2 ξ � + ρej ρej + 1 1 ρi + 1M + ρi ρi + 1 1 ρej + 1M + 1 ρej + 1 1 ρi + 1M (71) 31 APPENDIX C Using Jensen inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' the ergodic rate can be written as E {Rbk} ≈ log2 (1 + E {γbk}) (72) We will follow similar steps as in Appendix A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHG¯ΘΘ = 1 ρb + 1hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kA¯Θ (73) where A = ΘH � ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G � Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Last expression can be written as hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHG¯ΘΘ = 1 (ρb + 1) � (ρk + 1) �√ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k + ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k � A¯Θ = 1 (ρb + 1) � (ρk + 1) \uf8eb \uf8ec \uf8ed√ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kA¯Θ � �� � ∆1 + ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kA¯Θ � �� � ∆2 \uf8f6 \uf8f7 \uf8f8 (74) The average can be written as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' E ���hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHGHG¯ΘΘ ��2� = 1 (ρb + 1)2 (ρk + 1) E \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρr ������� ¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kA¯Θ � �� � ∆1 ������� 2 + ������� ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kA¯Θ � �� � ∆2 ������� 2\uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe (75) ∆1 = √ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH � ρb ¯GH ¯G + √ρb ¯GH ˜G + √ρb ˜GH ¯G + ˜GH ˜G � Θ¯Θ = \uf8eb \uf8ec \uf8edρb √ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ¯GΘ¯Θ � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 + √ρb √ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ˜GΘ¯Θ � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 +√ρb √ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ¯GΘ¯Θ � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3 + √ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ˜GΘ¯Θ � �� � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 \uf8f6 \uf8f7 \uf8f8 (76) E � |∆1|2� = E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� + E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� + 2E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆H 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � (77) 32 where ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 = ρb √ρk � aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � � aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯Θ � (78) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� = ρ2 bρk ��� aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) ���2 × E ���aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯Θ ��2� (79) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� = ρ2 bρk ��� aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) ���2 × M (80) The second term can be expressed as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 = √ρb √ρkaH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) M � m=1 N � n=1 aH N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='n (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) ˜gnmejϕmej ¯ ϕm (81) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� = ρbρk ��aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ��2 NM (82) The third term can be written as ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3 = √ρb √ρk M � m=1 N � n=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ˜gH nme−jϕmaN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='n (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) M � m=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ej ¯ ϕmejϕm (83) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbρkMN \uf8eb \uf8edE ����� M � m=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ej ¯ ϕmejϕm ����� 2\uf8f6 \uf8f8 (84) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbρkMN � M + ρ (κ)2 M � m1=1 M � m2̸=m1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ejϕm1aM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m2 (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) e−jϕm2 � (85) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbρkMN � ρ (κ)2 M + � 1 − ρ (κ)2� M � (86) 33 The forth term can be represented as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 = √ρk N � n=1 M � m1=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m1 (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) e−jϕm˜gH nm1 M � m2=1 ˜gnm2ejϕmej ¯ ϕm (87) E � |∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� = ρk � N2M + NM2� (88) Now the last term can be written as E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � = ρbρk � aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � × � aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯Θ � ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ˜GΘ¯Θ (89) E � ∆1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � = ρbρk � aH M (φa kr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe kr) ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � (aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ) ρk¯hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHNΘ (90) We will repeat similar steps for ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ∆2 = \uf8eb \uf8ec \uf8edρb˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ¯GΘ¯Θ � �� � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 + √ρb˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ˜GΘ¯Θ � �� � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 +√ρb˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ¯GΘ¯Θ � �� � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3 + ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ˜GΘ¯Θ � �� � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 \uf8f6 \uf8f7 \uf8f8 (91) E � |∆2|2� = E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� + E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� + 2E � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆H 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � (92) where ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1 = ρb˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ¯GH ¯GΘ¯Θ (93) 34 E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1|2� = ρ2 b ��ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ ��2 F (94) and ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2 = √ρb˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) M � m=1 N � n=1 aH N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='n (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) ˜gnmejϕmej ¯ ϕm (95) E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='2|2� = ρb ���˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHaM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ��� 2 NM (96) while ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3 = √ρb M � m=1 N � n=1 ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='nm˜gH nme−jϕmaN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='n (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) M � m=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ej ¯ ϕmejϕm (97) E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='3|2� = ρbMN \uf8eb \uf8edE ����� M � m=1 aH M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='m (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) ej ¯ ϕmejϕm ����� 2\uf8f6 \uf8f8 = ρbMN � ρ (κ)2 M + � 1 − ρ (κ)2� M � (98) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 = ρk N � n=1 M � m1=1 ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='nm1e−jϕm˜gH nm1 M � m2=1 ˜gnm2ejϕmej ¯ ϕm (99) E � |∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4|2� = ρ2 k � N2M + NM2� (100) and E � ∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='1∆∗ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='4 � = E � ρbρk � ˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � � aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ¯Θ � ρk˜hH r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content='kΘH ˜GH ˜GΘ¯Θ � = ρk � ΘHaH M (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) aH N (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) aN (φa b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe b) � (aM (φa r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' φe r) Θ) ρkΘNΘH (101) 35 REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Di Renzo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf'} +page_content=' Zappone, M.' metadata={'source': 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Alfaro, Ricardo L. Soto† +Dpto. Matem´aticas, Universidad Cat´olica del Norte, Casilla 1280 +Antofagasta, Chile. +Abstract +A list Λ = {λ1, λ2, . . . , λn} of complex numbers is said to be real- +izable if it is the spectrum of a nonnegative matrix. Λ is said to be +universally realizable (UR) if it is realizable for each possible Jordan +canonical form allowed by Λ. In this paper, using companion matrices +and applying a procedure by ˇSmigoc, is provides a sufficient condi- +tion for the universal realizability of left half-plane spectra, that is, +Λ = {λ1, . . . , λn} with λ1 > 0, Re λi ≤ 0, i = 2, . . . , n. It is also shown +how the effect of adding a negative real number to a not UR left half- +plane list of complex numbers, makes the new list UR, and a family +of left half-plane lists that are UR is characterized. +AMS classification: +15A18, 15A20, 15A29 +Key words: Nonnegative matrix; companion matrix; Universal realizabil- +ity; ˇSmigoc’s glue. +1 +Introduction +A list Λ = {λ1, λ2, . . . , λn} of complex numbers is said to be realizable if it +is the spectrum of an n-by-n nonnegative matrix A, and A is said to be a +realizing matrix for Λ. The problem of the realizability of spectra is called the +∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, N´ucleo 6 UCN +VRIDT 083-2020, Chile. +†E-mail addresses: rsoto@ucn.cl (R. L. Soto), jaime.alfaro@ucn.cl (J. H. Alfaro). +1 + +nonnegative inverse eigenvalue problem (NIEP). From the Perron-Frobenius +Theorem we know that if Λ = {λ1, λ2, . . . , λn} is the spectrum of an n- +by-n nonnegative matrix A, then the leading eigenvalue of A equals to the +spectral radius of A, ρ(A) =: max +1≤i≤n |λi| . This eigenvalue is called the Perron +eigenvalue, and we shall assume in this paper, that ρ(A) = λ1. +A matrix is said to have constant row sums, if each one of its rows sums +up to the same constant α. The set of all matrices with constant row sums +equal to α, is denoted by CSα. Then, any matrix A ∈ CSα has the eigenvector +eT = [1, 1, . . . , 1], corresponding to the eigenvalue α. The real matrices with +constant row sums are important because it is known that the problem of +finding a nonnegative matrix with spectrum Λ = {λ1, . . . , λn}, is equivalent +to the problem of finding a nonnegative matrix in CSλ1 with spectrum Λ (see +[3]). We denote by ek, the n-dimensional vector, with 1 in the kth position +and zeros elsewhere. If Λ = {λ1, . . . , λn}, then sk(Λ) = +n +� +i=1 +λk +i , k = 1, 2, . . . . +A list Λ = {λ1, λ2, . . . , λn} of complex numbers, is said to be diagonaliz- +ably realizable (DR), if there is a diagonalizable realizing matrix for Λ The +list Λ is said to be universally realizable (UR), if it is realizable for each possi- +ble Jordan canonical form (JCF) allowed by Λ. The problem of the universal +realizability of spectra, is called the universal realizability problem (URP). +The URP contains the NIEP, and both problems are equivalent if the given +numbers λ1, λ2, . . . , λn are distinct. In terms of n, both problems remain +unsolved for n ≥ 5. It is clear that if Λ is UR, then Λ must be DR. The +first known results on the URP are due to Minc [7, 8]. In terms of the URP, +Minc [7] showed that if a list Λ = {λ1, λ2, . . . , λn} of complex numbers is the +spectrum of a diagonalizable positive matrix, then Λ is UR. The positivity +condition is necessary for Minc’s proof, and the question set by Minc himself, +whether the result holds for nonnegative realizations was open for almost 40 +years. Recently, two extensions of Minc’s result have been obtained in [1, 4]. +In [1], Collao et al. showed that a nonnegative matrix A ∈ CSλ1, with a pos- +itive column, is similar to a positive matrix. Note that if A is nonnegative +with a positive row and AT has a positive eigenvector, then AT is also similar +to a positive matrix. Besides, if Λ is diagonalizably realizable by a matrix +A ∈ CSλ1 having a positive column, then Λ is UR. In [4], Johnson et al. in- +troduced the concept of ODP matrices, that is, nonnegative matrices with all +positive off-diagonal entries (zero diagonal entries are permitted) and proved +2 + +that if Λ is diagonalizably ODP realizable, then Λ is UR. Note that both +extensions contain, as a particular case, Minc’s result in [7]. Both extensions +allow us to significantly increase the set of spectra that can be proved to be +UR, as for instance, certain spectra Λ = {λ1, . . . , λn} with s1(Λ) = 0, which +is not possible from Minc’s result. In particular, we shall use the extension +in [1] to generate some of our results. +Remark 1.1 In [1], Section 2, Theorem 2.1 and Corollary 2.1, there is an +error in assuming that if A is nonnegative with a positive row, then AT, which +has a positive column, is similar to a positive matrix. The reason is that we +cannot guarantee that AT has a positive eigenvector. +Regarding non-positive universal realizations, we mention that in [10, +2] the authors proved, respectively, that lists of complex numbers Λ = +{λ1, . . . , λn}, of Suleimanova type, that is, +λ1 > 0, Re λi ≤ 0, |Re λi| ≥ |Im λi| , i = 2, 3, . . . , n, +or of ˇSmigoc type, that is, +λ1 > 0, Re λi ≤ 0, +√ +3 |Re λi| ≥ |Im λi| , i = 2, 3, . . . , n, +(1) +are UR if and only if they are realizable if and only if +n +� +i=1 +λi ≥ 0. +Outline of the paper: The paper is organized as follows: In Section 2, we +present the mathematical tools that will be used to generate our results. In +Section 3, we study the URP for a left half-plane list and we give a sufficient +condition for it to be UR. In Section 4, we discuss the effect of adding a +negative real number −c to a left half-plane list Λ = {λ1, −a±bi, . . . , −a±bi}, +which is not UR (or even not realizable), or we do not know whether it is, +and we show how Λ ∪ {−c} becomes UR. We also characterize a family of +left half-plane lists that are UR. In Section 5, we show that the merge of two +lists diagonalizably realizable Γ1 ∈ CSλ1 and Γ2 ∈ CSµ1 is UR. Examples +are shown to illustrate the results. +2 +Preliminaries +Throughout this paper we use the following results: The first one, by ˇSmigoc +[9], gives a procedure that we call ˇSmigoc’s glue technique, to obtain from two +3 + +matrices A and B of size n-by-n and m-by-m, respectively, a new (n+m−1)- +by-(n + m − 1) matrix C, preserving in certain way, the corresponding JCFs +of A and B. The second one, by Laffey and ˇSmigoc [6] solves the NIEP for +lists of complex numbers on the left half-plane, that is, lists with λ1 > 0, +Re λi ≤ 0, i = 2, . . . , n. Moreover, we also use Lemma 5 in [6]. +Theorem 2.1 [9] Suppose B is an m-by-m matrix with a JCF that contains +at least one 1-by-1 Jordan block corresponding to the eigenvalue c: +J(B) = +� c +0 +0 +I(B) +� +. +Let t and s, respectively, be the left and the right eigenvectors of B associated +with the 1-by-1 Jordan block in the above canonical form. Furthermore, we +normalize vectors t and s so that t +Ts = 1. Let J(A) be a JCF for the n-by-n +matrix +A = +� +A1 +a +bT +c +� +, +where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C +n-1. +Then the matrix +C = +� +A1 +at +T +sb +T +B +� +has JCF +J(C) = +� +J(A) +0 +0 +I(B) +� +. +Theorem 2.2 [6] Let Λ = {λ1, λ2, . . . , λn} be a list of complex numbers with +λ1 ≥ |λi| and Re λi ≤ 0, i = 2, . . . , n. Then Λ is realizable if and only if +s1 = s1(Λ) ≥ 0, +s2 = s2(Λ) ≥ 0, +s2 +1 ≤ ns2. +Lemma 2.1 [6] Let t be a nonnegative real number and let λ2, λ3, . . . , λn be +complex numbers with real parts less than or equal to zero, such that the list +{λ2, λ3, . . . , λn} is closed under complex conjugation. Set ρ = 2t−λ2−· · ·−λn +and +f(x) = (x − ρ) +n +� +j=2 +(x − λj) = xn − 2txn−1 + b2xn−2 + · · · + bn. +(2) +Then b2 ≤ 0 implies bj ≤ 0 for j = 3, 4, . . . , n. +4 + +3 +Companion matrices and the ˇSmigoc’s glue. +We say that a list Λ = {λ1, λ2, . . . , λn} of complex numbers is on the left half- +plane if λ1 > 0, Re λi ≤ 0, i = 2, 3, . . . , n. In this section we give a sufficient +condition for a left half-plane list of complex numbers to be UR. Of course, +it is our interest to consider lists of complex numbers containing elements +out of realizability region of lists of ˇSmigoc type. Our strategy consists in to +decompose the given list Λ = {λ1, λ2, . . . , λn} into sub-lists +Λk = {λk1, λk2, . . . , λkpk}, λ11 = λ1, k = 1, 2, . . . , t, +with auxiliary lists +Γ1 += +Λ1 +Γk += +{s1(Γk−1), λk1, λk2, . . . , λkpk}, +k = 2, , . . . , t, +each one of them being the spectrum of a nonnegative companion matrix +Ak, in such a way that it be possible to apply ˇSmigoc’s glue technique to +the matrices Ak, to obtain an n-by-n nonnegative matrix with spectrum Λ +for each possible JCF allowed by Λ. In the case s1(Λ) > 0, with λi ̸= 0, +i = 2, . . . , n, we may choose, if they exist, sub-lists Γk being the spectrum +of a diagonalizable nonnegative companion matrix with a positive column. +Then, after ˇSmigoc’s glue, we obtain a diagonalizable nonnegative n-by-n +matrix A with spectrum Λ and a positive column, which is similar to a +diagonalizable positive matrix. Thus, from the extension in [1], Λ is UR. +Next we have the following corollary from Theorem 2.1: +Corollary 3.1 Let Λ = {λ1, λ2, . . . , λn} be a realizable left half-plane list of +complex numbers. Suppose that for each JCF J allowed by Λ, there exists a +decomposition of Λ as +Λ += +Λ1 ∪ Λ2 ∪ · · · ∪ Λt, where +Λk += +{λk1, λk2, . . . , λkpk}, k = 1, 2, . . . , t, λ11 = λ1, +with auxiliary lists +Γ1 += +Λ1, +Γk += +{s1(Γk−1), λk1, λk2 . . . , λkpk}, k = 2, . . . , t, +being the spectrum of a nonnegative companion matrix Ak with JCF J(Ak) +as a sub-matrix of J, k = 1, 2, . . . , t. +Then Λ is universally realizable. +5 + +Proof. +Since each matrix Ak, k = 1, 2, . . . , t, is nonnegative companion +with JCF J(Ak) being a submatrix of J, then, from ˇSmigoc’s glue applied to +matrices Ak, we obtain an n-by-n nonnegative matrix with spectrum Λ and +JCF J. As J is any JCF allowed by Λ, then Λ is UR. +The following result is well known and useful. +Lemma 3.1 Let A be a diagonalizable irreducible nonnegative matrix with +spectrum Λ = {λ1, . . . , λn} and a positive row or column. Then A is similar +to a diagonalizable nonnegative matrix B ∈ CSλ1, with a positive row or +column. +Proof. If A is irreducible nonnegative, it has a positive eigenvector xT = +[x1, . . . , xn]. Then if D = dig{x1, . . . , xn}, the matrix +B = D−1AD = +�xj +xi +ai,j +� +∈ CSλ1 +is nonnegative with a positive row or column. +Suppose all lists Γk in Corollary 3.1, can be taken as the spectrum of a +diagonalizable nonnegative companion matrix Ak with a positive column (the +last one). Then, since the glue of matrices Ak gives an n-by-n diagonalizable +irreducible nonnegative matrix A with a positive column and spectrum Λ, A +is similar to a diagonalizable positive matrix with spectrum Λ and therefore +Λ is UR. This is what the next result shows. +Corollary 3.2 Let Λ = {λ1, λ2, . . . , λn}, λi ̸= 0, i = 2, . . . , n, s1(Λ) > 0, be +a realizable left half-plane list of complex numbers. If there is a decomposition +of Λ as in Corollary 3.1, with all lists Γk being the spectrum of a diagonal- +izable nonnegative companion matrix Ak, with a positive column, then Λ is +universally realizable. +Proof. It is enough to prove the result for two lists Γk of the decomposition +of Λ. Let Γk−1 and Γk, k = 2, . . . , t, be the spectrum, respectively, of matrices +Ak−1 and Ak, which are diagonalizable nonnegative companion with a posi- +tive column (the last one). Then Ak−1 and Ak are irreducible. In particular, +Ak has a positive eigenvector s and, since AT +k is also irreducible, Ak has also +a positive left eigenvector tT with tTs = 1. Now, let +Ak−1 = +� A1,k−1 +a +bT +s1(Γk−1) +� +. +6 + +Since the last column of Ak−1 is positive, the vector a is also positive and +atT is a positive submatrix. Therefore, the glue of Ak−1 with Ak, +Ck = +� A1,k−1 +atT +sbT +Ak +� +, +is a diagonalizable nonnegative matrix with its last column being positive. +Note that Ck is also irreducible. Then Ck has, besides, a positive eigenvector, +and from Lemma 3.1 Ck is similar to a matrix with constant row sums and +with its last column being positive. Thus, Ck is similar to a diagonalizable +positive matrix. Then, ˇSmigoc’s glue applied to all matrices Ak gives an n- +by-n diagonalizable irreducible nonnegative matrix A with a positive column +and spectrum Λ. Therefore, A is similar to a diagonalizable positive matrix +with spectrum Λ and from the extension in [1] Λ is UR. +Observe that if λi ̸= 0, i = 2, . . . , n; s1(Λ) > 0; b2(Ak) > 0 in Corollary +3.2, then we can guarantee the existence of an n-by-n diagonalizable nonneg- +ative irreducible matrix A with spectrum Λ and a positive column. Thus, +this is enough to show the universal realizability of Λ. +Example 3.1 Consider the list +Λ += +{23, −2, −2, −1 ± 5i, −1 ± 5i, −1 ± 5i, −2 ± 7i, −2 ± 7i}, with +Γ1 += +{23, −1 ± 5i}, Γ2 = {21, −2, −1 ± 5i, −2 ± 7i}, +Γ3 += +{13, −2, −1 ± 5i, −2 ± 7i}. +The diagonalizable companion matrices +A1 += + + +0 +0 +598 +1 +0 +20 +0 +1 +21 + + , A2 = + + +0 +0 +0 +0 +0 +57 876 +1 +0 +0 +0 +0 +35 002 +0 +1 +0 +0 +0 +6266 +0 +0 +1 +0 +0 +1695 +0 +0 +0 +1 +0 +69 +0 +0 +0 +0 +1 +13 + + +, +A3 += + + +0 +0 +0 +0 +0 +35 828 +1 +0 +0 +0 +0 +20 618 +0 +1 +0 +0 +0 +3194 +0 +0 +1 +0 +0 +903 +0 +0 +0 +1 +0 +5 +0 +0 +0 +0 +1 +5 + + +7 + +realize lists Γ1, Γ2 and Γ3, respectively. +ˇSmigoc’s glue technique applied to +matrices A1, A2 and A3 gives a 13-by-13 diagonalizable irreducible nonnega- +tive matrix with a positive column and spectrum Λ. Therefore, from Lemma +3.1 and [1], Λ is UR. +4 +The effect of adding a negative real number +to a not UR list +In this section we show how to add a negative real number −c to a list of +complex numbers +Λ = {λ, −a ± bi, . . . , −a ± bi}, λ, a, b > 0, with s1(Λ) > 0, +which is not UR or we do not know whether it is, makes +Λc = {λ, −c, −a ± bi, . . . , −a ± bi +� +�� +� +(n−2) complex numbers +} +UR. +For instance, the list Λ1 = {6, −1 ± 3i, −1 ± 3i} is realizable, but +we do not know whether it is UR, while Λ2 = {17, −3 ± 9i, −3 ± 9i} is not +realizable. However, both lists become UR if we add an appropriate negative +real number −c to each of them. +We start this section with a lemma which gives a formula to compute the +coefficient b2 in (2), Lemma 2.1, for lists Λc +Lemma 4.1 Let +Λc = {λ, −c, −a ± bi, . . . , −a ± bi +� +�� +� +(n−2) complex numbers +} +be a realizable left half-plane lists of complex numbers and let Λc = Λ1 ∪ Λ2 ∪ +· · ·∪Λt be a decomposition of Λc, −c ∈ Λt, with auxiliary lists Γk with realizing +companion matrices Ak, k = 1, 2, . . . , t, as in Corollary 3.1, associated with a +desired JCF allowed by Λc. Then the entry in position (n − 1, n) of a matrix +Ak, k = 1, 2, . . . , t, is +b2 = p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a), +(3) +8 + +where (k −1) is the number of pairs −a±bi of the last list Γt of the diagonal- +izable decomposition of Λc, plus the number of pairs −a ± bi of each previous +list Γk, k = 1, . . . , t − 1, of the decomposition, and p is the number of pairs +−a ± bi of the corresponding list Γk. Moreover, b2 increases if k increases. +Proof. It is well known that b2 = +� +1≤j1 0, be a list +of complex numbers with s1(Λ) > 0. If +(2n − 11)a2 + b2 +2a +≤ λ, +(4) +and there is a real number c > 0 such that +2a(na − λ) + b2 − 7a2 +λ − (n − 2)a +≤ c ≤ λ − (n − 2)a, +(5) +then +Λc = {λ, −c, −a ± bi, . . . , −a ± bi +� +�� +� +(n−2) complex numbers +} +becomes universally realizable. +Proof. Consider the decomposition Λc = Λ1 ∪ Λ2 ∪ · · · ∪ Λ n−2 +2 , with +Λ1 += +{λ, −a ± bi}, +Λk += +{−a ± bi}, k = 2, . . . ,n − 4 +2 +, +Λ n−2 +2 += +{−c, −a ± bi}. +We take the auxiliary sub-lists +Γ1 += +Λ1 = {λ, −a ± bi} +Γ2 += +{λ − 2a, −a ± bi} +Γ3 += +{λ − 4a, −a ± bi} +... +Γ n−4 +2 += +{λ − (n − 6)a, −a ± bi}, +Γ n−2 +2 += +{λ − (n − 4)a, −c, −a ± bi}, +10 + +where Γ n−4 +2 +and Γ n−2 +2 +are the spectrum of the diagonalizable companion ma- +trices +A n−4 +2 += + + +0 +0 +(a2 + b2)(λ − (n − 6)a) +1 +0 +2aλ − a2(2n − 11) − b2 +0 +1 +λ − (n − 4)a + + +and +A n−2 +2 += + + +0 +0 +0 +(a2 + b2)(λ − (n − 4)a)c +1 +0 +0 +(a2 + b2)(λ − (n − 4)a) + (7a2 − b2 + 2aλ − 2a2n)c +0 +1 +0 +(λ − (n − 2)a)c + (7a2 − b2 + 2aλ − 2a2n) +0 +0 +1 +λ − (n − 2)a − c + + , +respectively. Observe that sub-lists Γ n−6 +2 , . . . , Γ2, Γ1 have the same pair of +complex numbers that the list Γ n−4 +2 , but with a bigger Perron eigenvalue. +Then, if Γ n−4 +2 +is diagonalizably companion realizable, Γ n−6 +2 , . . . , Γ2, Γ1 also +are. Thus, from Lemma 2.1 we only need to consider the entries in position +(2, 3) in A n−4 +2 +and in position (3, 4) in A n−2 +2 . From (4) and (5) these entries +are nonnegative and therefore A n−4 +2 +and A n−2 +2 +are diagonalizable companion +realizing matrices. Thus, after applying n−4 +2 +times ˇSmigoc’s glue to the ma- +trices A1, . . . , A n−2 +2 , we obtain an n-by-n diagonalizable nonnegative matrix +A with spectrum Λc. Thus Λc is DR. +To obtain an n-by-n nonnegative matrix A with spectrum Λc and a non- +diagonal JCF J, we take Λc = Λ1 ∪ · · · ∪ Λt with auxiliary lists Γk being the +spectrum of a companion matrix Ak with JCF as a sub-matrix of J. Next we +need to prove that all Ak are nonnegative. To do that, we compute b2(At) +from the formula in (3), where At (with Γt containing −c) is the last diago- +nalizable matrix in the diagonalizable decomposition of Λc. From (4) and (5) +b2(At) ≥ 0. From Lemma 4.1 all b2(Ak), k = 1, . . . , t − 1, are nonnegative. +Therefore the glue of matrices Ak gives an n-by-n nonnegative matrix A with +the desired JCF J. +Example 4.2 i) Λ = {6, −1 ± 3i, −1 ± 3i} is realizable by the companion +matrix +C = + + +0 +0 +0 +0 +600 +1 +0 +0 +0 +140 +0 +1 +0 +0 +104 +0 +0 +1 +0 +0 +0 +0 +0 +1 +2 + + +, +11 + +with a non-diagonal JCF. We do not know whether Λ has a diagonalizable +realization. Then, consider the list +Λc = {6, −c, −1 ± 3i, −1 ± 3i}. +Condition (4) is satisfied and from (5) we have 1 ≤ c ≤ 2. Then for c = 2, +we have that +Γ1 = {6, −1 ± 3i}, +Γ2 = {4, −2, −1 ± 3i} +are the spectrum of diagonalizable nonnegative companion matrices +A1 = + + +0 +0 +60 +1 +0 +2 +0 +1 +4 + + , and A2 = + + +0 +0 +0 +80 +1 +0 +0 +36 +0 +1 +0 +2 +0 +0 +1 +0 + + , +respectively. Then, from ˇSmigoc’s glue we obtain a diagonalizable nonnegative +matrix with spectrum Λc. It is clear that, from the characteristic polynomial +associated to Λc, Λc has also a companion realization A3, +A3 = + + +0 +0 +0 +0 +0 +1200 +1 +0 +0 +0 +0 +880 +0 +1 +0 +0 +0 +348 +0 +0 +1 +0 +0 +104 +0 +0 +0 +1 +0 +4 +0 +0 +0 +0 +1 +0 + + +, +with a JCF with blocks of maximum size. Note that the formula in (3) gives +(k = 2, p = 1, t = 2) b2(A2) = 2, while (k = 3, p = 2) gives b2(A3) = 4. +Therefore Λc is UR. Observe that if 1 ≤ c ≤ 2, then +Λc = {6, −c, −1 ± 3i, −1 ± 3i} +is also UR. +ii) Consider the list Λ = {17, −3±9i, −3±9i}. Since s1(Λ) = 5 and s2(Λ) = +1, Λ is not realizable. From condition (5), 24 +5 ≤ c ≤ 5. Then for c = 5, +Λc = {17, −5, −3 ± 9i, −3 ± 9i} +12 + +is UR. In fact, +Γ1 = {17, −3 ± 9i} and Γ2 = {11, −5, −3 ± 9i} +are the spectrum of diagonalizable nonnegative companion matrices, which +from ˇSmigoc’s glue give rise to a diagonalizable nonnegative matrix with +spectrum Λc. From the characteristic polynomial associated to Λc we obtain +a nonnegative companion matrix with spectrum Λc and non-diagonal JCF. +Therefore, Λc is UR. +Observe that in Theorem 4.1, in spite that s1(Λ) > 0, if s1(Λ) is small +enough, there are lists Λc, which are not UR or we cannot to prove they are +from our procedure. However, from Theorem 4.1 we may compute a Perron +eigenvalue λ, which guarantees that for a family of lists Λc, with c > 0 and +n ≥ 6, Λc will be UR. Then, the following result characterizes a family of +left half-plane lists, which are UR. +Corollary 4.1 The left half-plane lists of the family +Λc = { 1 +2a((2n − 7)a2 + b2), −c, −a ± bi, . . . , −a ± bi +� +�� +� +(n−2) complex numbers +}, +with 0 < +√ +3a < b, 0 < c ≤ b2−3a2 +2a +, are universally realizable.. +Proof. It is clear that for λ = +1 +2a ((2n − 7)a2 + b2) , conditions (4) and (5) +in Theorem 4.1 are satisfied. Moreover, from 0 < +√ +3a < b, λ − (n − 2)a = +b2−3a2 +2a +> 0. +Then, from Corollary 4.1 some left half-plane lists that are UR are: +i) Λc += +{2n − 3 +2 +a, −c, −a ± 2ai, . . . , −a ± 2ai +� +�� +� +(n−2) complex numbers +}, with 0 < c ≤ a +2 +ii) Λc += +{(n + 1)a, −c, −a ± 3ai, . . . , −a ± 3ai +� +�� +� +(n−2) complex numbers +}, with 0 < c ≤ 3a +. +iii) Λc += +{2n + 9 +2 +a, −c, −a ± 4ai, . . . , −a ± 4ai +� +�� +� +(n−2) complex numbers +}, with 0 < c ≤ 13 +2 a +iv) Λc += +{8n − 3 +8 +a, −c, −a ± 5 +2ai, . . . , −a ± 5 +2ai +� +�� +� +(n−2) complex numbers +}, with 0 < c ≤ 13 +8 a, +13 + +and so on. +Observe that in Corollary 4.1, if c is strictly less than its upper bound, +then Λc, as we have seen, can be realized by a diagonalizable matrix with its +last column being positive. Then, from the extension in [1], Λc is UR. +5 +The merge of spectra +Let Γ1 = {λ1, λ2, . . . , λn} and Γ2 = {µ1, µ2, . . . , µm} be lists of complex +numbers. In [5] the authors define the concept of the merge of the spectra Γ1 +with Γ2 as +Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm}, +and prove that if Γ1 and Γ2 are diagonalizably ODP realizable, then the +merge Γ1 with Γ2, is also diagonalizably ODP realizable, and therefore from +the extension in [4], Γ is UR. Here we set a similar result as follows: +Theorem 5.1 Let Γ1 = {λ1, λ2, . . . , λn}, λ1 > |λi| , i = 2, . . . , n, be the +spectrum of a diagonalizable nonnegative n-by-n matrix A ∈ CSλ1 with its last +column being positive. Let Γ2 = {µ1, µ2, . . . , µm}, µ1 > |µi| , i = 2, . . . , m, be +the spectrum of a diagonalizable nonnegative m-by-m matrix B ∈ CSµ1 with +its last column being positive. Then +Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm} +is universally realizable.. +Proof. Let A ∈ CSλ1 be a diagonalizable nonnegative matrix with spec- +trum Γ1 and with its last column being positive. Then A is similar to a +diagonalizable positive matrix A′. If α1, . . . , αn are the diagonal entires of A′, +then +A1 = A′ + e[0, 0, . . . , µ1] = +� A′ +11 +a +bT +αn + µ1 +� +∈ CSλ1+µ1 +is diagonalizable positive with spectrum {λ1 + µ1, λ2, . . . , λn} and diagonal +entries α1, α2, . . . , αn + µ1. Let B ∈ CSµ1 be a diagonalizable nonnegative +matrix with spectrum Γ2 and with its last column being positive. Then B is +similar to a diagonalizable positive matrix B′ and +B1 = B′ + e[αn, 0, . . . , 0] +14 + +is diagonalizable positive with spectrum {µ1 + αn, µ2, . . . , µm}. Now, by ap- +plying the ˇSmigoc’s glue to matrices A1 and B1, we obtain a diagonalizable +positive matrix C with spectrum Γ. Hence, Γ is UR +Theorem 5.1 is useful to decide, in many cases, about the universal real- +izability of left half-plane list of complex numbers, as for instance: +Example 5.1 Is the list +Γ = {30, −1, −5, −1 ± 3i, −1 ± 3i, −1 ± 3i, −3 ± 9i, −3 ± 9i} UR? +Observe that from the results in Section 4, +Γ1 += +{21, −5, −3 ± 9i, −3 ± 9i}. +Γ2 += +{9, −1, −1 ± 3i, −1 ± 3i, −1 ± 3i} +are the spectrum of a diagonalizably nonnegative matrix with constant row +sums and a positive column (the last one). Then, they are similar to diag- +onalizable positive matrices and from Theorem 5.1, the merge Γ is also the +spectrum of a diagonalizable positive matrix. Therefore, Γ is UR. +References +[1] M. Collao, M. Salas, R. L. Soto, Spectra universally realizable by doubly +stochastic matrices, Special Matrices 6 (2018) 301-309. +[2] R. C. Diaz, R. L. Soto, Nonnegative inverse elementary divisors problem +in the left half-plane, Linear Multilinear Algebra 64 (2016) 258-268. +[3] C. R. Johnson, Row stochastic matrices similar to doubly stochastic +matrices, Linear Multilinear Algebra 10 (1981) 113-130. +[4] C. R. Johnson, A. I. Julio, R. L. Soto, Nonnegative realizability with +Jordan structure, Linear Algebra Appl. 587 (2020) 302-313. +[5] C. R. Johnson, A. I. Julio, R. L. Soto, Indices of diagonalizable and +universal realizability of spectra, submitted. +[6] T. J. Laffey, H. ˇSmigoc, Nonnegative realization of spectra having neg- +ative real parts, Linear Algebra Appl. 416 (2006) 148-159. +15 + +[7] H. Minc, Inverse elementary divisor problem for nonnegative matrices, +Proc. of the Amer. Math. Society 83 (1981) 665-669. +[8] H. Minc, Inverse elementary divisor problem for doubly stochastic ma- +trices, Linear Multilinear Algebra 11 (1982) 121-131. +[9] H. ˇSmigoc, The inverse eigenvalue problem for nonnegative matrices, +Linear Algebra Appl. 393 (2004) 365-374. +[10] R. L. Soto, R. C. D´ıaz, H. Nina, M. Salas, Nonnegative matrices with +prescribed spectrum and elementary divisors, Linear Algebra Appl. 439 +(2013) 3591-3604. +16 + diff --git a/5NFIT4oBgHgl3EQf7itm/content/tmp_files/load_file.txt b/5NFIT4oBgHgl3EQf7itm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..38949d1a5ec997b424365d890b43aae7d7cc4c08 --- /dev/null +++ b/5NFIT4oBgHgl3EQf7itm/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf,len=531 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='11398v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='SP] 26 Jan 2023 ˇSmigoc’s glue for universal realizability in the left half-plane∗ Jaime H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Alfaro, Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Soto† Dpto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Matem´aticas, Universidad Cat´olica del Norte, Casilla 1280 Antofagasta, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Abstract A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} of complex numbers is said to be real- izable if it is the spectrum of a nonnegative matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In this paper, using companion matrices and applying a procedure by ˇSmigoc, is provides a sufficient condi- tion for the universal realizability of left half-plane spectra, that is, Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} with λ1 > 0, Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' It is also shown how the effect of adding a negative real number to a not UR left half- plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' AMS classification: 15A18, 15A20, 15A29 Key words: Nonnegative matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' companion matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Universal realizabil- ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' ˇSmigoc’s glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 1 Introduction A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} of complex numbers is said to be realizable if it is the spectrum of an n-by-n nonnegative matrix A, and A is said to be a realizing matrix for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The problem of the realizability of spectra is called the ∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, N´ucleo 6 UCN VRIDT 083-2020, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' †E-mail addresses: rsoto@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='cl (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Soto), jaime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='alfaro@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='cl (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Alfaro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 1 nonnegative inverse eigenvalue problem (NIEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' From the Perron-Frobenius Theorem we know that if Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} is the spectrum of an n- by-n nonnegative matrix A, then the leading eigenvalue of A equals to the spectral radius of A, ρ(A) =: max 1≤i≤n |λi| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' This eigenvalue is called the Perron eigenvalue, and we shall assume in this paper, that ρ(A) = λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' A matrix is said to have constant row sums, if each one of its rows sums up to the same constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The set of all matrices with constant row sums equal to α, is denoted by CSα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then, any matrix A ∈ CSα has the eigenvector eT = [1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , 1], corresponding to the eigenvalue α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The real matrices with constant row sums are important because it is known that the problem of finding a nonnegative matrix with spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn}, is equivalent to the problem of finding a nonnegative matrix in CSλ1 with spectrum Λ (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' We denote by ek, the n-dimensional vector, with 1 in the kth position and zeros elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' If Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn}, then sk(Λ) = n � i=1 λk i , k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' A list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} of complex numbers, is said to be diagonaliz- ably realizable (DR), if there is a diagonalizable realizing matrix for Λ The list Λ is said to be universally realizable (UR), if it is realizable for each possi- ble Jordan canonical form (JCF) allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The problem of the universal realizability of spectra, is called the universal realizability problem (URP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The URP contains the NIEP, and both problems are equivalent if the given numbers λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In terms of n, both problems remain unsolved for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' It is clear that if Λ is UR, then Λ must be DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The first known results on the URP are due to Minc [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In terms of the URP, Minc [7] showed that if a list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} of complex numbers is the spectrum of a diagonalizable positive matrix, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The positivity condition is necessary for Minc’s proof, and the question set by Minc himself, whether the result holds for nonnegative realizations was open for almost 40 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Recently, two extensions of Minc’s result have been obtained in [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In [1], Collao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' showed that a nonnegative matrix A ∈ CSλ1, with a pos- itive column, is similar to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Note that if A is nonnegative with a positive row and AT has a positive eigenvector, then AT is also similar to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Besides, if Λ is diagonalizably realizable by a matrix A ∈ CSλ1 having a positive column, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In [4], Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' in- troduced the concept of ODP matrices, that is, nonnegative matrices with all positive off-diagonal entries (zero diagonal entries are permitted) and proved 2 that if Λ is diagonalizably ODP realizable, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Note that both extensions contain, as a particular case, Minc’s result in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Both extensions allow us to significantly increase the set of spectra that can be proved to be UR, as for instance, certain spectra Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} with s1(Λ) = 0, which is not possible from Minc’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In particular, we shall use the extension in [1] to generate some of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 In [1], Section 2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1, there is an error in assuming that if A is nonnegative with a positive row, then AT, which has a positive column, is similar to a positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The reason is that we cannot guarantee that AT has a positive eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Regarding non-positive universal realizations, we mention that in [10, 2] the authors proved, respectively, that lists of complex numbers Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn}, of Suleimanova type, that is, λ1 > 0, Re λi ≤ 0, |Re λi| ≥ |Im λi| , i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n, or of ˇSmigoc type, that is, λ1 > 0, Re λi ≤ 0, √ 3 |Re λi| ≥ |Im λi| , i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n, (1) are UR if and only if they are realizable if and only if n � i=1 λi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Outline of the paper: The paper is organized as follows: In Section 2, we present the mathematical tools that will be used to generate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In Section 3, we study the URP for a left half-plane list and we give a sufficient condition for it to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In Section 4, we discuss the effect of adding a negative real number −c to a left half-plane list Λ = {λ1, −a±bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , −a±bi}, which is not UR (or even not realizable), or we do not know whether it is, and we show how Λ ∪ {−c} becomes UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' We also characterize a family of left half-plane lists that are UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In Section 5, we show that the merge of two lists diagonalizably realizable Γ1 ∈ CSλ1 and Γ2 ∈ CSµ1 is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Examples are shown to illustrate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 2 Preliminaries Throughout this paper we use the following results: The first one, by ˇSmigoc [9], gives a procedure that we call ˇSmigoc’s glue technique, to obtain from two 3 matrices A and B of size n-by-n and m-by-m, respectively, a new (n+m−1)- by-(n + m − 1) matrix C, preserving in certain way, the corresponding JCFs of A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The second one, by Laffey and ˇSmigoc [6] solves the NIEP for lists of complex numbers on the left half-plane, that is, lists with λ1 > 0, Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Moreover, we also use Lemma 5 in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 [9] Suppose B is an m-by-m matrix with a JCF that contains at least one 1-by-1 Jordan block corresponding to the eigenvalue c: J(B) = � c 0 0 I(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Let t and s, respectively, be the left and the right eigenvectors of B associated with the 1-by-1 Jordan block in the above canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Furthermore, we normalize vectors t and s so that t Ts = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Let J(A) be a JCF for the n-by-n matrix A = � A1 a bT c � , where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C n-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then the matrix C = � A1 at T sb T B � has JCF J(C) = � J(A) 0 0 I(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='2 [6] Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} be a list of complex numbers with λ1 ≥ |λi| and Re λi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then Λ is realizable if and only if s1 = s1(Λ) ≥ 0, s2 = s2(Λ) ≥ 0, s2 1 ≤ ns2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 [6] Let t be a nonnegative real number and let λ2, λ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn be complex numbers with real parts less than or equal to zero, such that the list {λ2, λ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} is closed under complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Set ρ = 2t−λ2−· · ·−λn and f(x) = (x − ρ) n � j=2 (x − λj) = xn − 2txn−1 + b2xn−2 + · · · + bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' (2) Then b2 ≤ 0 implies bj ≤ 0 for j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 4 3 Companion matrices and the ˇSmigoc’s glue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' We say that a list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} of complex numbers is on the left half- plane if λ1 > 0, Re λi ≤ 0, i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In this section we give a sufficient condition for a left half-plane list of complex numbers to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Of course, it is our interest to consider lists of complex numbers containing elements out of realizability region of lists of ˇSmigoc type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Our strategy consists in to decompose the given list Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} into sub-lists Λk = {λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λkpk}, λ11 = λ1, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, with auxiliary lists Γ1 = Λ1 Γk = {s1(Γk−1), λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λkpk}, k = 2, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, each one of them being the spectrum of a nonnegative companion matrix Ak, in such a way that it be possible to apply ˇSmigoc’s glue technique to the matrices Ak, to obtain an n-by-n nonnegative matrix with spectrum Λ for each possible JCF allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In the case s1(Λ) > 0, with λi ̸= 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n, we may choose, if they exist, sub-lists Γk being the spectrum of a diagonalizable nonnegative companion matrix with a positive column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then, after ˇSmigoc’s glue, we obtain a diagonalizable nonnegative n-by-n matrix A with spectrum Λ and a positive column, which is similar to a diagonalizable positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Thus, from the extension in [1], Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Next we have the following corollary from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} be a realizable left half-plane list of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Suppose that for each JCF J allowed by Λ, there exists a decomposition of Λ as Λ = Λ1 ∪ Λ2 ∪ · · · ∪ Λt, where Λk = {λk1, λk2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λkpk}, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, λ11 = λ1, with auxiliary lists Γ1 = Λ1, Γk = {s1(Γk−1), λk1, λk2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λkpk}, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, being the spectrum of a nonnegative companion matrix Ak with JCF J(Ak) as a sub-matrix of J, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then Λ is universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Since each matrix Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, is nonnegative companion with JCF J(Ak) being a submatrix of J, then, from ˇSmigoc’s glue applied to matrices Ak, we obtain an n-by-n nonnegative matrix with spectrum Λ and JCF J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' As J is any JCF allowed by Λ, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The following result is well known and useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 Let A be a diagonalizable irreducible nonnegative matrix with spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn} and a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then A is similar to a diagonalizable nonnegative matrix B ∈ CSλ1, with a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' If A is irreducible nonnegative, it has a positive eigenvector xT = [x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then if D = dig{x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , xn}, the matrix B = D−1AD = �xj xi ai,j � ∈ CSλ1 is nonnegative with a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Suppose all lists Γk in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1, can be taken as the spectrum of a diagonalizable nonnegative companion matrix Ak with a positive column (the last one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then, since the glue of matrices Ak gives an n-by-n diagonalizable irreducible nonnegative matrix A with a positive column and spectrum Λ, A is similar to a diagonalizable positive matrix with spectrum Λ and therefore Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' This is what the next result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='2 Let Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , λn}, λi ̸= 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n, s1(Λ) > 0, be a realizable left half-plane list of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' If there is a decomposition of Λ as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1, with all lists Γk being the spectrum of a diagonal- izable nonnegative companion matrix Ak, with a positive column, then Λ is universally realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' It is enough to prove the result for two lists Γk of the decomposition of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Let Γk−1 and Γk, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, be the spectrum, respectively, of matrices Ak−1 and Ak, which are diagonalizable nonnegative companion with a posi- tive column (the last one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then Ak−1 and Ak are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' In particular, Ak has a positive eigenvector s and, since AT k is also irreducible, Ak has also a positive left eigenvector tT with tTs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Now, let Ak−1 = � A1,k−1 a bT s1(Γk−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 6 Since the last column of Ak−1 is positive, the vector a is also positive and atT is a positive submatrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Therefore, the glue of Ak−1 with Ak, Ck = � A1,k−1 atT sbT Ak � , is a diagonalizable nonnegative matrix with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Note that Ck is also irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then Ck has, besides, a positive eigenvector, and from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 Ck is similar to a matrix with constant row sums and with its last column being positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Thus, Ck is similar to a diagonalizable positive matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then, ˇSmigoc’s glue applied to all matrices Ak gives an n- by-n diagonalizable irreducible nonnegative matrix A with a positive column and spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Therefore, A is similar to a diagonalizable positive matrix with spectrum Λ and from the extension in [1] Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Observe that if λi ̸= 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' s1(Λ) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' b2(Ak) > 0 in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='2, then we can guarantee the existence of an n-by-n diagonalizable nonneg- ative irreducible matrix A with spectrum Λ and a positive column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Thus, this is enough to show the universal realizability of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 Consider the list Λ = {23, −2, −2, −1 ± 5i, −1 ± 5i, −1 ± 5i, −2 ± 7i, −2 ± 7i}, with Γ1 = {23, −1 ± 5i}, Γ2 = {21, −2, −1 ± 5i, −2 ± 7i}, Γ3 = {13, −2, −1 ± 5i, −2 ± 7i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' The diagonalizable companion matrices A1 = \uf8ee \uf8f0 0 0 598 1 0 20 0 1 21 \uf8f9 \uf8fb , A2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 57 876 1 0 0 0 0 35 002 0 1 0 0 0 6266 0 0 1 0 0 1695 0 0 0 1 0 69 0 0 0 0 1 13 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , A3 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 35 828 1 0 0 0 0 20 618 0 1 0 0 0 3194 0 0 1 0 0 903 0 0 0 1 0 5 0 0 0 0 1 5 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb 7 realize lists Γ1, Γ2 and Γ3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' ˇSmigoc’s glue technique applied to matrices A1, A2 and A3 gives a 13-by-13 diagonalizable irreducible nonnega- tive matrix with a positive column and spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Therefore, from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 and [1], Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' 4 The effect of adding a negative real number to a not UR list In this section we show how to add a negative real number −c to a list of complex numbers Λ = {λ, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , −a ± bi}, λ, a, b > 0, with s1(Λ) > 0, which is not UR or we do not know whether it is, makes Λc = {λ, −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , −a ± bi � �� � (n−2) complex numbers } UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' For instance, the list Λ1 = {6, −1 ± 3i, −1 ± 3i} is realizable, but we do not know whether it is UR, while Λ2 = {17, −3 ± 9i, −3 ± 9i} is not realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' However, both lists become UR if we add an appropriate negative real number −c to each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' We start this section with a lemma which gives a formula to compute the coefficient b2 in (2), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1, for lists Λc Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1 Let Λc = {λ, −c, −a ± bi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , −a ± bi � �� � (n−2) complex numbers } be a realizable left half-plane lists of complex numbers and let Λc = Λ1 ∪ Λ2 ∪ · ·∪Λt be a decomposition of Λc, −c ∈ Λt, with auxiliary lists Γk with realizing companion matrices Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, as in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content='1, associated with a desired JCF allowed by Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Then the entry in position (n − 1, n) of a matrix Ak, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t, is b2 = p(2aλ − 2a2n + (4k − 2p + 1)a2 − b2) + c(λ − (n − 2)a), (3) 8 where (k −1) is the number of pairs −a±bi of the last list Γt of the diagonal- izable decomposition of Λc, plus the number of pairs −a ± bi of each previous list Γk, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' , t − 1, of the decomposition, and p is the number of pairs −a ± bi of the corresponding list Γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Moreover, b2 increases if k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf'} +page_content=' It is well known that b2 = � 1≤j1> |Ws|. +Given a new test image feature vector x∗ the goal is then to +learn a function z∗ = f(x∗), using all available information, +that predicts a class label z∗. Note that the form of the problem +changes drastically depending on the label set assumed for z∗: +• Supervised learning: z∗ ∈ Ws; +• Zero-shot learning: z∗ ∈ Wt ; +• Generalized zero-shot learning: z∗ ∈ {Ws, Wt}; +• Open set recognition: z∗ ∈ W. +Note that open set recognition is similar to generalized zero- +shot learning, however, in open set setting additional distractor +classes that do not exist in either source or target datasets are +present. We posit that a single unified f(x∗) can be learned for +all cases. We formalize the definition of vocabulary-informed +learning (Voc) as follows: +Definition 3.1. Vocabulary-informed Learning (Voc): +is a +learning setting that makes use of complete vocabulary data +(W) during training. Unlike a more traditional ZSL that typi- +cally makes use of the vocabulary (e.g., semantic embedding) +at test time, Voc utilizes exactly the same data during training. +Notably, Voc requires no additional annotations or semantic +knowledge; it simply shifts the burden from testing to training, +leveraging the vocabulary to learn a better model. +The vocabulary W can be represented by semantic embed- +ding space learned by word2vec [52] or GloVec [57] on large- +scale corpus; each vocabulary entity w ∈ W is represented as +a distributed semantic vector u ∈ Rd. Semantics of embedding +space help with knowledge transfer among classes, and allow +ZSL, G-ZSL and open set image recognition. Note that such +semantic embedding spaces are equivalent to the “semantic +knowledge base” for ZSL defined in [54] and hence make it +appropriate to use Vocabulary-informed Learning in ZSL. +3.2 +Learning Embedding and Recognition +Assuming we can learn a mapping g : Rp → Rd, from image +features to this semantic space, recognition can be carried out +using simple nearest neighbor distance, e.g., f(x∗) = car if +g(x∗) is closer to ucar than to any other word vector; uj in +this context can be interpreted as the prototype of the class +j. Essentially, the attribute or semantic word vector of the +class name can be taken as the class prototype [30]. The +core question is then how to learn the mapping g(x) and +what form of inference is optimal in the semantic space. +For learning we propose the discriminative maximum margin +criterion that ensures that labeled samples xi project closer +to their corresponding class prototypes uzi than to any other +prototype ui in the open set vocabulary i ∈ W \ zi. +Learning Embedding: To learn the function f(x), one needs +to establish the correspondence between visual feature space +and semantic space. Particularly, in the training step, each +image sample xi is regressed towards its corresponding class +prototype uzi by minimizing +W = arg min +W +Ns +� +i=1 +L (xi, uzi) + λ ∥ W ∥2 +F +(1) +where L (xi, uzi) = ∥g (xi) − uzi∥2 +2 ; and g : Rp → Rd is +the mapping from image features to semantic space; ∥ · ∥F +indicates the Frobenius Norm. If g (x) = W T x is a linear +mapping, we have the closed form solution for Eq. (1). The +loss function in Eq. (1) can be interperted as a variant of SVR +embedding. However, this is too limiting. To learn the linear +embedding matrix W, we introduce and discuss two sets of +methods in Section 3.3 and Section 3.4. +Recognition: The recognition step can be formulated using +the nearest neighbor classifier. Given a testing instance x⋆, +z⋆ = arg min +i +��W T x⋆ − ui +��2 +2 . +(2) +Eq. (2) measures the distance between predicted vector and the +class prototypes in the semantic space. In terms of different +label set, we can do supervise, zero-shot, generalized zero-shot +or open set recognition without modifications. +In particular, we explore a simple variant of Eq. (2) to +classify the testing instance x⋆, +z∗ = arg min +i +∥ W T x∗ − ω (ui) ∥2 +2, +(3) +where the Nearest Neighbor (NN) classifier measures distance +between the predicted semantic vectors and a function of pro- +totypes in the semantic space, e.g., ω (ui) = ui is equivalent +to Eq (2). In practice, we employ semantic vector prototype +averaging to define ω (·). For example, sometimes, there might +be more than one positive prototype, such as pig, pigs and hog. +In such the circumstance, choosing the most likely prototype +and using NN may not be sensible, hance we introduce the +averaging strategy to consider more prototypes for robustness. +Note that this strategy is known as Rocchio algorithm in infor- +mation retrieval. Rocchio algorithm is a method for relevance +feedback that uses more relevant instances to update the query +for better recall and possibly precision in the vector space +(Chap 14 in [51]). It was first suggested for use in ZSL in [27]; +more sophisticated algorithms [25], [58] are also possible. +3.3 +Maximum Margin Voc Embedding (MM-Voc) +The maximum margin vocabulary-informed embedding learns +the mapping g(x) : Rp → Rd, from low-level features x to the +semantic word space by utilizing maximum margin strategy. +Specifically, consider g(x) = W T x, where1 W ⊆ Rp×d. +Ideally we want to estimate W such that uzi = W T xi for all +labeled instances in Ds. Note that we would obviously want +this to hold for instances belonging to unobserved classes as +well, but we cannot enforce this explicitly in the optimization +as we have no labeled samples for them. +Data Term: The easiest way to enforce the above objective +is to minimize Euclidian distance between sample projections +1. Generalizing to a kernel version is straightforward, see [76]. + +5 +and appropriate prototypes in the embedding space, +D (xi, uzi) = +��W T xi − uzi +��2 +2 . +(4) +Where we need to minimize this term with respect to each +instance (xi, uzi), where zi is the class label of xi in Ds. Such +embedding is also known, in the literature, as data embedding +[35] or compatibility function [3]. +To make the embedding more comparable to support vector +regression (SVR), we employ the maximal margin strategy – +ϵ−insensitive smooth SVR (ϵ−SSVR) [46] in Eq. (1). That is, +L (xi, uzi) = Lϵ (xi, uzi) + λ ∥ W ∥2 +F +(5) +where Lϵ (xi, uzi) = 1T | ξ |2 +ϵ; λ is regularization coefficient. +(|ξ|ϵ)j = max +� +0, +���W T +⋆jxi − (uzi)j +��� − wzi · ϵ +� +(6) +|ξ|ϵ ∈ Rd; ()j indicates the j-th value of corresponding vector; +W⋆j is the j-th column of W, and wzi is the scaling weight +derived from the density of class zi and it’s neighboring +classes. In our conference version [29], equal weight wzi is +used for all classes. Here we notice that it is beneficial to +use the density/coverage of each labeled training class as the +constraint in learning the projection from visual feature space +to semantic space. We introduce a specific weighting strategy +to compute wzi in Section 3.4. +The conventional ϵ−SVR is formulated as a constrained +minimization problem, i.e., convex quadratic programming +problem, while ϵ−SSVR employs quadratic smoothing [89] to +make Eq. (5) differentiable everywhere, and thus ϵ−SSVR can +be solved as an unconstrained minimization problem directly2. +Triplet Term: Data term above only ensures that labelled +samples project close to their correct prototypes. However, +since it is doing so for many samples and over a number +of classes, it is unlikely that all the data constraints can be +satisfied exactly. Specifically, consider the following case, if +uzi is in the part of the semantic space where no other entities +live (i.e., distance from uzi to any other prototype in the em- +bedding space is large), then projecting xi further away from +uzi is asymptomatic, i.e., will not result in misclassification. +However, if the uzi is close to other prototypes then minor +error in regression may result in misclassification. +To embed this intuition into our learning, we enforce more +discriminative constraints in the learned semantic embedding +space. Specifically, the distance of D (xi, uzi) should not only +be as small as possible, but should also be smaller than the +distance D (xi, ua), ∀a ̸= zi. Formally, we define the triplet +term +MV (xi, uzi) = 1 +2 +AV +� +a=1 +� +C + 1 +2D (xi, uzi) − 1 +2D (xi, ua) +�2 ++ +, +(7) +where a ∈ Wt (or more precisely a ∈ W \ Ws) is selected +from the open vocabulary; C is the margin gap constant. Here, +[·]2 ++ indicates the quadratically smooth hinge loss [89] which +2. In practice, our tentative experiments shows that the Eq. (4) and Eq. (5) +will lead to the similar results, on average; but formulation in Eq. (5) is more +stable and has lower variance. +is convex and has the gradient at every point. To speedup +computation, we use the closest AV target prototypes to each +source/auxiliary prototype uzi in the semantic space. We also +define similar constraints for the source prototype pairs: +MS (xi, uzi) = 1 +2 +BS +� +b=1 +� +C + 1 +2D (xi, uzi) − 1 +2D (xi, ub) +�2 ++ +(8) +where b ∈ Ws is selected from source/auxiliary dataset +vocabulary. This term enforces that D (xi, uzi) should be +smaller than the distance D (xi, ub), ∀b ̸= zi. To facilitate the +computation, we similarly use closest BS prototypes that are +closest to each prototype uzi in the source classes. Note that, +the Crammer and Singer loss [13], [75] is the upper bound of +Eq. (7) and (8) which we use to tolerate slight variants of uzi +(e.g., the prototypes of ’pigs’ Vs. ’pig’). +To sum up, the complete triplet maximum margin term is: +M (xi, uzi) = MV (xi, uzi) + MS (xi, uzi) . +(9) +We note that the form of rank hinge loss in Eq. (7) and (8) is +similar to DeViSE [24], but DeViSE only considers loss with +respect to source/auxiliary data and prototypes. +Maximum Margin Vocabulary-informed Embedding: The +complete combined objective can now be written as: +W = argmin +W +nT +� +i=1 +(αLϵ (xi, uzi) + +(1 − α)M (xi, uzi)) + λ ∥ W ∥2 +F , (10) +where α ∈ [0, 1] is the coefficient that controls contribution +of the two terms. One practical advantage is that the objective +function in Eq. (10) is an unconstrained minimization problem +which is differentiable and can be solved with L-BFGS. W is +initialized with all zeros and converges in 10 − 20 iterations. +3.4 +Weighted +Maximum +Margin +Voc +Embedding +(WMM-Voc) +We note that there is no previous method that directly estimates +the density of source training classes in the semantic space. +However, doing so may lead to several benefits. First, the num- +ber of training instances in source classes may be unbalanced. +In such a case, an estimate of the density of samples in a +training class can be utilized as a constraint in learning the +embedding characterized by Eq. (6). Second, in the semantic +space, the instances from the classes whose data samples +span a large radius [62] may reside in the neighborhood +of many other classes or open vocabulary. This can happen +when the embedding is not well learned. We can interpret this +phenomenon as hubness [45], [67]3. Adding a penalty based +on the density of each training class may be helpful in better +learning the embedding and alleviating the hubness problem. +This subsection introduces a strategy for estimating the +density of each known class in the semantic space (i.e., +wzi in Eq. (6)). Generally, we know the prototype of each +known and novel class in the semantic space. To estimate +3. However, the causes for hubness are still under investigation [16], [67]. + +6 +Fig. 2. Illustration of margin distribution of prototypes in +the semantic space. +the density/coverage of a known class, one needs to look at +pairwise distance between a prototype and the nearest negative +instance and the furthest positive instance. This intuition leads +us to introduce the concept of margin distribution. +Margin Distribution: The concept of margin is fundamen- +tal to maximum margin classifiers (e.g., SVMs) in machine +learning. The margin enables an intuitive interpretation of such +classifiers in searching for the maximum margin separator in a +Reproducing Kernel Hilbert Space. Previous margin classifiers +[92] aim to maximize a single margin across all training +instances. In contrast, some recent studies [17], [62], [64], [90] +suggest that the knowledge of margin distribution of instances, +rather than a single margin across all instances, is crucial for +improving the generalization performance of a classifier. +The “instance margin” is defined as the distance between +one instance and the separating hyperplane. Formally, for one +instance i in the semantic space g (xi) and sufficiently many4 +samples g (xj) (zi ̸= zj) drawn from well behaved class +distributions5. We define the distance dij = ∥g (xi) − g (xj)∥. +For instance i, we can obtain a set of distances Di += +{dij, zj ̸= zi} with the minimal values ¯di⋆ = minDi. As +shown in [62], the distribution for the minimal values of the +margin distance is characterized by a Weibull distribution. +Based on this finding, we can express the probability of g (x) +being included in the boundary estimated by g (xi): +ψ (g (x) ; g (xi)) = exp +� +− +�∥g (x) − g (xi)∥ +λi +�κi� +, +(11) +where κi and λi are Weibull shape and scale parameters +obtained by fitting Di using Maximum Likelihood Estimate +(MLE), which is summarized6 in Alg. 1. Equation (11) quan- +titatively describes the margin of one specific class, probabilis- +4. In our experiments, we use all available training instances here. +5. The well behaved indicates that the moments of the distribution should +be well-defined. For example, Cauchy distribution is not well-behaved [39]. +6. codes released in https://github.com/xiaomeiyy/WMM-Voc. +tically, in our semantic embedding space. Note that Eq. (11) +requires ψ (·) to be non-degenerate margin distribution, which +is essentially guaranteed by Extreme Value Theorem [40]. +Algorithm 1 EVT estimator by Weibull distribution. +Input: Extreme values x1, · · · , xn +Output: Estimated parameters ˆκ, ˆλ +If n == 1: +ˆκ = ∞, ˆλ = x1. +Else: +1. Sort x1, · · · , xn to get x[1] ≥ · · · ≥ x[n] +(where x[i] is the re-ordered value). +2. Maximum likelihood estimator for κ: +n � � +xκ +[i] log x[i] − xκ +[n] log x[n] +� +� � +xκ +[i] − xκ +[n] +� += +� +log x[i] +(12) +3. Solve Eq. (12), and numerically estimate ˆκ. +(e.g., using fzero function in MATLAB) +4. Compute ˆλ = +�� � +xˆκ +[i] − xˆκ +[n] +� +/n +�1/ˆκ +. +Margin Distribution of Prototypes: Consider a class zi +which in the embedding space is represented by a prototype +uzi. In accordance with above formalism, we can also assume +sufficiently many samples g (xj) drawn from other (zi ̸= zj) +well behaved class distributions. We can also consider the +prototypes of vast open vocabulary uzj (zi ̸= zj, zj ∈ Wt). +Under these assumptions, we can obtain a set of distances +Duzi = { +��uzi − gzj +�� , zj ̸= zi, gzj ∈ +� +g (xj) , uzj +� +} for the +prototype uzi. As a result, the distribution for the minimal +values of the margin distance for uzi is given by a Weibull +distribution. The probability that gzi is included in the bound- +ary estimated by uzi is given by +ψ (gzi; uzi) = exp +� +− +� +∥gzi − uzi∥ +λuzi +�κuzi � +. +(13) +The above equation models the distribution of minimum value; +thus it can be used to estimate the boundary density (or more +specifically, the boundary distribution) of class zi. +We set significant level to 0.05 to approximately esti- +mate the minimal value ¯duzi⋆. As illustrated in Figure 2, if +ψ (gzi; uzi) < 0.05, we will assume gzi does not belong to +the prototype uzi; otherwise, gzi is included in the boundary +estimated by uzi. In term of the significant level of 0.05, +we can further denote the minimal values as ¯d(0.05) +uzi⋆ , i.e., +exp +� +− +� ¯d(0.05) +uzi ⋆ +λuzi +�κuzi � += 0.05. Thus we have +¯d(0.05) +uzi⋆ = λuzi · log1/κuzi +� 1 +0.05 +� +(14) +Coverage Distribution of Prototypes: Now, for class zi +consider the nearest instance from another class g (xj) where +zi ̸= zj; with sufficient many instances g (xk) from class zi, +we have pairwise unique ("within class") distance: +cuzik = ∥g (xk) − uzi∥. +(15) + +Positiveinstances +Negativeinstances +★ Prototypes +id7 +We consider outliers those instances g (xk) that have larger +distance to uzi than the nearest instance g (xj) (zi ̸= zj) +of another class. To remove the outliers we hence consider +Cuzi = +� +cuzik|cuzik ≤ minzj̸=zk ∥g (xj) − uzi∥ +� +. As illus- +trated in Figure 2, we only consider positive instances within +the orange circle and all other instances with larger distance +are removed. Then the distribution of the largest distance +¯cuzi⋆ = max Cuzi will follow a reversed Weibull distribution. +This allows us to get the probability distribution to describe +positive instances, +φ (g (xk) ; uzi) = 1 − exp +� +� +�− +� +∥g (xk) − uzi∥ +λ′uzi +�κ +′ +uzi +� +� +� +(16) +where κ +′ +i and λ +′ +i are reverse Weibull shape and scale pa- +rameters individually obtained from fitting the largest Cuzi , +¯cuzi⋆ is the distance between instance and prototype, φ is the +probability that the instance is in the class. +Similar to the margin distribution, we can estimate the +coverage by setting the significant level to 0.05. As shown in +Figure 2, we establish two boundaries to estimate the scale of +each class probabilistically. If φ (g (xk) ; uzi) ⩾ 0.05, g (xk) +is included in the coverage distribution uzi. The maximum +values ¯c(0.05) +uzi⋆ +can be computed as φ (g (xk) ; uzi) = 0.05. It +results in, +¯c(0.05) +uzi⋆ = λ′ +uzi · log +1/κ +′ +uzi +� +1 +1 − 0.05 +� +. +(17) +By combining the terms computed in Eq. (14) and (17), we +can obtain the weight wzi for class zi in Eq. (6), +wzi ∝ +� +¯d(0.05) +uzi⋆ + ¯c(0.05) +uzi⋆ +� +(18) +As explained in Algorithm 1, we set ˆκ = ∞, ˆλ = x1 in +one-shot setting. In few-shot learning setting, we can estimate +ˆκ and ˆλ directly. In addition, such an initialization of weights +(ˆκ and ˆλ) intrinsically helps learn the embedding weight W. +The learning process of parameters: The process could +be interpreted as a form of block coordinate descent where +we estimate the embedding/mapping; then density within that +embedding and so on. In practice, the weights wzi are initially +randomized. But they do not play an important role at the +beginning of the optimization, since +���W T +⋆jxi − (uzi)j +��� is very +large in the first few iterations. In other words, the optimization +is initially dominated by the data term and maximum margin +terms play little role. However, once we can get a relative +good mapping (i.e., smaller +���W T +⋆jxi − (uzi)j +���) after several +training iterations, the weight wzi starts becoming significant. +By virtue of such an optimization, the weighted version can +achieve better performance than the previous non-weighted +version in our conference paper [29]. +Deep Weighted Maximum Margin Voc Embedding (Deep +WMM-Voc). In practice, we extend WMM-Voc to include +a deep network for feature extraction. Rather than extracting +low-level features using an off-the-shelf pre-trained model in +Eq. (10), we use an integrated deep network to extract xi +from the raw images. As a result, the loss function in Eq. (10) +is also used to optimize the parameters of the deep network. +In particular, we fix the convolutional layers of corresponding +network and fine-tune the last fully connected layer in our task. +The network was trained using stochastic gradient descendent. +4 +EXPERIMENTS +4.1 +Experimental setup +We conduct our experiments on Animals with Attributes +(AwA) dataset, and ImageNet 2012/2010 dataset. +AwA dataset: AwA consists of 50 classes of animals (30, 475 +images in total). In [43] standard split into 40 source/auxiliary +classes (|Ws| = 40) and 10 target/test classes (|Wt| = 10) +is introduced. We follow this split for supervised and zero- +shot learning. We use ResNet101 features (downloaded from +[54]) on AwA to make the results more easily comparable to +state-of-the-art. +ImageNet 2012/2010 dataset: ImageNet is a large-scale +dataset. We use 1000 (|Ws| = 1000) classes of ILSVRC 2012 +as the source/auxiliary classes and 360 (|Wt| = 360) classes +of ILSVRC 2010 that are not used in ILSVRC 2012 as target +data. We use pre-trained VGG-19 model [12] to extract deep +features for ImageNet. +Recognition tasks: We consider several different settings in +a variety of experiments. We first divide the two datasets into +source and target splits. On source classes, we can validate +whether our framework can be used to solve one-shot and +supervised recognition. By using both the source and target +classes, transfer learning based settings can be evaluated. +1) SUPERVISED recognition: learning is on source classes; +test instances come from the same classes with Ws as +recognition vocabulary. In particular, under this setting, +we also validate the one- and few-shot recognition sce- +narios, i.e., classes have one or few training examples. +2) ZERO-SHOT recognition: In ZSL, learning is on the +source classes with Ws vocabulary; test instances come +from target dataset with Wt as recognition vocabulary. +3) GENERAL-ZERO-SHOT +recognition: +G-ZSL +uses +source classes to learn, with test instances coming from +either target Wt or original Ws recognition vocabulary. +4) OPEN-SET recognition: Again source classes are used +for learning, but the entire open vocabulary with |W| ≈ +310K atoms is used at test time. In practice, test images +come from both source and target splits (similar to +G-ZSL); however, unlike G-ZSL there are additional +distractor classes. In other words, chance performance +for open-set recognition is much lower than for G-ZSL. +We test both our Voc variants – MM-Voc and WMM-Voc. +Additionally, we also validate the Deep WMM-Voc by fine- +tuning the WMM-Voc on VGG-19 architecture and optimizing +the weights with respect to the loss in Eq. (10). +Competitors: We compare to a variety of the models in the +literature, including: +1) SVM: SVM classifier trained directly on the training +instances of source data, without the use of semantic +embedding. This is the standard (SUPERVISED/ONE- +SHOT) learning setting and the learned classifier can only + +8 +predict the labels within the source classes. Hence, SVM +is inapplicable in ZSL, G-ZSL, and open-set recognition +settings. +2) SVR-Map: SVR is used to learn W and the recognition +is done, similar to our method, in the resulting semantic +manifold. This corresponds to only optimizing Eq. (5). +3) Deep-SVR: This is a variant SVR, which further allows +fine-tuning of the underlying neural network generating +the features. In this case, W is expressed as the last +linear layer and the entire network is fine-tuned with +respect to the loss encoding only the data term (Eq. (5)). +4) SAE: SAE is a semantic encoder-decoder paradigm that +projects visual features into a semantic space and then +reconstructs the original visual feature representation +[38]. The SAE has two variants in learning the embed- +ding space, i.e., semantic space to feature space (S→F), +and feature space to semantic space (F→S). By default, +the best result of these two variants are reported. +5) ESZSL: ESZSL first learns the mapping between visual +features and attributes, then models the relationship +between attributes and classes [61]. +6) DeVise, ConSE, AMP: To compare with state-of-the-art +large-scale zero-shot learning approaches we implement +DeViSE [24] and ConSE [53]7. ConSE uses a multi- +class logistic regression classifier for predicting class +probabilities of source instances; and the parameter +T (number of top-T nearest embeddings for a given +instance) was selected from {1, 10, 100, 1000} that gives +the best results. ConSE method in supervised setting +works the same as SVR. We use the AMP code provided +on the author webpage [31]. +Metrics: Classification accuracies are reported as the eval- +uation metrics on most of tasks. In our conference version +[29], we further introduce an evaluation setting for OPEN- +SET tasks where we do not assume that test data comes from +either source/auxiliary domain or target domain. Thus we split +the two cases (i.e., SUPERVISED-like, and ZERO-SHOT-like +settings), to mimic SUPERVISED and ZERO-SHOT scenarios +for easier analysis. Particularly, in G-ZSL task, this newly +introduced evaluation setting is corresponding to the evaluation +metrics defined in [85]: (1) S → T: Test instances from seen +classes, the prediction candidates include both seen and unseen +classes; (2) U → T: Test instances from unseen classes, the +prediction candidates include both seen and unseen classes. +(3) The harmonic mean is used as the main evaluation metric +to further combine the results of both S → T and U → T: +H = 2·(Acc(U → T) × Acc(S → T)) +(Acc(U → T) + Acc(S → T)). +(19) +Setting of Parameters: For the recognition tasks, we learn +classifiers by using various number of training instances. +We compare relevant baselines with results of our method +variants: MM-Voc, WMM-Voc, Deep WMM-Voc. Each setting +is repeated/tested 10 times. The averaged results are reported +to reduce the variance. For each setting, our Voc methods are +trained by a single model to be capable of solving the tasks +7. Codes for [24] and [53] are not publicly available. +of supervised, zero-shot, G-ZSL and open-set recognition. +Specifically, +1) In Deep WMM-Voc, we fix λ to 0.01 and α = 0.6 with +the learning rate initially set to 1e−5 and is reduced by +1 +2 every 10 epochs. AV and BS are set to 5 in order to +balance performance and computational cost of pairwise +constraints. +2) To solve Eq. (10) at a scale, one can use Stochastic +Gradient Descent (SGD) which makes great progress ini- +tially, but often is slow when approaching convergence. +In contrast, the L-BFGS method mentioned above can +achieve steady convergence at the cost of computing the +full objective and gradient at each iteration. L-BFGS +can usually achieve better results than SGD with good +initialization, however, is computationally expensive. To +leverage benefits of both of these methods, we utilize a +hybrid method to solve Eq. (10) in large-scale datasets: +the solver is initialized with few instances to approx- +imate the gradients using SGD first; then gradually +more instances are used and switch to L-BFGS is made +with iterations. This solver is motivated by Friedlander +et al. [23], who theoretically analyzed and proved the +convergence for the hybrid optimization methods. In +practice, we use L-BFGS and the Hybrid algorithms for +AwA and ImageNet respectively. The hybrid algorithm +can save between 20 ∼ 50% training time as compared +with L-BFGS. +Open set vocabulary. We use Google word2vec to learn +the open set vocabulary set from a large text corpus of +around 7 billion words: UMBC WebBase (3 billion words), +the latest Wikipedia articles (3 billion words) and other web +documents (1 billion words). Some rare (low frequency) +words and high frequency stopping words were pruned in the +vocabulary set: we remove words with the frequency < 300 +or > 10 million times. The result is a vocabulary of around +310K words/phrases with openness ≈ 1, which is defined as +openness = 1 − +� +(2 × |Ws|) / (|W|) [66]. +4.2 +Experimental results on AwA dataset +4.2.1 +Learning Classifiers from Few Source Training +Instances +We are particularly interested in learning of classifiers from +few source training instances. This is inclined to mimic human +performance of learning from few examples and illustrate +ability of our model to learn with little data8. We show +that, our vocabulary-informed learning is able to improve the +recognition accuracy on all settings. +By only using 200 training instances, we report the results +on standard supervised (on source classes), zero-shot (on +target classes), and generalized zero-shot recognition (both +on source and target classes) as shown in Table 2. Note that +for ZSL and G-ZSL, our settings is a more realistic and yet +8. As for feature representations, the ResNet100 features from [54] are +trained from ImageNet 2012 dataset, which potentially have some overlapped +classes with AwA dataset. + +9 +Methods +S. Sp +Features +Acc. +WMM-Voc +W +CNNresnet101 +90.79 +WMM-Voc: closed +W +CNNresnet101 +84.51 +Deep WMM-Voc +W +CNNresnet101 +90.65 +Deep WMM-Voc: closed +W +CNNresnet101 +83.85 +SAE +W +CNNresnet101 +71.42 +ESZSL +W +CNNresnet101 +74.17 +Deep-SVR +W +CNNresnet101 +67.22 +Akata et al. [3] +A+W +CNNGoogleNet +73.90 +TMV-BLP [25] +A+W +CNNOverFeat +69.90 +AMP (SR+SE) [31] +A+W +CNNOverFeat +66.00 +PST [58] +A+W +CNNOverFeat +54.10 +Latem [83] +A+W +CNNresnet101 +74.80 +SJE [3] +A+W +CNNresnet101 +76.70 +DeViSE [24] +W +CNNresnet101 +72.90 +ConSE [53] +W +CNNresnet101 +63.60 +CMT [68] +W +CNNresnet101 +58.90 +SSE [91] +W +CNNresnet101 +54.50 +SSE [91] +W +CNNVGG19 +57.49 +TASTE [87] +W +CNNVGG19 +89.40 +KLDA+KRR [48] +W +CNNGoogleNet +79.30 +CLN+KRR [48] +W +CNNVGG19 +81.00 +UVDS [50] +W +CNNVGG19 +62.88 +DEM [10] +W +CNNInception-V2 +86.70 +DS [60] +W/A +CNNOverFeat +52.70 +SYNC [9] +W/A +CNNresnet101 +72.20 +Relation Net [69] +A +CNNInception-V2 +84.50 +ESZSL [61] +A +CNNresnet101 +74.70 +UVDS [50] +A +CNNGoogleNet +80.28 +GFZSL [78] +A +CNNVGG19 +80.50 +DEM [10] +A +CNNInception-V2 +78.80 +SE-GZSL [77] +A +CNNVGG19 +69.50 +cycle-CLSWGAN [21] +A +CNNresnet101 +66.30 +f-CLSWGAN [84] +A +CNNresnet101 +68.20 +PTMCA [47] +A +CNNresnet101 +66.20 +Jayaraman et al. [36] +A +low-level +48.70 +DAP [43] +A +CNNVGG19 +57.50 +DAP [43] +A +CNNresnet101 +57.10 +DAP [43] +A +CNNOverFeat +53.20 +ALE [2] +A +CNNresnet101 +78.60 +Yu et al. [86] +A +low-level +48.30 +IAP [43] +A +CNNOverFeat +44.50 +HEX [14] +A +CNNDECAF +44.20 +AHLE [2] +A +low-level +43.50 +TABLE 1 +Zero-shot comparison on AwA. We compare the +state-of-the-art ZSL results using different semantic +spaces (S. Sp) including word vector (W) and attribute +(A). 1000 dimension word2vec dictionary is used for our +model. (Chance-level =10%). Different types of features +are used by different methods. WMM-Voc: closed and +Deep WMM-Voc: closed are the two variants of our +model obtained by learning the vocabulary-informed +constraints only from known classes (i.e., closed set), +similar to our conference version [29]. +ZSL results by 100-d word vector +40 +200 +800 +1600 +12139 +24295 +Training Instance Number +0 +20 +40 +60 +80 +100 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSL +ZSL results by 1000-d word vector +40 +200 +800 +1600 +12139 +24295 +Training Instance Number +0 +20 +40 +60 +80 +100 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSL +Fig. 3. The ZSL results on AwA by different settings. +more challenging than those in previous methods [38], [61], +since the source classes have few training instances. We also +compare using 100/1000-dimensional word2vec representation +(i.e., d = 100/1000). Both the Top-1 and Top-5 classification +accuracy is reported. Note that the key novelty of our WMM- +Voc comes from directly estimating the density of source train- +ing classes. Such an approach would be helpful in alleviating +the hubness problem and should lead to better performance +in zero-shot learning. As shown in Table 2, the improvement +from MM-Voc to WMM-Voc and then further to Deep WMM- +Voc validate this point. +We highlight the following observations: (1) Deep WMM- +Voc achieves the best zero-shot learning accuracy compared +with the other state-of-art methods. It is 18.45% and 21.02% +higher than SAE and ESZSL respectively on Top-1 accuracy. +Our WMM-Voc can still beat the state-of-the-art SAE and +ESZSL by outperforming 17.67% and 20.24% individually on +Top-1 accuracy. (2) In supervised learning task, the ESZSL +and Deep WMM-Voc have almost the same performance, if we +consider the variances in sampling the 200 training instances. +Our WMM-Voc is slightly better than these two methods. +(3) In G-ZSL setting, our two models get significantly better +performance compared with the other competitors. Notably, +the Top-1 accuracy of SAE and ESZSL is 0. While Deep +WMM-Voc and WMM-Voc both have higher accuracy. This +shows the effectiveness of our two models. (4) As expected, +the deep models that fine-tune features along with classifiers +(Deep-SVR and Deep WMM-Voc) are better than counterparts +with pre-extracted representations (SVR-Map, WMM-Voc). +4.2.2 +Results on different training/testing splits +We conduct experiments using different number of training +instances and compare results on tasks of supervised, zero- +shot and generalized zero-shot learning. On each split, we use +both 100 and 1000 dimensional word vectors. We use 12,156 +testing instances from source classes in supervised, G-ZSL +and open-set setting as well as 6,180 testing instances from + +10 +Dimension +SVR-Map +Deep-SVR +SAE +ESZSL +MM-Voc +WMM-Voc +Deep WMM-Voc +Supervised +100-dim +51.4/- +71.59/91.98 +70.22/92.60 +74.86/94.85 +58.01/87.88 +75.57/94.31 +76.23/94.85 +1000-dim +57.1/- +76.32/95.22 +75.32/94.17 +75.08/94.27 +59.1/77.73 +79.44/96.01 +76.55/96.22 +Zero-shot +100-dim +52.1/- +53.12/84.24 +67.96/95.08 +73.69/95.83 +61.10/96.02 +82.78/98.92 +84.87/98.87 +1000-dim +58.0/- +64.29/88.71 +71.42/97.18 +74.17/97.12 +83.84/96.74 +89.09/99.21 +88.07/99.40 +G-ZSL +100-dim +- +5.65/54.45 +2.15/52.7 +2.88/68.37 +19.74/85.79 +28.92/88.01 +33.04/89.11 +1000-dim +- +0/39.84 +0/35.91 +0/33.09 +8.54/59.79 +27.98/90.47 +34.77/90.76 +TABLE 2 +Classification accuracy (Top-1 / Top-5) on AwA dataset for SUPERVISED, GENERAL ZERO-SHOT and ZERO-SHOT settings for +100-dim and 1000-dim word2vec representation (200 instances). +target classes in zero-shot, G-ZSL and open-set setting. All the +competitors are using the same types of features – ResNet101. +Supervised learning: The results are compared in Figure 4. +As shown in the figure, we observe that our method shows +significant improvements over the competitors in few-shot +setting; however, as the number of instance increasing, the +visual semantic mapping, g(x), can be well learned, and the +effects of additional vocabulary-informed constraints, become +less pronounced. +Zero-shot learning: The ZSL results are compared in Figure +3. On all the settings, our two Voc methods – Deep WMM- +Voc and WMM-Voc outperforms all the other baselines. This +validates the importance of information learning from the +open vocabulary. Further, we compare our results with the +state-of-the-art ZSL results on AwA dataset in Table 1. Our +two models achieve 90.79% and 90.65% accuracy, which +is markedly higher than all previous methods. This is par- +ticularly impressive, if we take into account the fact that +we use only a semantic space and no additional attribute +representations (unlike many other competitor methods). We +argue that much of our success and improvement comes from +a more discriminative information obtained using the open +set vocabulary and corresponding large margin constraints, +rather than from the features. Varying the number of training +instances may slightly affect accuracy of methods reported +in Table 1. Therefore we report the best results of each +competitor and our own method at different number of train- +ing instances {200, 800, 1600, 121319, 24295}.All competing +methods in Figure 3 use the same features. +General zero-shot learning:. The general zero-shot learning +results are compared in Table 3. We consider the accuracies of +both U → T (ZERO-SHOT-like) and S → T (SUPERVISED-like). +In term of the harmonic mean H, our methods have signifi- +cantly better performance in the general zero-shot setting. This +again shows that our framework can have better generalization +by learning from the open vocabulary. On the other hand, in the +terms of Area Under Seen-Unseen accuracy Curve (AUSUC), +the performance of Deep-SVR is very weak and the scores +of ESZSL and SAE are lower than our method. Overall, the +results of AUSUC still support the superiority of our methods +on G-ZSL tasks. Notably, since the source domain only have +24295 instances (including training and testing images), we +are unable to obtain the results of SUPERVISED-like setting +(S → T), H and AUSUC with all source instances. +Supervised results by 100-d word vector +200 +800 +1600 +12139 +Training Instance Number +0 +20 +40 +60 +80 +100 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSL +Supervised results by 1000-d word vector +200 +800 +1600 +12139 +Training Instance Number +0 +20 +40 +60 +80 +100 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSL +Fig. 4. Supervised learning results of AwA datasets. +4.2.3 +Large-scale open set recognition +We also compare the results on OPEN-SET310K setting with +the large vocabulary of approximately 310K entities; as such +the chance performance is much lower. We use 100-dim word +vector representations as the semantic space. While our OPEN- +SET variants do not assume that test data comes from either +source/auxiliary domain or target domain, we split the two +cases to mimic SUPERVISED and ZERO-SHOT scenarios for +easier analysis. The results are shown in Figure 5. +On SUPERVISED-like setting, Figure 5 (left), our Deep +WMM-Voc and WMM-Voc have better performance than the +other baselines. The better results are largely due to the better +embedding matrix W learned by enforcing maximum margins +between training class name and open set vocabulary on source +training data. This validates the effectiveness of proposed +framework. In particular, we find that (1) The “deep” version +always has better performance than their corresponding “non- +deep” counterparts. For example, the Deep-SVR and Deep +WMM-Voc achieve higher open-set recognition accuracy than +SVR-Map and WMM-Voc. (2) The WMM-Voc has better +performance than MM-Voc; this shows that the weighting +strategy introduced in Section 3.4 can indeed help better learn + +11 +TABLE 3 +The G-ZSL results (100-dim/1000-dim) of AwA dataset. We compare the results by varying the number of training +instances (No.of Tr. Ins.) of each class. Where H is defined in Eq. (19) without calibrated stacking, AUSUC means +the Area Under Seen-Unseen accuracy Curve with calibrated stacking and ’-’ represents the unavailable results. +Metrics +ESZSL +SAE +Deep-SVR +WMM-Voc +Deep WMM-Voc +200 +U → T +2.88/0 +2.15/0 +5.65/0 +28.92/27.98 +33.04/34.77 +S → T +75.76/76.08 +70.13/75.32 +71.22/76.32 +70.20/74.20 +71.16/69.48 +H +5.55/0 +4.17/0 +10.47/0 +40.96/40.64 +45.13/46.35 +AUSUC +0.4231/0.4344 +0.3885/0.4556 +0.3048/0.3939 +0.4840/0.5190 +0.5028/0.4776 +800 +U → T +0.19/0 +0.78/0 +5.34/0.02 +25.57/25.68 +27.59/27.77 +S → T +81.14/83.95 +78.02/83.41 +78.92/81.46 +74.23/77.33 +75.53/77.19 +H +0.38/0 +1.54/0 +10.00/0.04 +38.04/38.56 +40.42/40.85 +AUSUC +0.4409/0.4710 +0.3870/0.4483 +0.3452/0.4400 +0.4764/0.5387 +0.4953/0.5353 +1600 +U → T +0.71/0 +0.87/0 +4.69/0 +24.63/27.22 +33.66/32.86 +S → T +85.62/86.24 +81.08/85.48 +83.30/86.02 +74.99/77.67 +78.96/78.64 +H +1.41/0 +1.72/0 +8.88/0 +37.08/40.31 +47.20/46.35 +AUSUC +0.4507/0.5139 +0.4190/0.4740 +0.3776/0.4780 +0.5016/0.5572 +0.5554/0.5733 +12139 +U → T +0.37/0 +0.44/0 +5.19/0 +27.80/30.53 +32.23/28.19 +S → T +89.98/91.16 +88.18/91.26 +85.37/85.64 +77.36/78.34 +80.64/78.32 +H +0.74/0 +0.88/0 +9.79/0 +40.90/43.94 +46.05/41.46 +AUSUC +0.5096/0.5294 +0.4493/0.5120 +0.3353/0.4397 +0.5144/0.5319 +0.5525/0.5394 +24295 +U → T +0.83/0 +0.37/0 +5.39/0 +27.15/29.42 +35.65/31.78 +S → T +- +- +- +- +- +H +- +- +- +- +- +AUSUC +- +- +- +- +- +the embedding from visual to semantic space. +On ZERO SHOT-like setting, our method still has a notable +advantage over that of SVR-Map, Deep-SVR methods on Top- +k (k > 3) accuracy, again thanks to the better embedding +W learned by Eq. (10). However, we notice that our top- +1 accuracy on ZERO SHOT-like setting is lower than Deep +SVR method. We find that our method tends to label some +instances from target data with their nearest classes from +within source label set. For example, “humpback whale” from +testing data is more likely to be labeled as “blue whale”. +However, when considering Top-k (k > 3) accuracy, our +method still has advantages over baselines. It suggests that +the semantic embeddings may be suffering from the problem +that density of source classes is more concentrated than that +of target classes. To show the effectiveness of WMM-Voc, as +opposed to MM-Voc, we employ the False Positive Rate as the +metric, rfp = Ne/Nun, where Ne means the number of testing +unseen instances predicted as seen ones and Nun defines +the number of testing unseeen instances. Experiments are +conducted on AwA dataset with all training instances, and 100- +dim word vector prototypes. The false positive rates are 0.16, +0.10, 0.12, 0.05 and 0.06 by using SVR, Deep-SVR, MM-Voc, +WMM-Voc and Deep WMM-Voc, respectively. They further +validates that WMM-Voc outperforms MM-Voc. +4.3 +Experimental results on ImageNet dataset +We validate our methods on large-scale ImageNet 2012/2010 +dataset; the 1000-dimensional word2vec representation is used +here since this dataset has larger number of classes than AwA. +Testing Classes +AwA dataset +Aux. +Targ. +Total +Vocab +OPEN-SET310K +(left) +(right) +40 / 10 +310K +0 +5 +10 +15 +20 +Hit@k +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Accuracy (%) +SUPERVISED-like(Aux) +0 +5 +10 +15 +20 +Hit@k +0 +10 +20 +30 +40 +50 +60 +70 +Accuracy (%) +ZERO SHOT-like(Tag) +SVR-Map +MM-Voc +Deep-SVR +Deep WMM-Voc +WMM-Voc +Fig. 5. +Openset results of AwA datasets. We use +1600 training instances equally sampled from all source +classes to train the model. +The instances of testing classes are equally sampled; making +experiment less sensitive to the problem of unbalanced data. +To be specific, 50×1, 000 testing instances from source classes +are used in supervised, G-ZSL and open-set setting as well as + +12 +TABLE 4 +The classification accuracy (Top-1 / Top-5) of ImageNet 2012/2010 dataset on ZERO-SHOT and SUPERVISED settings using +3000 source training instances. +Settings +SVR-Map +Deep-SVR +ESZSL +SAE +MM-Voc +WMM-Voc +Deep WMM-Voc +Supervised +25.6/– +31.26/50.51 +38.26/64.38 +32.95/54.44 +37.1/62.35 +35.95/62.77 +38.92/65.35 +Zero-shot +4.1/– +5.29/13.32 +5.86/13.71 +5.11/12.62 +8.90/14.90 +8.50/20.73 +9.26/21.99 +360 × 100 testing instances from target classes are used in +zero-shot, G-ZSL and open-set setting. The VGG-19 features +of ImageNet pre-trained network are utilized as the input of +all algorithms to make a fair comparison. We employ the +Deep-SVR, SAE, ESZSL as baselines under the SUPERVISED, +ZERO-SHOT and GENERAL ZERO-SHOT settings respectively. +4.3.1 +Pseudo-few-shot Source Training instances +The standard few-shot learning assumes disjoint instance set +on source and target domains, as discussed in Sec. 4.3.2. +As an ablation study, we would like to simulate a few-shot- +like learning task on source domain by slightly violating +the standard few-shot learning assumption. We name this +setting “Pseudo-few-shot learning”: only few source training +instances are used here and the feature extractor – VGG- +19 model is pre-trained on ILSVRC 2012 dataset [12]. The +“Pseudo-” here indicates that large amount of instances are +used to train the feature extractor, but not used in training +classifiers. Thus the experiments in this section can be served +as an additional ablation study to reveal the insights of our +model in addressing the few-shot-like task on source domain. +Particularly, we conduct the experiments of using few-shot +source training instance, i.e., 3,000 training instances used +here. The results are listed in Table 4. We introduce this +setting to particularly focus on learning from few training +samples per class, in order to mimic human capability and +performance in learning from few examples. We show that +our vocabulary-informed learning framework enables learning +with little data. In particular, we highlight that the Top-5 +performance of WMM-Voc is much higher (>5%) than that +of MM-Voc, despite the slightly worse performance on Top- +1 accuracy. Note that the degradation of Top-1 results on +ImageNet is also understandable. Note, WMM-Voc is only +fitting the 3000 training instances on ImageNet dataset, and +the features of these training instances may not be fine- +tuned/optimized for the newly introduced penalty term of +WMM-Voc. Once the features of training instances are fine- +tuned by the deep version; we can show that the Deep WMM- +Voc can improve from MM-Voc and WMM-Voc. +Critically, with different settings in Table 4, our vocabulary- +informed learning can beat the other baselines under all +settings. We highlight several findings: +(1) The supervised performance of our methods stands +out from the state-of-art. Specifically, our Deep WMM-Voc +achieves the highest supervised recognition accuracy, with +ESZSL following closely. SVR-Map appears to be the worst. +(2) On Zero-shot learning task, our proposed Deep WMM- +Voc gets 9.26% Top-1 and 21.99% Top-5 accuracy. It outper- +forms all the other baselines. Comparing with our previous +Supervised Learning results of ImageNet +3k +5k +10k +20k +50k +Training Instance Number +0 +20 +40 +60 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSl +Zero-shot Learning results of ImageNet +3k +5k +10k +20k +50k +Training Instance Number +0 +2 +4 +6 +8 +10 +Top1 Accuracy(%) +WMM-Voc +Deep WMM-Voc +Deep-SVR +SAE(F->S) +SAE(S->F) +ESZSL +Fig. 6. The supervised and zero-shot learning results on +ImageNet 2012/2010 dataset. +MM-Voc result in [29], our result is 0.36% higher than MM- +Voc. This improvement is statistically significant due to the +few number of training instances and large number of testing +instances. Additionally, the Top-1 result of WMM-Voc is +8.5% which is also comparable to that of MM-Voc, and far +higher than those of SVR, Deep-SVR, ESZSL and SAE. This +validates the effectiveness of learning from open vocabulary +proposed in our two variants. +(3) In G-ZSL setting, we observe that both Deep WMM- +Voc and WMM-Voc outperform all the other baselines. The +full set of experiments on G-ZSL under different settings are +reported in Table 5. +4.3.2 +Few-shot Target Training instances +We further introduce few-shot learning experiments on target +instances to validate the performance of our methods. The +experiments are conducted on ImageNet dataset. In total, there +are 360 target classes from ImageNet 2010 data split with +100 instances per class; the feature extractor – VGG-19 is +trained on the 1000 classes from ImageNet 2012. The 1 or +3 training instances are sampled from each target class. The +other instances of the target classes are utilized as the test +set. This is the few-shot learning setting, which is consistent +with general definition [20]. We compare to SVM, KNN, +Deep SVR, and SAE. The results are shown in Table 6. We +can see that our method (WMM-Voc) can beat all the other +competitors. Particularly, we have an obvious advantage in 1- +shot target setting. Our Deep variant (Deep WMM-Voc) has + +13 +TABLE 5 +The G-ZSL results (1000-dim) of ImageNet datasets. We +compare the results of using different number of training +instances (No.) D-SVR, W-V and D-W-V indicate Deep +SVR, WWM-Voc, and Deep WWM-Voc, respectively. +No. +Metrics +ESZSL +SAE +D-SVR +W-V +D-W-V +3000 +U → T +0.46 +0.24 +0.20 +2.02 +1.93 +S → T +38.07 +32.86 +31.06 +32.40 +36.61 +H +0.91 +0.48 +0.40 +3.80 +3.67 +10000 +U → T +0.38 +0.18 +0.18 +2.01 +1.99 +S → T +49.65 +46.23 +33.54 +32.87 +43.53 +H +0.75 +0.36 +0.36 +3.79 +3.81 +50000 +U → T +0.37 +0.19 +0.20 +2.11 +2.15 +S → T +57.43 +54.55 +36.75 +33.16 +47.28 +H +0.74 +0.38 +0.40 +3.97 +4.11 +TABLE 6 +Results of few-shot target training instances on +ImageNet dataset. +Method +1-instance +3-instance +SVM +2.65 +9.81 +KNN +5.23 +13.3 +Deep SVR +14.01 +25.00 +SAE +14.93 +26.42 +WMM-Voc +17.26 +26.59 +Deep WMM-Voc +17.95 +30.44 +better performance both in 1- and 3-shot setting. This shows +the efficacy of proposed methods in few-shot learning task. +4.3.3 +Results on different training/testing splits +We further validate our findings on ImageNet 2012/2010 +dataset. In general, our framework has advantages over the +baselines since open vocabulary helps inform the learning +process when few training instances or limited training data is +available. The results are compared in Figure 6. +Supervised learning: As shown in Figure 6, we compare the +supervised results by increasing the training instances from +3,000 to 50,000. With 3,000 training instances, the results of +Deep WMM-Voc are better than all the other baselines with +the help of learning from free vocabulary. We further evaluate +our models with larger number of training instances (> 3 +per class). We observe that for standard supervised learning +setting, the improvements achieved using vocabulary-informed +learning tend to somewhat diminish as the number of training +instances substantially grows. With large number of training +instances, the mapping between low-level image features and +semantic words, g(x), becomes better behaved and effect of +additional constraints, due to the open-vocabulary, becomes +less pronounced. +Zero-shot Learning: We further validate the results on zero- +shot learning setting. Figure 6 shows that our models can beat +all other baselines. Our Deep WMM-Voc always performs +the best with the source training instances increased from +TABLE 7 +ImageNet comparison to state-of-the-art on ZSL: We +compare the results of using 3, 000/all training instances +for all methods; T-1 (top 1) and T-5 (top 5) classification +in (%) is reported. The VGG-19 features are used for all +methods. +Methods +S. Sp +T-1 +T-5 +Deep WMM-Voc +W +9.26/10.29 +21.99/23.12 +WMM-Voc +W +8.5/8.76 +20.30/21.36 +MM-Voc +W +8.9/9.5 +14.9/16.8 +SAE +W +5.11/9.32 +12.26/21.04 +ESZSL +W +5.86/8.3 +13.71/18.2 +Deep-SVR +W +5.29/5.7 +13.32/14.12 +Embed [88] +W +–/11.00 +–/25.70 +ConSE [53] +W +5.5/7.8 +13.1/15.5 +DeViSE [24] +W +3.7/5.2 +11.8/12.8 +AMP [31] +W +3.5/6.1 +10.5/13.1 +Chance +– +2.78e-3 +– +3,000 to 50,000. The WMM-Voc always has the second +best performance; especially when only few source training +instances are available, i.e., 3,000 and 5,000 training instances. +Our Deep WMM-Voc and WMM-Voc demonstrate significant +improvements over the competitors in ZSL task. The good +performance of Deep WMM-Voc and WMM-Voc is largely +due to our vocabulary-informed learning framework which can +leverage the discriminative information from open vocabulary +and max-margin constraints, helping to improve performance. +General Zero-shot Learning: In G-ZSL, our methods still +have the best performance compared to the baselines, as seen +from Table 5. The Top-1 results of WMM-Voc and Deep +WMM-Voc are beyond 2%; in contrast, the performance of +other state-of-art methods are lower than 0.5%. +Varying training set size: In Figure 6 we also evaluate our +model with the larger number of training instances (> 3 per +class) in all settings. The results are inline with prior findings. +The state-of-the-art on ZSL: We compare our results to sev- +eral state-of-the-art large-scale zero-shot recognition models. +Our results are better than those of ConSE, DeViSE, Deep- +SVR, SAE, ESZSL and AMP on both T-1 and T-5 metrics +with a very significant margin. Poor results of DeViSE with +3, 000 training instances are largely due to the inefficient +learning of visual-semantic embedding matrix. AMP algorithm +also relies on the embedding matrix from DeViSE, which +explains similar poor performance of AMP with 3, 000 training +instances. Table 7 shows that our Deep WMM-Voc obtains +good performance with (all) 50,000 training instances. Top-5 +accuracy of our methods are beyond 20%. This again validates +that our proposed methods can have the advantages of learning +from limited available training instances by leveraging the +discriminative information from open vocabulary. Embed [1] +has slightly better ZSL performance compared to our models. +However, unlike the other works that directly use word vector +representations of class names, [1] require additional textual +descriptions of each class to learn better class prototypes. +Open-set recognition: The open set image recognition results + +14 +Testing Classes +ImageNet Data +Aux. +Tag. +Total +Vocab +OPEN-SET310K +(left) +(right) +1000 / 360 +310K +0 +5 +10 +15 +20 +Hit@k +5 +10 +15 +20 +25 +30 +Accuracy (%) +SUPERVISED-like(Aux) +0 +5 +10 +15 +20 +Hit@k +0 +1 +2 +3 +4 +5 +6 +Accuracy (%) +ZERO SHOT-like(Tag) +MM-Voc +Deep-SVR +Deep WMM-Voc +WMM-Voc +Fig. 7. +Open set recognition results on ImageNet +2012/2010 dataset: Openness=0.9839. Chance=3.2e − +4%. We use the synsets of each class— a set of synony- +mous (word or prhase) terms as the ground truth names +for each instance. We use the model trained with 50,000 +instances sampled equally from source classes. +are shown in Figure 7. On SUPERVISED-like settings, we +notice the MM-Voc and WMM-Voc have similar open set +recognition accuracy. Since this dataset is very large, linear +mapping g(x) may not have enough capacity to model the +embedding mapping from visual space to semantic space. Thus +adding constraints on source training classes in WMM-Voc +may slightly hinder the learning such an embedding. That +explains why the results of WMM-Voc are slightly inferior +to MM-Voc. Deep WMM-Voc has the best performance, due +to its ability to fine-tune low-level feature representation while +learning the embedding. On the ZERO-SHOT-like setting, our +WMM-Voc and Deep WMM-Voc have the best performance. +Qualitative visualization: We illustrate the embedding space +learned +by +our +Deep +WMM-Voc +model +for +the +Ima- +geNet2012/2010 dataset in Figure 1. In particular, we have +4 source/auxiliary and 2 target/zero-shot classes in this figure. +The better separation among classes is largely attributed to +open-set max-margin constraints introduced in our vocabulary- +informed learning model. We further visualize the semantic +space in Figure 8. Critically, we list seven target classes +on AwA dataset, as well as their surrounding neighbor- +hood open vocabulary. For example, “orcas” is very near to +“killer_whale”. While “orcas” are semantically different from +“killer_whale”, the difference is much smaller if we compare +the “orcas” with the other classes, such as “spider monkey”, +“grizzly_bear” and so on. Hence the “orcas” can be used +to help learn the class of “killer_whale” in our vocabulary- +informed learning framework. +killer_whale +chihuahua_dog +dalmatian_dog +buffalo +grizzly_bear +collie +spider_monkey +Word prototype: +golden_retriever +collies +fox_terrier +sheepdog +shetland_pony +jack_russell_terrier +cattle_dog +wellard +drover +collie +killer_whale +tilikum +orcas +brancheau +orca +seaworld +bottlenose_dolphin +sea_lion +whale +seaworld_orlando +spider_monkey +capybara +giant_anteater +marmoset +tamarin +macaw +howler_monkeys +squirrel_monkeys +macaque +macaws +grizzly_bear +grizzly +polar_bear +elk +grizzly_bears +coyote +caribou +mountain_lion +mountain_goats +bighorn_sheep +buffalo +minneapolis +detroit +chicago +kansas_city +pittsburgh +erie +duluth +minnesota +grand_rapids +chihuahua_dog +ciudad_ju_rez +sonora +sinaloa +coahuila +durango +jalisco +michoac_n +zacatecas +nuevo_leon +dalmatian_dog +istria +ragusa +dalmatian_coast +kor_ula +gradisca +sardinian +zadar +croatian +istrian +Fig. 8. +Visualization of the semantic space: We +show the t-SNE visualization of the semantic space. The +words in boxes are the mapping of training image in the +semantic space, and close neighbors are shown. The +neighborhoods extend the single training data to a space +semantically meaningful. +5 +CONCLUSION AND FUTURE WORK +This paper introduces the learning paradigm of vocabulary- +informed learning, by utilizing open set semantic vocabulary +to help train better classifiers for observed and unobserved +classes in supervised learning, ZSL, G-ZSL, and open set +image recognition settings. We formulate vocabulary-informed +learning in the maximum margin frameworks. Extensive ex- +perimental results illustrate the efficacy of such learning +paradigm. Strikingly, it achieves competitive performance with +very few training instances and is relatively robust to a large +open set vocabulary of up to 310, 000 class labels. +ACKNOWLEDGMENTS +This work was supported in part by NSFC Project (61702108, +61622204), STCSM Project (16JC1420400), Eastern Scholar +(TP2017006), Shanghai Municipal Science and Technol- +ogy Major Project (2017SHZDZX01, 2018SHZDZX01) and +ZJLab. +REFERENCES +[1] +2.4, 4.3.3 +[2] +Z. Akata, F. Perronnin, Z. Harchaoui, and C. Schmid. Label-embedding +for image classification. +IEEE transactions on pattern analysis and +machine intelligence, 38(7):1425–1438, 2015. 2, 2.4, 4.2 +[3] +Z. Akata, S. Reed, D. Walter, H. Lee, and B. Schiele. Evaluation of +output embeddings for fine-grained image classification. In Proceedings +of the IEEE Conference on Computer Vision and Pattern Recognition, +pages 2927–2936, 2015. 2, 2.3, 2.4, 3.3, 4.2 +[4] +Y. Amit, M. Fink, N. Srebro, and S. 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Springer, +2014. 3.4 +Yanwei Fu received the Ph.D. degree from +Queen Mary University of London in 2014, and +the M.Eng. degree from the Department of Com- +puter Science and Technology, Nanjing Univer- +sity, China, in 2011. He held a post-doctoral po- +sition at Disney Research, Pittsburgh, PA, USA, +from 2015 to 2016. He is currently a tenure-track +Professor with Fudan University. His research +interests are image and video understanding, +and life-long learning. + +17 +Xiaomei Wang is a PhD student in the School of +Computer Science of Fudan University. She re- +ceived the Master degree of communication and +information system from Shanghai University in +2016 and the Bachelor degree of electronic infor- +mation engineering from Shandong University of +Technology in 2012. Her reaseach interests in- +clude zero-shot/few-shot learning, image/video +captioning and visual question answering. +Hanze Dong is an undergraduate student ma- +joring in mathematics (data science track) at +the School of Data Science, Fudan University. +He works in Shanghai Key Lab of Intelligent +Information Processing under the supervision +of Professor Yanwei Fu. His current research +interests include both machine learning theory +and its applications. +Yu-Gang Jiang is Professor of Computer Sci- +ence at Fudan University and Director of Fudan- +Jilian Joint Research Center on Intelligent Video +Technology, Shanghai, China. He is interested +in all aspects of extracting high-level informa- +tion from big video data, such as video event +recognition, object/scene recognition and large- +scale visual search. His work has led to many +awards, including the inaugural ACM China Ris- +ing Star Award, the 2015 ACM SIGMM Rising +Star Award, and the research award for out- +standing young researchers from NSF China. He is currently an as- +sociate editor of ACM TOMM, Machine Vision and Applications (MVA) +and Neurocomputing. He holds a PhD in Computer Science from City +University of Hong Kong and spent three years working at Columbia +University before joining Fudan in 2011. +Meng Wang is a professor at the Hefei Univer- +sity of Technology, China. He received his B.E. +degree and Ph.D. degree in the Special Class +for the Gifted Young and the Department of +Electronic Engineering and Information Science +from the University of Science and Technology of +China (USTC), Hefei, China, in 2003 and 2008, +respectively. His current research interests in- +clude multimedia content analysis, computer vi- +sion, and pattern recognition. He has authored +more than 200 book chapters, journal and con- +ference papers in these areas. He is the recipient of the ACM SIGMM +Rising Star Award 2014. He is an associate editor of IEEE Transactions +on Knowledge and Data Engineering (IEEE TKDE), IEEE Transactions +on Circuits and Systems for Video Technology (IEEE TCSVT), IEEE +Transactions on Multimedia (IEEE TMM), and IEEE Transactions on +Neural Networks and Learning Systems (IEEE TNNLS). +Xiangyang Xue received the BS, MS, and PhD +degrees in communication engineering from Xi- +dian University, Xi’an, China, in 1989, 1992, and +1995, respectively. He is currently a professor of +computer science with Fudan University, Shang- +hai, China. His research interests include com- +puter vision, multimedia information processing +and machine learning. +Leonid Sigal is an Associate Professor in the +Department of Computer Science at the Univer- +sity of British Columbia and a Faculty Member +of the Vector Institute for Artificial Intelligence. +He is a recipient of Canada CIFAR AI Chair +and NSERC Canada Research Chair (CRC) in +Computer Vision and Machine Learning. Prior +to this he was a Senior Research Scientist at +Disney Research. He completed his Ph.D. at +Brown University in 2008; received his M.A. +from Boston University in 1999, and M.Sc. from +Brown University in 2003. Leonid’s research interests lie in the areas +of computer vision, machine learning, and computer graphics. Leonid’s +research emphasis is on machine learning and statistical approaches +for visual recognition, reasoning, understanding and analytics. He has +published more than 70 papers in venues and journals in these fields +(including TPAMI, IJCV, CVPR, ICCV and NeurIPS). + diff --git a/69AzT4oBgHgl3EQfEvpR/content/tmp_files/load_file.txt b/69AzT4oBgHgl3EQfEvpR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..141cc0b5def80d6e2307448c80b7461b7ad8599c --- /dev/null +++ b/69AzT4oBgHgl3EQfEvpR/content/tmp_files/load_file.txt @@ -0,0 +1,2052 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf,len=2051 +page_content='1 Vocabulary-informed Zero-shot and Open-set Learning Yanwei Fu, Xiaomei Wang, Hanze Dong, Yu-Gang Jiang, Meng Wang, Xiangyang Xue, Leonid Sigal Abstract—Despite significant progress in object categorization, in recent years, a number of important challenges remain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' mainly, the ability to learn from limited labeled data and to recognize object classes within large, potentially open, set of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zero-shot learning is one way of addressing these challenges, but it has only been shown to work with limited sized class vocabularies and typically requires separation between supervised and unsupervised classes, allowing former to inform the latter but not vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We propose the notion of vocabulary-informed learning to alleviate the above mentioned challenges and address problems of supervised, zero- shot, generalized zero-shot and open set recognition using a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, we propose a weighted maximum margin framework for semantic manifold-based recognition that incorporates distance constraints from (both supervised and unsupervised) vocabulary atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Distance constraints ensure that labeled samples are projected closer to their correct prototypes, in the embedding space, than to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We illustrate that resulting model shows improvements in supervised, zero-shot, generalized zero-shot, and large open set recognition, with up to 310K class vocabulary on Animal with Attributes and ImageNet datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Index Terms—Vocabulary-informed learning, Generalized zero-shot learning, Open-set recognition, Zero-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 1 INTRODUCTION Object recognition, more specifically object categorization, has seen unprecedented advances in recent years with development of convolutional neural networks (CNNs) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, most successful recognition models, to date, are formulated as su- pervised learning problems, in many cases requiring hundreds, if not thousands, labeled instances to learn a given concept class [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This exuberant need for large labeled instances has limited recognition models to domains with hundreds to thousands of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Humans, on the other hand, are able to distinguish beyond 30, 000 basic level categories [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Even more impressive is the fact that humans can learn from few examples, by effectively leveraging information from other object category classes, and even recognize objects without ever seeing them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', by reading about them on the Internet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This ability has spawned the research in few-shot and zero- shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zero-shot learning (ZSL) has now been widely studied in a variety of research areas including neural decoding of fMRI images [54], character recognition [44], face verification [42], object recognition [43], and video understanding [27], [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Typically, zero-shot learning approaches aim to recog- nize instances from the unseen or unknown testing target Yanwei Fu and Hanze Dong are with the School of Data Science, Fudan University, Shanghai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Email: {yanweifu,hzdong15}@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Xiaomei Wang, Yu-Gang Jiang and Xiangyang Xue are with the School of Computer Science, Shanghai Key Lab of Intelligent Information Processing, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Email: {17110240025,ygj,xyxue}@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Yu-Gang Jiang is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Yanwei Fu is also with AITRICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Meng Wang is with the School of Computer and Information Science, Hefei University of Technology, Hefei, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Email: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='wangmeng@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Leonid Sigal is with the Department of Computer Science, University of British Columbia, BC, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Email: lsigal@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='ubc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' categories by transferring information through intermediate- level semantic representations, from known observed source (or auxiliary) categories for which many labeled instances exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In other words, supervised classes/instances, are used as context for recognition of classes that contain no visual instances at training time, but that can be put in some correspondence with supervised classes/instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Therefore, a general experimental setting of ZSL is that the classes in target and source (auxiliary) dataset are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Typically, the learning is done on the source dataset and then information is transferred to the target dataset, with performance measured on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This setting has a few important drawbacks: (1) it assumes that target classes cannot be mis-classified as source classes and vice versa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' this greatly and unrealistically simplifies the problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) the target label set is often relatively small, between ten [43] and several thousand unknown labels [24], compared to at least 30, 000 entry level categories that humans can distinguish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (3) large amounts of data in the source (auxiliary) classes are required, which is problematic as it has been shown that most object classes have very few instances (long-tailed distribution of objects in the world [72]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' and (4) the vast open set vocabulary and corresponding semantic knowledge, defined as part of ZSL [54], is not leveraged in any way to inform the learning or source class recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' A few works recently looked at resolving (1) through class-incremental learning [66], [68] or generalized zero-shot learning (G-ZSL) [11], [54] which are designed to distinguish between seen (source) and unseen (target) classes at the testing time and apply an appropriate model – supervised for the former and ZSL for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, (2)–(4) remain largely unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, while (2) and (3) are artifacts of the ZSL setting, (4) is more fundamental;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', a recent study [34] argues that concepts, in our own brains, are represented in the form of a continuous semantic space mapped smoothly arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='00998v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='CV] 3 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Illustration of the semantic embeddings learned (left) using support vector regression (SVR) and (right) using the proposed vocabulary-informed learning (Deep WMM-Voc) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In both cases, t-SNE visualization is used to illustrate samples from 4 source/auxiliary classes (denoted by ×) and 2 target/zero-shot classed (denoted by ) from the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Decision boundaries, illustrated by dashed lines, are drawn by hand for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The large margin constraints, both among the source/target classes and the external vocabulary atoms, are denoted by arrows and words on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that the WMM-Voc approach on the right leads to a better embedding with more compact and separated classes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', see truck and car or unicycle and tricycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' across the cortical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, consider learning about a car by looking at image instances in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Not knowing that other motor vehicles exist in the world, one may be tempted to call anything that has 4-wheels a car.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As a result, the zero-shot class truck may have a large overlap with the car class (see Figure 1 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, imagine knowing that there also exist many other motor vehicles (trucks, mini- vans, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Even without having visually seen such objects, the very basic knowledge that they exist in the world and are closely related to a car should, in principle, alter the criterion for recognizing instance as a car (making the recognition criterion stricter in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Encoding this in our vocabulary- informed learning model results in better separation among classes (see Figure 1 (right)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To tackle the limitations of ZSL and towards the goal of generic recognition, we propose the idea of vocabulary- informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' assuming we have few labeled training instances and a large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' potentially open set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' vocab- ulary/semantic dictionary (along with textual sources from which statistical semantic relations among vocabulary atoms can be learned),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' the task of vocabulary-informed learning is to learn a unified model that utilizes this semantic dictionary to help train better classifiers for observed (source) classes and unobserved (target) classes in supervised,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' zero-shot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' general- ized zero-shot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' and open set image recognition settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we formulate Weighted Maximum Margin Vocabulary-informed Embedding (WMM-Voc), which learns a joint embedding for visual features and semantic words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In this formulation, two maximum margin sets of constraints are simultaneously optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The first set ensures that la- beled training visual instances, belonging to a particular class, project close to semantic word vector prototype corresponding to the class name in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The second set ensures that these instances are closer to the correct class word vector prototype than to any of the incorrect ones in the embedding space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' including those that may not contain training data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', zero-shot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The constraints in the first set further take into the account the distribution of training samples for each class, and nearby classes, to dynamically set appropriate margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In other words, for some classes the distance, between the projected training sample and the word vector prototype, is explicitly penalized more (or less) than for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This weighting is derived using extreme values theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Contributions: Our main contribution is to propose a novel paradigm for potentially open set image recognition: vocabulary-informed learning (Voc), which is capable of uti- lizing vocabulary over unsupervised items, during training, to improve recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We extend the model initially proposed by us in a conference paper [29] to include class-specific weighting in the data term, as well as the ability to run the models as an end-to-end network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Particularly, classification is done through the nearest-neighbor distance to class prototypes in the semantic embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Semantic embedding is learned subject to constraints ensuring that labeled images project into semantic space such that they end up closer to the correct class prototypes than to incorrect ones (whether those prototypes are part of the source or target classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We show that word embedding (word2vec) can be used effectively mini van X××o×0 roller 0 skate % 00 XOX 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' x8 X % X X to 8 8 8 8 to 00 88 skate 8 board 8 8 xo helicopter motorcycle dirigible motor Q scooter3 to initialize the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Experimentally, we illustrate that through this paradigm: we can achieve very competitive supervised (on source classes), ZSL (on target classes) and G-ZSL performance, as well as open set image recognition performance with a large number of unobserved vocabulary entities (up to 300, 000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' effective learning with few samples is also illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Critically, our models can be directly utilized in G-ZSL scenario and still has much better results than the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2 RELATED WORK Our model belongs to a class of transfer learning approaches [55], also sometimes called meta-learning [79] or learning to learn [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The key idea of transfer learning is to transfer the knowledge from previously learned categories to recognize new categories with no training examples (zero-shot learning [43], [59]), few examples (one-shot learning [19], [71]) or from vast open set vocabulary [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The process of knowledge transfer can be done by sharing features [4], [5], [22], [33], [73], [81], semantic attributes [43], [58], [60], or contextual information [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Visual-semantic embeddings have been widely used for transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Such models embed visual features into a semantic space by learning projections of different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Examples include WSABIE [80], ALE [2], SJE [3], DeViSE [24], SVR [18], [43], kernel embedding [33] and Siamese networks [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Open-set Recognition The term “open set recognition” was initially defined in [65], [66] and formalized in [6], [7], [63] which mainly aim at identifying whether an image belongs to a seen or unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The problem is also known as class-incremental learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, none of these methods can further identify classes for unseen instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The exceptions are [24], [53] which augment zero-shot (unseen) class labels with source (seen) labels in some of their experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Similarly, we define the open set image recognition as the problems of recognizing the class name of an image from a potentially very large open set vocabulary (including, but not limited to source and target labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that methods like [65], [66] are orthogonal but potentially useful here – it is still worth identifying seen or unseen instances to be recognized with different label sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Conceptually similar, but different in formulation and task, open-vocabulary object retrieval [32] focused on retrieving objects using natural language open- vocabulary queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 One-shot Learning While most of machine learning-based object recognition algorithms require a large amount of training data, one-shot learning [20] aims to learn object classifiers from one, or very few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To compensate for the lack of training instances and enable one-shot learning, knowledge must be transferred from other sources, for example, by sharing features [5], semantic attributes [27], [43], [58], [60], or contextual infor- mation [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, none of the previous work had used the open set vocabulary to help learn the object classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Zero-shot Learning Zero-shot Learning (ZSL) aims to recognize novel classes with no training instance by transferring knowledge from source classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ZSL was first explored with use of attribute-based semantic representations [18], [26], [27], [28], [42], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This required pre-defined attribute vector prototypes for each class, which is costly to obtain for a large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Recently, semantic word vectors were proposed as a way to embed any class name without human annotation effort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' they can therefore serve as an alternative semantic representation [3], [24], [31], [53] for ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Semantic word vectors are learned from large-scale text corpus by language models, such as word2vec [52] or GloVec [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, most of the previous work only use word vectors as semantic representations in ZSL setting, but have neither (1) utilized semantic word vectors explicitly for learning better classifiers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' nor (2) for extending ZSL setting towards open set image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' A notable exception is [53] which aims to recognize 21K zero-shot classes given a modest vocabulary of 1K source classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' we explore vocabularies that are up to an order of the magnitude larger – 310K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Generalized zero-shot recognition (G-ZSL) [11] relaxed the problem setup of conventional zero-shot learning by consider- ing the training classes in the recognition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Chao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [11] investigated the G-ZSL task and found that it is less effective to directly extend the existing zero-shot learning algorithms to deal with G-ZSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Recently, Xian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [54] systematically compared the evaluation settings for ZSL and G-ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Comparing against existing ZSL models, which are inferior in the G-ZSL scenario, we show that our vocabulary-informed frameworks can be directly utilized for G-ZSL and achieve very competitive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4 Visual-semantic Embedding Mapping between visual features and semantic entities has been explored in three ways: (1) directly learning the embed- ding by regressing from visual features to the semantic space using Support Vector Regressors (SVR) [18], [43] or neural network [68];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) projecting visual features and semantic entities into a common new space, such as SJE [3], WSABIE [80], ALE [2], DeViSE [24], and CCA [25], [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (3) learning the embeddings by regressing from the semantic space to visual features, including [1], [10], [38], [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In contrast to other embedding methods, our model trains a better visual-semantic embedding from only few training in- stances with the help of a large amount of open set vocabulary items (using a maximum margin strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our formulation is inspired by the unified semantic embedding model of [35], however, unlike [35], our formulation is built on word vector representation, contains a data term, and incorporates constraints to unlabeled vocabulary prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3 VOCABULARY-INFORMED LEARNING 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Problem setup Assume a labeled source dataset Ds = {xi, zi}Ns i=1 of Ns samples, where xi ∈ Rp is the image feature representation 4 of image i and zi ∈ Ws is a class label taken from a set of English words or phrases W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' consequently, |Ws| is the number of source classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Further, suppose another set of class labels for target classes Wt, also taken from W, such that Ws ∩Wt = ∅, for which no labeled samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We note that potentially |Wt| >> |Ws|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Given a new test image feature vector x∗ the goal is then to learn a function z∗ = f(x∗), using all available information, that predicts a class label z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that the form of the problem changes drastically depending on the label set assumed for z∗: Supervised learning: z∗ ∈ Ws;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zero-shot learning: z∗ ∈ Wt ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Generalized zero-shot learning: z∗ ∈ {Ws, Wt};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Open set recognition: z∗ ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that open set recognition is similar to generalized zero- shot learning, however, in open set setting additional distractor classes that do not exist in either source or target datasets are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We posit that a single unified f(x∗) can be learned for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We formalize the definition of vocabulary-informed learning (Voc) as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Vocabulary-informed Learning (Voc): is a learning setting that makes use of complete vocabulary data (W) during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Unlike a more traditional ZSL that typi- cally makes use of the vocabulary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', semantic embedding) at test time, Voc utilizes exactly the same data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Notably, Voc requires no additional annotations or semantic knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' it simply shifts the burden from testing to training, leveraging the vocabulary to learn a better model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The vocabulary W can be represented by semantic embed- ding space learned by word2vec [52] or GloVec [57] on large- scale corpus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' each vocabulary entity w ∈ W is represented as a distributed semantic vector u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Semantics of embedding space help with knowledge transfer among classes, and allow ZSL, G-ZSL and open set image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that such semantic embedding spaces are equivalent to the “semantic knowledge base” for ZSL defined in [54] and hence make it appropriate to use Vocabulary-informed Learning in ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 Learning Embedding and Recognition Assuming we can learn a mapping g : Rp → Rd, from image features to this semantic space, recognition can be carried out using simple nearest neighbor distance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', f(x∗) = car if g(x∗) is closer to ucar than to any other word vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uj in this context can be interpreted as the prototype of the class j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Essentially, the attribute or semantic word vector of the class name can be taken as the class prototype [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The core question is then how to learn the mapping g(x) and what form of inference is optimal in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For learning we propose the discriminative maximum margin criterion that ensures that labeled samples xi project closer to their corresponding class prototypes uzi than to any other prototype ui in the open set vocabulary i ∈ W \\ zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Learning Embedding: To learn the function f(x), one needs to establish the correspondence between visual feature space and semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Particularly, in the training step, each image sample xi is regressed towards its corresponding class prototype uzi by minimizing W = arg min W Ns � i=1 L (xi, uzi) + λ ∥ W ∥2 F (1) where L (xi, uzi) = ∥g (xi) − uzi∥2 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' and g : Rp → Rd is the mapping from image features to semantic space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ∥ · ∥F indicates the Frobenius Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' If g (x) = W T x is a linear mapping, we have the closed form solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The loss function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (1) can be interperted as a variant of SVR embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, this is too limiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To learn the linear embedding matrix W, we introduce and discuss two sets of methods in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Recognition: The recognition step can be formulated using the nearest neighbor classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Given a testing instance x⋆, z⋆ = arg min i ��W T x⋆ − ui ��2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) measures the distance between predicted vector and the class prototypes in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In terms of different label set, we can do supervise, zero-shot, generalized zero-shot or open set recognition without modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we explore a simple variant of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) to classify the testing instance x⋆, z∗ = arg min i ∥ W T x∗ − ω (ui) ∥2 2, (3) where the Nearest Neighbor (NN) classifier measures distance between the predicted semantic vectors and a function of pro- totypes in the semantic space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', ω (ui) = ui is equivalent to Eq (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, we employ semantic vector prototype averaging to define ω (·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, sometimes, there might be more than one positive prototype, such as pig, pigs and hog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In such the circumstance, choosing the most likely prototype and using NN may not be sensible, hance we introduce the averaging strategy to consider more prototypes for robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that this strategy is known as Rocchio algorithm in infor- mation retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Rocchio algorithm is a method for relevance feedback that uses more relevant instances to update the query for better recall and possibly precision in the vector space (Chap 14 in [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' It was first suggested for use in ZSL in [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' more sophisticated algorithms [25], [58] are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Maximum Margin Voc Embedding (MM-Voc) The maximum margin vocabulary-informed embedding learns the mapping g(x) : Rp → Rd, from low-level features x to the semantic word space by utilizing maximum margin strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, consider g(x) = W T x, where1 W ⊆ Rp×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Ideally we want to estimate W such that uzi = W T xi for all labeled instances in Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that we would obviously want this to hold for instances belonging to unobserved classes as well, but we cannot enforce this explicitly in the optimization as we have no labeled samples for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Data Term: The easiest way to enforce the above objective is to minimize Euclidian distance between sample projections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Generalizing to a kernel version is straightforward, see [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 5 and appropriate prototypes in the embedding space, D (xi, uzi) = ��W T xi − uzi ��2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (4) Where we need to minimize this term with respect to each instance (xi, uzi), where zi is the class label of xi in Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Such embedding is also known, in the literature, as data embedding [35] or compatibility function [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To make the embedding more comparable to support vector regression (SVR), we employ the maximal margin strategy – ϵ−insensitive smooth SVR (ϵ−SSVR) [46] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' That is, L (xi, uzi) = Lϵ (xi, uzi) + λ ∥ W ∥2 F (5) where Lϵ (xi, uzi) = 1T | ξ |2 ϵ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' λ is regularization coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (|ξ|ϵ)j = max � 0, ���W T ⋆jxi − (uzi)j ��� − wzi · ϵ � (6) |ξ|ϵ ∈ Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ()j indicates the j-th value of corresponding vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' W⋆j is the j-th column of W, and wzi is the scaling weight derived from the density of class zi and it’s neighboring classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In our conference version [29], equal weight wzi is used for all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Here we notice that it is beneficial to use the density/coverage of each labeled training class as the constraint in learning the projection from visual feature space to semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We introduce a specific weighting strategy to compute wzi in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The conventional ϵ−SVR is formulated as a constrained minimization problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', convex quadratic programming problem, while ϵ−SSVR employs quadratic smoothing [89] to make Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (5) differentiable everywhere, and thus ϵ−SSVR can be solved as an unconstrained minimization problem directly2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Triplet Term: Data term above only ensures that labelled samples project close to their correct prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, since it is doing so for many samples and over a number of classes, it is unlikely that all the data constraints can be satisfied exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, consider the following case, if uzi is in the part of the semantic space where no other entities live (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', distance from uzi to any other prototype in the em- bedding space is large), then projecting xi further away from uzi is asymptomatic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', will not result in misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, if the uzi is close to other prototypes then minor error in regression may result in misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To embed this intuition into our learning, we enforce more discriminative constraints in the learned semantic embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, the distance of D (xi, uzi) should not only be as small as possible, but should also be smaller than the distance D (xi, ua), ∀a ̸= zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Formally, we define the triplet term MV (xi, uzi) = 1 2 AV � a=1 � C + 1 2D (xi, uzi) − 1 2D (xi, ua) �2 + , (7) where a ∈ Wt (or more precisely a ∈ W \\ Ws) is selected from the open vocabulary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' C is the margin gap constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Here, [·]2 + indicates the quadratically smooth hinge loss [89] which 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, our tentative experiments shows that the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (5) will lead to the similar results, on average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' but formulation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (5) is more stable and has lower variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' is convex and has the gradient at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To speedup computation, we use the closest AV target prototypes to each source/auxiliary prototype uzi in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We also define similar constraints for the source prototype pairs: MS (xi, uzi) = 1 2 BS � b=1 � C + 1 2D (xi, uzi) − 1 2D (xi, ub) �2 + (8) where b ∈ Ws is selected from source/auxiliary dataset vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This term enforces that D (xi, uzi) should be smaller than the distance D (xi, ub), ∀b ̸= zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To facilitate the computation, we similarly use closest BS prototypes that are closest to each prototype uzi in the source classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that, the Crammer and Singer loss [13], [75] is the upper bound of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (7) and (8) which we use to tolerate slight variants of uzi (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', the prototypes of ’pigs’ Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ’pig’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To sum up, the complete triplet maximum margin term is: M (xi, uzi) = MV (xi, uzi) + MS (xi, uzi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (9) We note that the form of rank hinge loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (7) and (8) is similar to DeViSE [24], but DeViSE only considers loss with respect to source/auxiliary data and prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Maximum Margin Vocabulary-informed Embedding: The complete combined objective can now be written as: W = argmin W nT � i=1 (αLϵ (xi, uzi) + (1 − α)M (xi, uzi)) + λ ∥ W ∥2 F , (10) where α ∈ [0, 1] is the coefficient that controls contribution of the two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' One practical advantage is that the objective function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10) is an unconstrained minimization problem which is differentiable and can be solved with L-BFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' W is initialized with all zeros and converges in 10 − 20 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4 Weighted Maximum Margin Voc Embedding (WMM-Voc) We note that there is no previous method that directly estimates the density of source training classes in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, doing so may lead to several benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' First, the num- ber of training instances in source classes may be unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In such a case, an estimate of the density of samples in a training class can be utilized as a constraint in learning the embedding characterized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Second, in the semantic space, the instances from the classes whose data samples span a large radius [62] may reside in the neighborhood of many other classes or open vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This can happen when the embedding is not well learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We can interpret this phenomenon as hubness [45], [67]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Adding a penalty based on the density of each training class may be helpful in better learning the embedding and alleviating the hubness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This subsection introduces a strategy for estimating the density of each known class in the semantic space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', wzi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Generally, we know the prototype of each known and novel class in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To estimate 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, the causes for hubness are still under investigation [16], [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Illustration of margin distribution of prototypes in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' the density/coverage of a known class, one needs to look at pairwise distance between a prototype and the nearest negative instance and the furthest positive instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This intuition leads us to introduce the concept of margin distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Margin Distribution: The concept of margin is fundamen- tal to maximum margin classifiers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', SVMs) in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The margin enables an intuitive interpretation of such classifiers in searching for the maximum margin separator in a Reproducing Kernel Hilbert Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Previous margin classifiers [92] aim to maximize a single margin across all training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In contrast, some recent studies [17], [62], [64], [90] suggest that the knowledge of margin distribution of instances, rather than a single margin across all instances, is crucial for improving the generalization performance of a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The “instance margin” is defined as the distance between one instance and the separating hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Formally, for one instance i in the semantic space g (xi) and sufficiently many4 samples g (xj) (zi ̸= zj) drawn from well behaved class distributions5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We define the distance dij = ∥g (xi) − g (xj)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For instance i, we can obtain a set of distances Di = {dij, zj ̸= zi} with the minimal values ¯di⋆ = minDi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As shown in [62], the distribution for the minimal values of the margin distance is characterized by a Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Based on this finding, we can express the probability of g (x) being included in the boundary estimated by g (xi): ψ (g (x) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' g (xi)) = exp � − �∥g (x) − g (xi)∥ λi �κi� , (11) where κi and λi are Weibull shape and scale parameters obtained by fitting Di using Maximum Likelihood Estimate (MLE), which is summarized6 in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Equation (11) quan- titatively describes the margin of one specific class, probabilis- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In our experiments, we use all available training instances here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The well behaved indicates that the moments of the distribution should be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, Cauchy distribution is not well-behaved [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' codes released in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='com/xiaomeiyy/WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' tically, in our semantic embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (11) requires ψ (·) to be non-degenerate margin distribution, which is essentially guaranteed by Extreme Value Theorem [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Algorithm 1 EVT estimator by Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Input: Extreme values x1, · · · , xn Output: Estimated parameters ˆκ, ˆλ If n == 1: ˆκ = ∞, ˆλ = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Else: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Sort x1, · · · , xn to get x[1] ≥ · · · ≥ x[n] (where x[i] is the re-ordered value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Maximum likelihood estimator for κ: n � � xκ [i] log x[i] − xκ [n] log x[n] � � � xκ [i] − xκ [n] � = � log x[i] (12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (12), and numerically estimate ˆκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', using fzero function in MATLAB) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Compute ˆλ = �� � xˆκ [i] − xˆκ [n] � /n �1/ˆκ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Margin Distribution of Prototypes: Consider a class zi which in the embedding space is represented by a prototype uzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In accordance with above formalism, we can also assume sufficiently many samples g (xj) drawn from other (zi ̸= zj) well behaved class distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We can also consider the prototypes of vast open vocabulary uzj (zi ̸= zj, zj ∈ Wt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Under these assumptions, we can obtain a set of distances Duzi = { ��uzi − gzj �� , zj ̸= zi, gzj ∈ � g (xj) , uzj � } for the prototype uzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As a result, the distribution for the minimal values of the margin distance for uzi is given by a Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The probability that gzi is included in the bound- ary estimated by uzi is given by ψ (gzi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uzi) = exp � − � ∥gzi − uzi∥ λuzi �κuzi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (13) The above equation models the distribution of minimum value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' thus it can be used to estimate the boundary density (or more specifically, the boundary distribution) of class zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We set significant level to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05 to approximately esti- mate the minimal value ¯duzi⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As illustrated in Figure 2, if ψ (gzi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uzi) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05, we will assume gzi does not belong to the prototype uzi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' otherwise, gzi is included in the boundary estimated by uzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In term of the significant level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05, we can further denote the minimal values as ¯d(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', exp � − � ¯d(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi ⋆ λuzi �κuzi � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Thus we have ¯d(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ = λuzi · log1/κuzi � 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05 � (14) Coverage Distribution of Prototypes: Now, for class zi consider the nearest instance from another class g (xj) where zi ̸= zj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' with sufficient many instances g (xk) from class zi, we have pairwise unique ("within class") distance: cuzik = ∥g (xk) − uzi∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (15) Positiveinstances Negativeinstances ★ Prototypes id7 We consider outliers those instances g (xk) that have larger distance to uzi than the nearest instance g (xj) (zi ̸= zj) of another class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To remove the outliers we hence consider Cuzi = � cuzik|cuzik ≤ minzj̸=zk ∥g (xj) − uzi∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As illus- trated in Figure 2, we only consider positive instances within the orange circle and all other instances with larger distance are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Then the distribution of the largest distance ¯cuzi⋆ = max Cuzi will follow a reversed Weibull distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This allows us to get the probability distribution to describe positive instances, φ (g (xk) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uzi) = 1 − exp � � �− � ∥g (xk) − uzi∥ λ′uzi �κ ′ uzi � � � (16) where κ ′ i and λ ′ i are reverse Weibull shape and scale pa- rameters individually obtained from fitting the largest Cuzi , ¯cuzi⋆ is the distance between instance and prototype, φ is the probability that the instance is in the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Similar to the margin distribution, we can estimate the coverage by setting the significant level to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As shown in Figure 2, we establish two boundaries to estimate the scale of each class probabilistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' If φ (g (xk) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uzi) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05, g (xk) is included in the coverage distribution uzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The maximum values ¯c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ can be computed as φ (g (xk) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' uzi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' It results in, ¯c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ = λ′ uzi · log 1/κ ′ uzi � 1 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (17) By combining the terms computed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (14) and (17), we can obtain the weight wzi for class zi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (6), wzi ∝ � ¯d(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ + ¯c(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05) uzi⋆ � (18) As explained in Algorithm 1, we set ˆκ = ∞, ˆλ = x1 in one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In few-shot learning setting, we can estimate ˆκ and ˆλ directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In addition, such an initialization of weights (ˆκ and ˆλ) intrinsically helps learn the embedding weight W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The learning process of parameters: The process could be interpreted as a form of block coordinate descent where we estimate the embedding/mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' then density within that embedding and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, the weights wzi are initially randomized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' But they do not play an important role at the beginning of the optimization, since ���W T ⋆jxi − (uzi)j ��� is very large in the first few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In other words, the optimization is initially dominated by the data term and maximum margin terms play little role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, once we can get a relative good mapping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', smaller ���W T ⋆jxi − (uzi)j ���) after several training iterations, the weight wzi starts becoming significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' By virtue of such an optimization, the weighted version can achieve better performance than the previous non-weighted version in our conference paper [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Deep Weighted Maximum Margin Voc Embedding (Deep WMM-Voc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, we extend WMM-Voc to include a deep network for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Rather than extracting low-level features using an off-the-shelf pre-trained model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10), we use an integrated deep network to extract xi from the raw images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As a result, the loss function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10) is also used to optimize the parameters of the deep network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we fix the convolutional layers of corresponding network and fine-tune the last fully connected layer in our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The network was trained using stochastic gradient descendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4 EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Experimental setup We conduct our experiments on Animals with Attributes (AwA) dataset, and ImageNet 2012/2010 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' AwA dataset: AwA consists of 50 classes of animals (30, 475 images in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In [43] standard split into 40 source/auxiliary classes (|Ws| = 40) and 10 target/test classes (|Wt| = 10) is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We follow this split for supervised and zero- shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use ResNet101 features (downloaded from [54]) on AwA to make the results more easily comparable to state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ImageNet 2012/2010 dataset: ImageNet is a large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use 1000 (|Ws| = 1000) classes of ILSVRC 2012 as the source/auxiliary classes and 360 (|Wt| = 360) classes of ILSVRC 2010 that are not used in ILSVRC 2012 as target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use pre-trained VGG-19 model [12] to extract deep features for ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Recognition tasks: We consider several different settings in a variety of experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We first divide the two datasets into source and target splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On source classes, we can validate whether our framework can be used to solve one-shot and supervised recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' By using both the source and target classes, transfer learning based settings can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 1) SUPERVISED recognition: learning is on source classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' test instances come from the same classes with Ws as recognition vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, under this setting, we also validate the one- and few-shot recognition sce- narios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', classes have one or few training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2) ZERO-SHOT recognition: In ZSL, learning is on the source classes with Ws vocabulary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' test instances come from target dataset with Wt as recognition vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3) GENERAL-ZERO-SHOT recognition: G-ZSL uses source classes to learn, with test instances coming from either target Wt or original Ws recognition vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4) OPEN-SET recognition: Again source classes are used for learning, but the entire open vocabulary with |W| ≈ 310K atoms is used at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, test images come from both source and target splits (similar to G-ZSL);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' however, unlike G-ZSL there are additional distractor classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In other words, chance performance for open-set recognition is much lower than for G-ZSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We test both our Voc variants – MM-Voc and WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Additionally, we also validate the Deep WMM-Voc by fine- tuning the WMM-Voc on VGG-19 architecture and optimizing the weights with respect to the loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Competitors: We compare to a variety of the models in the literature, including: 1) SVM: SVM classifier trained directly on the training instances of source data, without the use of semantic embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This is the standard (SUPERVISED/ONE- SHOT) learning setting and the learned classifier can only 8 predict the labels within the source classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Hence, SVM is inapplicable in ZSL, G-ZSL, and open-set recognition settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2) SVR-Map: SVR is used to learn W and the recognition is done, similar to our method, in the resulting semantic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This corresponds to only optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3) Deep-SVR: This is a variant SVR, which further allows fine-tuning of the underlying neural network generating the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In this case, W is expressed as the last linear layer and the entire network is fine-tuned with respect to the loss encoding only the data term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4) SAE: SAE is a semantic encoder-decoder paradigm that projects visual features into a semantic space and then reconstructs the original visual feature representation [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The SAE has two variants in learning the embed- ding space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', semantic space to feature space (S→F), and feature space to semantic space (F→S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' By default, the best result of these two variants are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 5) ESZSL: ESZSL first learns the mapping between visual features and attributes, then models the relationship between attributes and classes [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 6) DeVise, ConSE, AMP: To compare with state-of-the-art large-scale zero-shot learning approaches we implement DeViSE [24] and ConSE [53]7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ConSE uses a multi- class logistic regression classifier for predicting class probabilities of source instances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' and the parameter T (number of top-T nearest embeddings for a given instance) was selected from {1, 10, 100, 1000} that gives the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ConSE method in supervised setting works the same as SVR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use the AMP code provided on the author webpage [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Metrics: Classification accuracies are reported as the eval- uation metrics on most of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In our conference version [29], we further introduce an evaluation setting for OPEN- SET tasks where we do not assume that test data comes from either source/auxiliary domain or target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Thus we split the two cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', SUPERVISED-like, and ZERO-SHOT-like settings), to mimic SUPERVISED and ZERO-SHOT scenarios for easier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Particularly, in G-ZSL task, this newly introduced evaluation setting is corresponding to the evaluation metrics defined in [85]: (1) S → T: Test instances from seen classes, the prediction candidates include both seen and unseen classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) U → T: Test instances from unseen classes, the prediction candidates include both seen and unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (3) The harmonic mean is used as the main evaluation metric to further combine the results of both S → T and U → T: H = 2·(Acc(U → T) × Acc(S → T)) (Acc(U → T) + Acc(S → T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (19) Setting of Parameters: For the recognition tasks, we learn classifiers by using various number of training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We compare relevant baselines with results of our method variants: MM-Voc, WMM-Voc, Deep WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Each setting is repeated/tested 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The averaged results are reported to reduce the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For each setting, our Voc methods are trained by a single model to be capable of solving the tasks 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Codes for [24] and [53] are not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' of supervised, zero-shot, G-ZSL and open-set recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, 1) In Deep WMM-Voc, we fix λ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='01 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='6 with the learning rate initially set to 1e−5 and is reduced by 1 2 every 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' AV and BS are set to 5 in order to balance performance and computational cost of pairwise constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 2) To solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10) at a scale, one can use Stochastic Gradient Descent (SGD) which makes great progress ini- tially, but often is slow when approaching convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In contrast, the L-BFGS method mentioned above can achieve steady convergence at the cost of computing the full objective and gradient at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' L-BFGS can usually achieve better results than SGD with good initialization, however, is computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To leverage benefits of both of these methods, we utilize a hybrid method to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10) in large-scale datasets: the solver is initialized with few instances to approx- imate the gradients using SGD first;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' then gradually more instances are used and switch to L-BFGS is made with iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This solver is motivated by Friedlander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [23], who theoretically analyzed and proved the convergence for the hybrid optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In practice, we use L-BFGS and the Hybrid algorithms for AwA and ImageNet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The hybrid algorithm can save between 20 ∼ 50% training time as compared with L-BFGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Open set vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use Google word2vec to learn the open set vocabulary set from a large text corpus of around 7 billion words: UMBC WebBase (3 billion words), the latest Wikipedia articles (3 billion words) and other web documents (1 billion words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Some rare (low frequency) words and high frequency stopping words were pruned in the vocabulary set: we remove words with the frequency < 300 or > 10 million times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The result is a vocabulary of around 310K words/phrases with openness ≈ 1, which is defined as openness = 1 − � (2 × |Ws|) / (|W|) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 Experimental results on AwA dataset 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Learning Classifiers from Few Source Training Instances We are particularly interested in learning of classifiers from few source training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This is inclined to mimic human performance of learning from few examples and illustrate ability of our model to learn with little data8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We show that, our vocabulary-informed learning is able to improve the recognition accuracy on all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' By only using 200 training instances, we report the results on standard supervised (on source classes), zero-shot (on target classes), and generalized zero-shot recognition (both on source and target classes) as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that for ZSL and G-ZSL, our settings is a more realistic and yet 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As for feature representations, the ResNet100 features from [54] are trained from ImageNet 2012 dataset, which potentially have some overlapped classes with AwA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 9 Methods S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Sp Features Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' WMM-Voc W CNNresnet101 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='79 WMM-Voc: closed W CNNresnet101 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='51 Deep WMM-Voc W CNNresnet101 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65 Deep WMM-Voc: closed W CNNresnet101 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='85 SAE W CNNresnet101 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='42 ESZSL W CNNresnet101 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='17 Deep-SVR W CNNresnet101 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='22 Akata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [3] A+W CNNGoogleNet 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90 TMV-BLP [25] A+W CNNOverFeat 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90 AMP (SR+SE) [31] A+W CNNOverFeat 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='00 PST [58] A+W CNNOverFeat 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='10 Latem [83] A+W CNNresnet101 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='80 SJE [3] A+W CNNresnet101 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 DeViSE [24] W CNNresnet101 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90 ConSE [53] W CNNresnet101 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='60 CMT [68] W CNNresnet101 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90 SSE [91] W CNNresnet101 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 SSE [91] W CNNVGG19 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='49 TASTE [87] W CNNVGG19 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='40 KLDA+KRR [48] W CNNGoogleNet 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='30 CLN+KRR [48] W CNNVGG19 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='00 UVDS [50] W CNNVGG19 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='88 DEM [10] W CNNInception-V2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 DS [60] W/A CNNOverFeat 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 SYNC [9] W/A CNNresnet101 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 Relation Net [69] A CNNInception-V2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 ESZSL [61] A CNNresnet101 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 UVDS [50] A CNNGoogleNet 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='28 GFZSL [78] A CNNVGG19 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 DEM [10] A CNNInception-V2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='80 SE-GZSL [77] A CNNVGG19 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 cycle-CLSWGAN [21] A CNNresnet101 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='30 f-CLSWGAN [84] A CNNresnet101 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 PTMCA [47] A CNNresnet101 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 Jayaraman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [36] A low-level 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 DAP [43] A CNNVGG19 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 DAP [43] A CNNresnet101 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='10 DAP [43] A CNNOverFeat 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 ALE [2] A CNNresnet101 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='60 Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' [86] A low-level 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='30 IAP [43] A CNNOverFeat 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 HEX [14] A CNNDECAF 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 AHLE [2] A low-level 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50 TABLE 1 Zero-shot comparison on AwA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We compare the state-of-the-art ZSL results using different semantic spaces (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Sp) including word vector (W) and attribute (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 1000 dimension word2vec dictionary is used for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (Chance-level =10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Different types of features are used by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' WMM-Voc: closed and Deep WMM-Voc: closed are the two variants of our model obtained by learning the vocabulary-informed constraints only from known classes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', closed set), similar to our conference version [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ZSL results by 100-d word vector 40 200 800 1600 12139 24295 Training Instance Number 0 20 40 60 80 100 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSL ZSL results by 1000-d word vector 40 200 800 1600 12139 24295 Training Instance Number 0 20 40 60 80 100 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The ZSL results on AwA by different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' more challenging than those in previous methods [38], [61], since the source classes have few training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We also compare using 100/1000-dimensional word2vec representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', d = 100/1000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Both the Top-1 and Top-5 classification accuracy is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that the key novelty of our WMM- Voc comes from directly estimating the density of source train- ing classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Such an approach would be helpful in alleviating the hubness problem and should lead to better performance in zero-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As shown in Table 2, the improvement from MM-Voc to WMM-Voc and then further to Deep WMM- Voc validate this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We highlight the following observations: (1) Deep WMM- Voc achieves the best zero-shot learning accuracy compared with the other state-of-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' It is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='45% and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='02% higher than SAE and ESZSL respectively on Top-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our WMM-Voc can still beat the state-of-the-art SAE and ESZSL by outperforming 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='67% and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='24% individually on Top-1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) In supervised learning task, the ESZSL and Deep WMM-Voc have almost the same performance, if we consider the variances in sampling the 200 training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our WMM-Voc is slightly better than these two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (3) In G-ZSL setting, our two models get significantly better performance compared with the other competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Notably, the Top-1 accuracy of SAE and ESZSL is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' While Deep WMM-Voc and WMM-Voc both have higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This shows the effectiveness of our two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (4) As expected, the deep models that fine-tune features along with classifiers (Deep-SVR and Deep WMM-Voc) are better than counterparts with pre-extracted representations (SVR-Map, WMM-Voc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 Results on different training/testing splits We conduct experiments using different number of training instances and compare results on tasks of supervised, zero- shot and generalized zero-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On each split, we use both 100 and 1000 dimensional word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use 12,156 testing instances from source classes in supervised, G-ZSL and open-set setting as well as 6,180 testing instances from 10 Dimension SVR-Map Deep-SVR SAE ESZSL MM-Voc WMM-Voc Deep WMM-Voc Supervised 100-dim 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4/- 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='59/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='98 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='22/92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='60 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='86/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='85 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='01/87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='88 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='57/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='31 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='23/94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='85 1000-dim 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1/- 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='32/95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='22 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='76 TABLE 2 Classification accuracy (Top-1 / Top-5) on AwA dataset for SUPERVISED, GENERAL ZERO-SHOT and ZERO-SHOT settings for 100-dim and 1000-dim word2vec representation (200 instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' target classes in zero-shot, G-ZSL and open-set setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' All the competitors are using the same types of features – ResNet101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Supervised learning: The results are compared in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As shown in the figure, we observe that our method shows significant improvements over the competitors in few-shot setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' however, as the number of instance increasing, the visual semantic mapping, g(x), can be well learned, and the effects of additional vocabulary-informed constraints, become less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zero-shot learning: The ZSL results are compared in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On all the settings, our two Voc methods – Deep WMM- Voc and WMM-Voc outperforms all the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This validates the importance of information learning from the open vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Further, we compare our results with the state-of-the-art ZSL results on AwA dataset in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our two models achieve 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='79% and 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65% accuracy, which is markedly higher than all previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This is par- ticularly impressive, if we take into account the fact that we use only a semantic space and no additional attribute representations (unlike many other competitor methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We argue that much of our success and improvement comes from a more discriminative information obtained using the open set vocabulary and corresponding large margin constraints, rather than from the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Varying the number of training instances may slightly affect accuracy of methods reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Therefore we report the best results of each competitor and our own method at different number of train- ing instances {200, 800, 1600, 121319, 24295}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='All competing methods in Figure 3 use the same features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' General zero-shot learning:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The general zero-shot learning results are compared in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We consider the accuracies of both U → T (ZERO-SHOT-like) and S → T (SUPERVISED-like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In term of the harmonic mean H, our methods have signifi- cantly better performance in the general zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This again shows that our framework can have better generalization by learning from the open vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On the other hand, in the terms of Area Under Seen-Unseen accuracy Curve (AUSUC), the performance of Deep-SVR is very weak and the scores of ESZSL and SAE are lower than our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Overall, the results of AUSUC still support the superiority of our methods on G-ZSL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Notably, since the source domain only have 24295 instances (including training and testing images), we are unable to obtain the results of SUPERVISED-like setting (S → T), H and AUSUC with all source instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Supervised results by 100-d word vector 200 800 1600 12139 Training Instance Number 0 20 40 60 80 100 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSL Supervised results by 1000-d word vector 200 800 1600 12139 Training Instance Number 0 20 40 60 80 100 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Supervised learning results of AwA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Large-scale open set recognition We also compare the results on OPEN-SET310K setting with the large vocabulary of approximately 310K entities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' as such the chance performance is much lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use 100-dim word vector representations as the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' While our OPEN- SET variants do not assume that test data comes from either source/auxiliary domain or target domain, we split the two cases to mimic SUPERVISED and ZERO-SHOT scenarios for easier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The results are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On SUPERVISED-like setting, Figure 5 (left), our Deep WMM-Voc and WMM-Voc have better performance than the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The better results are largely due to the better embedding matrix W learned by enforcing maximum margins between training class name and open set vocabulary on source training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This validates the effectiveness of proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we find that (1) The “deep” version always has better performance than their corresponding “non- deep” counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, the Deep-SVR and Deep WMM-Voc achieve higher open-set recognition accuracy than SVR-Map and WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) The WMM-Voc has better performance than MM-Voc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' this shows that the weighting strategy introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4 can indeed help better learn 11 TABLE 3 The G-ZSL results (100-dim/1000-dim) of AwA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We compare the results by varying the number of training instances (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='of Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=') of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Where H is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (19) without calibrated stacking, AUSUC means the Area Under Seen-Unseen accuracy Curve with calibrated stacking and ’-’ represents the unavailable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Metrics ESZSL SAE Deep-SVR WMM-Voc Deep WMM-Voc 200 U → T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='88/0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='15/0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65/0 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='92/27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='98 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='04/34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='77 S → T 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3776/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5016/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5554/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5733 12139 U → T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='37/0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='44/0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='19/0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='80/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='53 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='23/28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='19 S → T 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='98/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='16 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='18/91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='37/85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='64 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='36/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='34 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='64/78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='32 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='74/0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='88/0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='79/0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90/43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='94 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05/41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='46 AUSUC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5096/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4493/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3353/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5144/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5525/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5394 24295 U → T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='83/0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='37/0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='39/0 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='15/29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='42 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65/31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='78 S → T H AUSUC the embedding from visual to semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On ZERO SHOT-like setting, our method still has a notable advantage over that of SVR-Map, Deep-SVR methods on Top- k (k > 3) accuracy, again thanks to the better embedding W learned by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, we notice that our top- 1 accuracy on ZERO SHOT-like setting is lower than Deep SVR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We find that our method tends to label some instances from target data with their nearest classes from within source label set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, “humpback whale” from testing data is more likely to be labeled as “blue whale”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, when considering Top-k (k > 3) accuracy, our method still has advantages over baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' It suggests that the semantic embeddings may be suffering from the problem that density of source classes is more concentrated than that of target classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To show the effectiveness of WMM-Voc, as opposed to MM-Voc, we employ the False Positive Rate as the metric, rfp = Ne/Nun, where Ne means the number of testing unseen instances predicted as seen ones and Nun defines the number of testing unseeen instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Experiments are conducted on AwA dataset with all training instances, and 100- dim word vector prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The false positive rates are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='16, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='06 by using SVR, Deep-SVR, MM-Voc, WMM-Voc and Deep WMM-Voc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' They further validates that WMM-Voc outperforms MM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Experimental results on ImageNet dataset We validate our methods on large-scale ImageNet 2012/2010 dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' the 1000-dimensional word2vec representation is used here since this dataset has larger number of classes than AwA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Testing Classes AwA dataset Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Targ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Total Vocab OPEN-SET310K (left) (right) 40 / 10 310K 0 5 10 15 20 Hit@k 50 55 60 65 70 75 80 85 90 95 100 Accuracy (%) SUPERVISED-like(Aux) 0 5 10 15 20 Hit@k 0 10 20 30 40 50 60 70 Accuracy (%) ZERO SHOT-like(Tag) SVR-Map MM-Voc Deep-SVR Deep WMM-Voc WMM-Voc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Openset results of AwA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use 1600 training instances equally sampled from all source classes to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The instances of testing classes are equally sampled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' making experiment less sensitive to the problem of unbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' To be specific, 50×1, 000 testing instances from source classes are used in supervised, G-ZSL and open-set setting as well as 12 TABLE 4 The classification accuracy (Top-1 / Top-5) of ImageNet 2012/2010 dataset on ZERO-SHOT and SUPERVISED settings using 3000 source training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Settings SVR-Map Deep-SVR ESZSL SAE MM-Voc WMM-Voc Deep WMM-Voc Supervised 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='6/– 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='51 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26/64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='38 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='95/54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='44 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='35 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='95/62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='77 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='92/65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='35 Zero-shot 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1/– 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='29/13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='86/13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='11/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90/14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='50/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='73 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='99 360 × 100 testing instances from target classes are used in zero-shot, G-ZSL and open-set setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The VGG-19 features of ImageNet pre-trained network are utilized as the input of all algorithms to make a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We employ the Deep-SVR, SAE, ESZSL as baselines under the SUPERVISED, ZERO-SHOT and GENERAL ZERO-SHOT settings respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Pseudo-few-shot Source Training instances The standard few-shot learning assumes disjoint instance set on source and target domains, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' As an ablation study, we would like to simulate a few-shot- like learning task on source domain by slightly violating the standard few-shot learning assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We name this setting “Pseudo-few-shot learning”: only few source training instances are used here and the feature extractor – VGG- 19 model is pre-trained on ILSVRC 2012 dataset [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The “Pseudo-” here indicates that large amount of instances are used to train the feature extractor, but not used in training classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Thus the experiments in this section can be served as an additional ablation study to reveal the insights of our model in addressing the few-shot-like task on source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Particularly, we conduct the experiments of using few-shot source training instance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', 3,000 training instances used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The results are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We introduce this setting to particularly focus on learning from few training samples per class, in order to mimic human capability and performance in learning from few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We show that our vocabulary-informed learning framework enables learning with little data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we highlight that the Top-5 performance of WMM-Voc is much higher (>5%) than that of MM-Voc, despite the slightly worse performance on Top- 1 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note that the degradation of Top-1 results on ImageNet is also understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Note, WMM-Voc is only fitting the 3000 training instances on ImageNet dataset, and the features of these training instances may not be fine- tuned/optimized for the newly introduced penalty term of WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Once the features of training instances are fine- tuned by the deep version;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' we can show that the Deep WMM- Voc can improve from MM-Voc and WMM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Critically, with different settings in Table 4, our vocabulary- informed learning can beat the other baselines under all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We highlight several findings: (1) The supervised performance of our methods stands out from the state-of-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Specifically, our Deep WMM-Voc achieves the highest supervised recognition accuracy, with ESZSL following closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' SVR-Map appears to be the worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (2) On Zero-shot learning task, our proposed Deep WMM- Voc gets 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26% Top-1 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='99% Top-5 accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' It outper- forms all the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Comparing with our previous Supervised Learning results of ImageNet 3k 5k 10k 20k 50k Training Instance Number 0 20 40 60 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSl Zero-shot Learning results of ImageNet 3k 5k 10k 20k 50k Training Instance Number 0 2 4 6 8 10 Top1 Accuracy(%) WMM-Voc Deep WMM-Voc Deep-SVR SAE(F->S) SAE(S->F) ESZSL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The supervised and zero-shot learning results on ImageNet 2012/2010 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' MM-Voc result in [29], our result is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='36% higher than MM- Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This improvement is statistically significant due to the few number of training instances and large number of testing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Additionally, the Top-1 result of WMM-Voc is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5% which is also comparable to that of MM-Voc, and far higher than those of SVR, Deep-SVR, ESZSL and SAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This validates the effectiveness of learning from open vocabulary proposed in our two variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' (3) In G-ZSL setting, we observe that both Deep WMM- Voc and WMM-Voc outperform all the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The full set of experiments on G-ZSL under different settings are reported in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 Few-shot Target Training instances We further introduce few-shot learning experiments on target instances to validate the performance of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The experiments are conducted on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In total, there are 360 target classes from ImageNet 2010 data split with 100 instances per class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' the feature extractor – VGG-19 is trained on the 1000 classes from ImageNet 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The 1 or 3 training instances are sampled from each target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The other instances of the target classes are utilized as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This is the few-shot learning setting, which is consistent with general definition [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We compare to SVM, KNN, Deep SVR, and SAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The results are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We can see that our method (WMM-Voc) can beat all the other competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Particularly, we have an obvious advantage in 1- shot target setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our Deep variant (Deep WMM-Voc) has 13 TABLE 5 The G-ZSL results (1000-dim) of ImageNet datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We compare the results of using different number of training instances (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=') D-SVR, W-V and D-W-V indicate Deep SVR, WWM-Voc, and Deep WWM-Voc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Metrics ESZSL SAE D-SVR W-V D-W-V 3000 U → T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='93 S → T 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='07 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='86 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='06 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='40 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='61 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='67 10000 U → T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='99 S → T 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='23 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='54 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='87 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='53 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='36 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='81 50000 U → T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='15 S → T 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='43 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='55 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='75 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='16 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='28 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='11 TABLE 6 Results of few-shot target training instances on ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Method 1-instance 3-instance SVM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='81 KNN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='23 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Deep SVR 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='01 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='00 SAE 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='93 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='42 WMM-Voc 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='59 Deep WMM-Voc 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='95 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='44 better performance both in 1- and 3-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This shows the efficacy of proposed methods in few-shot learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 Results on different training/testing splits We further validate our findings on ImageNet 2012/2010 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In general, our framework has advantages over the baselines since open vocabulary helps inform the learning process when few training instances or limited training data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The results are compared in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Supervised learning: As shown in Figure 6, we compare the supervised results by increasing the training instances from 3,000 to 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' With 3,000 training instances, the results of Deep WMM-Voc are better than all the other baselines with the help of learning from free vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We further evaluate our models with larger number of training instances (> 3 per class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We observe that for standard supervised learning setting, the improvements achieved using vocabulary-informed learning tend to somewhat diminish as the number of training instances substantially grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' With large number of training instances, the mapping between low-level image features and semantic words, g(x), becomes better behaved and effect of additional constraints, due to the open-vocabulary, becomes less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zero-shot Learning: We further validate the results on zero- shot learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Figure 6 shows that our models can beat all other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our Deep WMM-Voc always performs the best with the source training instances increased from TABLE 7 ImageNet comparison to state-of-the-art on ZSL: We compare the results of using 3, 000/all training instances for all methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' T-1 (top 1) and T-5 (top 5) classification in (%) is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The VGG-19 features are used for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Methods S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Sp T-1 T-5 Deep WMM-Voc W 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='29 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='99/23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='12 WMM-Voc W 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='76 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='30/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='36 MM-Voc W 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='9/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='9/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='8 SAE W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='11/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='32 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='26/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='04 ESZSL W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='86/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='71/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 Deep-SVR W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='29/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='32/14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='12 Embed [88] W –/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='00 –/25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='70 ConSE [53] W 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1/15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5 DeViSE [24] W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='7/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='8/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='8 AMP [31] W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5/13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='1 Chance – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='78e-3 – 3,000 to 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The WMM-Voc always has the second best performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' especially when only few source training instances are available, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=', 3,000 and 5,000 training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our Deep WMM-Voc and WMM-Voc demonstrate significant improvements over the competitors in ZSL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The good performance of Deep WMM-Voc and WMM-Voc is largely due to our vocabulary-informed learning framework which can leverage the discriminative information from open vocabulary and max-margin constraints, helping to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' General Zero-shot Learning: In G-ZSL, our methods still have the best performance compared to the baselines, as seen from Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The Top-1 results of WMM-Voc and Deep WMM-Voc are beyond 2%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' in contrast, the performance of other state-of-art methods are lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Varying training set size: In Figure 6 we also evaluate our model with the larger number of training instances (> 3 per class) in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The results are inline with prior findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The state-of-the-art on ZSL: We compare our results to sev- eral state-of-the-art large-scale zero-shot recognition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Our results are better than those of ConSE, DeViSE, Deep- SVR, SAE, ESZSL and AMP on both T-1 and T-5 metrics with a very significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Poor results of DeViSE with 3, 000 training instances are largely due to the inefficient learning of visual-semantic embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' AMP algorithm also relies on the embedding matrix from DeViSE, which explains similar poor performance of AMP with 3, 000 training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Table 7 shows that our Deep WMM-Voc obtains good performance with (all) 50,000 training instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Top-5 accuracy of our methods are beyond 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' This again validates that our proposed methods can have the advantages of learning from limited available training instances by leveraging the discriminative information from open vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Embed [1] has slightly better ZSL performance compared to our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' However, unlike the other works that directly use word vector representations of class names, [1] require additional textual descriptions of each class to learn better class prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Open-set recognition: The open set image recognition results 14 Testing Classes ImageNet Data Aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Total Vocab OPEN-SET310K (left) (right) 1000 / 360 310K 0 5 10 15 20 Hit@k 5 10 15 20 25 30 Accuracy (%) SUPERVISED-like(Aux) 0 5 10 15 20 Hit@k 0 1 2 3 4 5 6 Accuracy (%) ZERO SHOT-like(Tag) MM-Voc Deep-SVR Deep WMM-Voc WMM-Voc Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Open set recognition results on ImageNet 2012/2010 dataset: Openness=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='9839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Chance=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2e − 4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use the synsets of each class— a set of synony- mous (word or prhase) terms as the ground truth names for each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We use the model trained with 50,000 instances sampled equally from source classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On SUPERVISED-like settings, we notice the MM-Voc and WMM-Voc have similar open set recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Since this dataset is very large, linear mapping g(x) may not have enough capacity to model the embedding mapping from visual space to semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Thus adding constraints on source training classes in WMM-Voc may slightly hinder the learning such an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' That explains why the results of WMM-Voc are slightly inferior to MM-Voc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Deep WMM-Voc has the best performance, due to its ability to fine-tune low-level feature representation while learning the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' On the ZERO-SHOT-like setting, our WMM-Voc and Deep WMM-Voc have the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Qualitative visualization: We illustrate the embedding space learned by our Deep WMM-Voc model for the Ima- geNet2012/2010 dataset in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In particular, we have 4 source/auxiliary and 2 target/zero-shot classes in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The better separation among classes is largely attributed to open-set max-margin constraints introduced in our vocabulary- informed learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We further visualize the semantic space in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Critically, we list seven target classes on AwA dataset, as well as their surrounding neighbor- hood open vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' For example, “orcas” is very near to “killer_whale”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' While “orcas” are semantically different from “killer_whale”, the difference is much smaller if we compare the “orcas” with the other classes, such as “spider monkey”, “grizzly_bear” and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Hence the “orcas” can be used to help learn the class of “killer_whale” in our vocabulary- informed learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='killer_whale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='chihuahua_dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='dalmatian_dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='buffalo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='grizzly_bear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='collie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='spider_monkey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='Word prototype: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='golden_retriever ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='collies ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='fox_terrier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='sheepdog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='shetland_pony ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='jack_russell_terrier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='cattle_dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='wellard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='drover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='collie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='killer_whale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='tilikum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='orcas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='brancheau ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='orca ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='seaworld ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='bottlenose_dolphin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='sea_lion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='whale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='seaworld_orlando ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='spider_monkey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='capybara ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='giant_anteater ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='marmoset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='tamarin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='macaw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='howler_monkeys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='squirrel_monkeys ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='macaque ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='macaws ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='grizzly_bear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='grizzly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='polar_bear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='elk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='grizzly_bears ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='coyote ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='caribou ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='mountain_lion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='mountain_goats ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='bighorn_sheep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='buffalo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='minneapolis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='detroit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='chicago ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='kansas_city ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='pittsburgh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='erie ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='duluth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='minnesota ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='grand_rapids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='chihuahua_dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='ciudad_ju_rez ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='sonora ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='sinaloa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='coahuila ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='durango ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='jalisco ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='michoac_n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='zacatecas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='nuevo_leon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='dalmatian_dog ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='istria ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='ragusa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='dalmatian_coast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='kor_ula ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='gradisca ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='sardinian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='zadar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='croatian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='istrian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Visualization of the semantic space: We show the t-SNE visualization of the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The words in boxes are the mapping of training image in the semantic space, and close neighbors are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' The neighborhoods extend the single training data to a space semantically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 5 CONCLUSION AND FUTURE WORK This paper introduces the learning paradigm of vocabulary- informed learning, by utilizing open set semantic vocabulary to help train better classifiers for observed and unobserved classes in supervised learning, ZSL, G-ZSL, and open set image recognition settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' We formulate vocabulary-informed learning in the maximum margin frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Extensive ex- perimental results illustrate the efficacy of such learning paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Strikingly, it achieves competitive performance with very few training instances and is relatively robust to a large open set vocabulary of up to 310, 000 class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported in part by NSFC Project (61702108, 61622204), STCSM Project (16JC1420400), Eastern Scholar (TP2017006), Shanghai Municipal Science and Technol- ogy Major Project (2017SHZDZX01, 2018SHZDZX01) and ZJLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' REFERENCES [1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='3 [2] Z.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='2 [92] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Large margin distribution learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' In IAPR Workshop on Artificial Neural Networks in Pattern Recognition, pages 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='4 Yanwei Fu received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' degree from Queen Mary University of London in 2014, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' degree from the Department of Com- puter Science and Technology, Nanjing Univer- sity, China, in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He held a post-doctoral po- sition at Disney Research, Pittsburgh, PA, USA, from 2015 to 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is currently a tenure-track Professor with Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' His research interests are image and video understanding, and life-long learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' 17 Xiaomei Wang is a PhD student in the School of Computer Science of Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' She re- ceived the Master degree of communication and information system from Shanghai University in 2016 and the Bachelor degree of electronic infor- mation engineering from Shandong University of Technology in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Her reaseach interests in- clude zero-shot/few-shot learning, image/video captioning and visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Hanze Dong is an undergraduate student ma- joring in mathematics (data science track) at the School of Data Science, Fudan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He works in Shanghai Key Lab of Intelligent Information Processing under the supervision of Professor Yanwei Fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' His current research interests include both machine learning theory and its applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Yu-Gang Jiang is Professor of Computer Sci- ence at Fudan University and Director of Fudan- Jilian Joint Research Center on Intelligent Video Technology, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is interested in all aspects of extracting high-level informa- tion from big video data, such as video event recognition, object/scene recognition and large- scale visual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' His work has led to many awards, including the inaugural ACM China Ris- ing Star Award, the 2015 ACM SIGMM Rising Star Award, and the research award for out- standing young researchers from NSF China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is currently an as- sociate editor of ACM TOMM, Machine Vision and Applications (MVA) and Neurocomputing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He holds a PhD in Computer Science from City University of Hong Kong and spent three years working at Columbia University before joining Fudan in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Meng Wang is a professor at the Hefei Univer- sity of Technology, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' degree and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' degree in the Special Class for the Gifted Young and the Department of Electronic Engineering and Information Science from the University of Science and Technology of China (USTC), Hefei, China, in 2003 and 2008, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' His current research interests in- clude multimedia content analysis, computer vi- sion, and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He has authored more than 200 book chapters, journal and con- ference papers in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is the recipient of the ACM SIGMM Rising Star Award 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is an associate editor of IEEE Transactions on Knowledge and Data Engineering (IEEE TKDE), IEEE Transactions on Circuits and Systems for Video Technology (IEEE TCSVT), IEEE Transactions on Multimedia (IEEE TMM), and IEEE Transactions on Neural Networks and Learning Systems (IEEE TNNLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Xiangyang Xue received the BS, MS, and PhD degrees in communication engineering from Xi- dian University, Xi’an, China, in 1989, 1992, and 1995, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is currently a professor of computer science with Fudan University, Shang- hai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' His research interests include com- puter vision, multimedia information processing and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Leonid Sigal is an Associate Professor in the Department of Computer Science at the Univer- sity of British Columbia and a Faculty Member of the Vector Institute for Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He is a recipient of Canada CIFAR AI Chair and NSERC Canada Research Chair (CRC) in Computer Vision and Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Prior to this he was a Senior Research Scientist at Disney Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He completed his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' at Brown University in 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' from Boston University in 1999, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' from Brown University in 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Leonid’s research interests lie in the areas of computer vision, machine learning, and computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' Leonid’s research emphasis is on machine learning and statistical approaches for visual recognition, reasoning, understanding and analytics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} +page_content=' He has published more than 70 papers in venues and journals in these fields (including TPAMI, IJCV, CVPR, ICCV and NeurIPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf'} diff --git a/6NE4T4oBgHgl3EQf1g37/content/tmp_files/2301.05292v1.pdf.txt b/6NE4T4oBgHgl3EQf1g37/content/tmp_files/2301.05292v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2638f67d21e2100401cda7957fd3def374b6b8d5 --- /dev/null +++ b/6NE4T4oBgHgl3EQf1g37/content/tmp_files/2301.05292v1.pdf.txt @@ -0,0 +1,659 @@ +A Novel Framework for Handling Sparse Data in Traffic Forecast +Nikolaos Zygouras +Huawei Amsterdam Research Center +Netherlands +nikolas.zygouras@huawei.com +Dimitrios Gunopulos +National and Kapodistrian University of Athens +Greece +dg@di.uoa.gr +ABSTRACT +The ever increasing amount of GPS-equipped vehicles provides in +real-time valuable traffic information for the roads traversed by +the moving vehicles. In this way, a set of sparse and time evolving +traffic reports is generated for each road. These time series are a +valuable asset in order to forecast the future traffic condition. In +this paper we present a deep learning framework that encodes the +sparse recent traffic information and forecasts the future traffic con- +dition. Our framework consists of a recurrent part and a decoder. +The recurrent part employs an attention mechanism that encodes +the traffic reports that are available at a particular time window. +The decoder is responsible to forecast the future traffic condition. +CCS CONCEPTS +• Information systems → Data stream mining; Location based +services; Geographic information systems. +KEYWORDS +travel time estimation, traffic forecasting, deep learning, trans- +former, GPS trajectories, mining mobility data +ACM Reference Format: +Nikolaos Zygouras and Dimitrios Gunopulos. 2022. A Novel Framework +for Handling Sparse Data in Traffic Forecast. In The 30th International +Conference on Advances in Geographic Information Systems (SIGSPATIAL +’22), November 1–4, 2022, Seattle, WA, USA. ACM, New York, NY, USA, 4 pages. +https://doi.org/10.1145/3557915.3560968 +1 +INTRODUCTION +In recent years, the wide usage of mobile devices and the corre- +sponding collection of vast amounts of spatiotemporal data have +resulted in the development of various novel Location Based Ser- +vices (LBS). The LBS are software services that integrate geographic +information providing appropriate services and information to the +users [7]. Traffic forecasting and travel time estimation are un- +doubtedly two of the widely used LBS and a lot of recent research +work has been conducted towards improving their performance. +The importance of such services is indicated by the fact that the +vast majority of drivers consults several times a day services that +Part of this work was done while N. Zygouras was at the National and Kapodistrian +University of Athens, Greece. +Permission to make digital or hard copies of part or all of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for third-party components of this work must be honored. +For all other uses, contact the owner/author(s). +SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA +© 2022 Copyright held by the owner/author(s). +ACM ISBN 978-1-4503-9529-8/22/11. +https://doi.org/10.1145/3557915.3560968 +Porto +Airport +Estádio do +Dragão +Timestamp now +? +? +Historical Data +Predictions +Travel +Times +... +... +? +r1 +r2 +Porto +Airport +Estádio +do +Dragão +r|P | +q +Figure 1: The travel time estimation problem for a given +query path 𝑃𝑞 (blue line) and time of departure 𝑡𝑞 in the city +of Porto, that starts at 10:00 from the airport of Porto and +ends at the Estádio do Dragão, the entire path is decomposed +by a set of |𝑃𝑞| road segments 𝑟1 → 𝑟2 → · · · → 𝑟 |𝑃 | and for +each road segment we have a time series of travel time re- +ports, received by the available probe vehicles. +perform travel time estimation in order to appropriately choose the +fastest route to follow. +Motivated by this, in this paper we propose a novel path based +travel time estimation technique that considers the available traffic +reports that have been received by the set of the available probe +vehicles. Each probe vehicle moves in the road network and reports +the time that was required to traverse each individual road segment. +In this way, for each road segment of the road network a time series +of the reported travel times are generated, illustrated at the right +part of Figure 1. Our technique receives a query path along with +a time of departure and estimates the time of arrival considering +the current traffic condition of the road network. Our problem is +illustrated in Figure 1. A query path 𝑃𝑞 and a time of departure 𝑡𝑞 +are received as input and the task is to estimate the time that is +required to traverse the whole path 𝑃𝑞 if the driver departs at 𝑡𝑞. +We propose a novel deep learning framework which is comprised +of a recurrent part and a decoder. The recurrent part encodes the +sparse traffic reports that are available at each time window using +an attention mechanism and an embedding representations for +each road segment. The decoder is responsible to forecast the traffic +condition of the next time window. +2 +RELATED WORK +In DeepGTT the travel time distribution for any route is learnt by +conditioning on the real-time traffic [5]. Initially, an embedding is +arXiv:2301.05292v1 [cs.LG] 12 Jan 2023 + +12:30Aveleda +Casteloda +Maia +Lavra +Gondim +EN542 +EN13 +Mioue +Silva Escura +Vila Novada +Barca +EN105-2 +Telha +Moreira +A28 +SaoPedro +Fins +160m +NTo7Maig-Este,Vermoim +PortoViaNorte) +A3Porto.Braqo +Alfen +Vermoim +Nogueira +ZonoIndustrial +A41 +Maiat +deifena +A.41 +OLIPORI +A3 +Perafita +A41 /Moia /(AE) Broga/A42 Felgueiras +EN14 +Milheiros +Santa Cruz +Gueifaes +Ermesinde +doBispo +Refinaria +Balio +deMatosinhos +EN15-1 +VILPL +dstoias +Aguas Santas +A4 +Guifoes +Sao Mamede +deInfesta +Pedroucos +Baguimdo +Senhora da +Monte +Hora +34m +Matosinhos +EN12 +RioTinto +Paranhos +EM612 +EN12 +Pargue +Aldoar +daCidode +Ranalde +EN15 +Bogvista +Fanzeres +Nevogilde +tecedc +For +Fan +Cedofeita +Areias/W12Circunva/acao +Lordelo.do +Campanttransavia +F-GZHUSUPERBOCKSIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA +Zygouras et al. +estimated for each link considering its characteristics, then a non- +linear factorization model generates the speed and finally an atten- +tion mechanism is used to generate the observed travel time. Also, +in HETETA [3] the road map is translated into a multi-relational +network, considering the traffic behavior patterns. Temporal and +graph convolutions are used in order to learn spatiotemporal het- +erogeneous information, considering recent, daily and weekly traf- +fic. CompactETA [2] provides an accurate ETA estimation with +low latency. Graph attention network was employed in order to +encode spatial and temporal dependencies of the weighted road +network and the sequential information of the route is encoded +with positional encoding. A multi-layer perceptron was used for +online inference. The authors in [6] proposed a multitask represen- +tation learning model which predicts the travel time of an origin- +destination pair extracting a representation that preserves trips +properties and road network structure. ConSTGAT [1] proposed +a spatiotemporal graph neural network exploiting the spatial and +temporal information with a 3D-attention mechanism and a model +with convolutions over local windows in order to capture route’s +contextual information. STGNN-TTE [4] adopted a spatial–temporal +module to capture the real-time traffic condition and a transformer +layer to estimate the links’ travel time and the total routes’ travel +time synchronously. +3 +OUR APPROACH +3.1 +Problem Definition +Road Network is represented as a directed graph 𝐺(𝑉, 𝐸), where +the nodes 𝑉 represent the junctions and the edges 𝐸 represent the +|𝐸| roads segments. A road segment 𝑟 ∈ 𝐸 is the part of the road net- +work between two consecutive junctions without any intermediate +junction between them. +Trip 𝑇 is a time ordered sequence of |𝑇 | points 𝑝1 → · · · → 𝑝 |𝑇 |; +each point 𝑝 contains the geospatial coordinates of the moving +object along with the corresponding timestamp 𝜏 that the vehicle +was at this particular location 𝑝 = (𝑙𝑜𝑛,𝑙𝑎𝑡,𝜏). +Map-matched Trip 𝑇𝐺 is a sequence of |𝑇𝐺 | consecutive points +𝑝′ +1 → · · · → 𝑝′ +|𝑇𝐺 | that comes from map matching trip 𝑇 on the +road network 𝐺. Each point 𝑝′ corresponds to a road segment that +was traversed by𝑇. Each point 𝑝′ of the map matched trip contains +a triplet (𝑟,𝑡𝑡,𝜏); 𝑟 is the traversed road segment, 𝑡𝑡 is the travel +time of the road segment 𝑟 and is computed assuming that the +vehicle moved with the same speed in the road network between +two consecutive GPS points and 𝜏 is the timestamp that the travel +time is reported to the system. +Travel time reports 𝐷 is the collection of travel times for the +road segments as they are extracted by the trips of all the available +probe vehicles that traverse the road network. Each travel time +report (𝑟,𝑡𝑡,𝑡,𝑇𝑖𝑑) contains the information of the map-matched +trips enriched by the id of the trip 𝑇𝑖𝑑. +Path 𝑃 is a sequence of |𝑃| consecutive road segments 𝑟1 → · · · → +𝑟 |𝑃 |, where 𝑟𝑖 is the 𝑖th road segment of 𝑃. +Below we define formally the traffic forecasting problem. +Traffic Forecasting: Given the available travel times of the last +𝐿 time windows T𝑡−𝐿+1:𝑡 a traffic forecasting model forecasts the +travel times of the next 𝐻 time windows T𝑡+1:𝑡+𝐻, where the vector +T𝑡 contains the travel times of the 𝐸 road segments at time 𝑡. The +Road +Network +Trajectories +Travel Times +Reports +D +Travel Times +ZScores +Aggregated +Travel Times +M +Time +Window +length +Roads +Embeddings +Matrix +Factorization +Map +Matching +Extracting +Roads Segs +Statistics +Figure 2: Data preparation. +input matrix T𝑡−𝐿+1:𝑡 ∈ R|𝐸 |×𝐿 has missing values for the roads +that were not traversed by any vehicle at a given time window. The +forecasted matrix T𝑡+1:𝑡+𝐻 ∈ R|𝐸 |×𝐻 contains forecasts for all the +road segments 𝐸 for the next 𝐻 time windows. +3.2 +Data Preparation +The first step of the proposed framework is to preprocess the raw +data and prepare them appropriately in order to feed them to the +neural network. The overview of the data preparation approach is +illustrated in Figure 2 and described below. +Map Matching. Firstly, we map-match the available trips matching +them to the road network 𝐺. Each trip 𝑇 is transformed into a map- +matched trip 𝑇𝐺. This procedure generates the set of the available +travel time reports 𝐷. This step is common to both the historical +data that are used to train our model and the streaming traffic data +that will be used to make forecasts in real time. +Modeling the periodicity of traffic. In order to model the peri- +odicity of traffic we estimate from the historical travel time reports +the average travel time 𝑎𝑣𝑔_𝑡𝑡𝑖,ℎ𝑜𝑢𝑟 for each road segment 𝑟𝑖 ∈ 𝐸 +and for different hours of day ℎ𝑜𝑢𝑟 ∈ [1 . . . 24]. Then, we subtract +from each travel time the historical average travel time for that road +segment at the given hour. In this way, we force the deep learning +framework to model, for each different road segment, the deviation +from the average travel time for the different hours of the day. +Standardizing Travel Times. Since road segments have different +lengths and speed limits we selected to standardize the travel time +reports, considering the average behaviour of each different road +segment. More specifically, for each road segment 𝑟𝑖 we compute +the historical average travel time 𝜇𝑖 and standard deviation of travel +times 𝜎𝑖 and we use these values in order to standardize the travel +times per road segment. For instance, if 𝑡𝑡5 is a travel time that is re- +ported for the road segment 𝑟5 then the corresponding Z-Score will +be 𝑡𝑡5−𝜇5 +𝜎5 +. In the rest of the paper we assume that travel times are +the Z-Scores of travel times with subtracted the average historical +travel time for the different hours of the day. +Aggregating travel times. The historical travel time reports 𝐷 +are grouped together generating a sparse matrix 𝑀 ∈ R|𝐸 |×𝑊 . The +rows of 𝑀 correspond to the |𝐸| road segments of the road network +𝐺 and the columns correspond to the𝑊 time windows. In this work +we use time windows of 15 minutes. If more than one travel time +reports are available for a particular road segment 𝑟𝑖 at the same +time window 𝑤𝑗 then 𝑀𝑖𝑗 contains the average travel time of the +available travel times. + +A Novel Framework for Handling Sparse Data in Traffic Forecast +SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA +Roads + Embeddings +Travel +Times +Scaled Dot-Product +Attention +Ki +Qi +Vi +headi +MatMul +MatMul +i=1...h +Concatenate +Concatenate +Linear +Linear +Linear +Linear +Linear +h +h +Figure 3: Multi-Head Scaled Dot-Product +Attention +Roads + Embeddings +Travel +Times +Roads + Embeddings +Travel +Times +Attention Mechanism ++ +Norm. ++ ++ +Linear +x2 +Roads + Embeddings +Travel +Times +Attention Mechanism ++ +Norm. ++ ++ +Conv1D +x2 +Norm. +V +K +Q +Encoder +Output +N +Figure 4: Encoder Block. +V +Roads + Embeddings +Travel +Times +Attention Mechanism ++ +Norm. ++ ++ +Linear +x2 +Norm. +V +K +Q +Attention Mechanism ++ +Norm. ++ +K +Q +Roads + Embeddings +Travel Times +Encoder +Output +Figure 5: Decoder Block. +Extracting Road Segments Embeddings. An embedding repre- +sentation 𝐸𝑖 is detected for each road segment 𝑟𝑖 considering its +historical travel time reports. Here, we follow the process intro- +duced by [9]. We perform matrix factorization in the sparse matrix +𝑀, learning a matrix P ∈ R|𝐸 |×𝑑 contains a 𝑑-dimensional embed- +ding representation of the available road segments +Feeding the Model. The deep learning model that is described +in Section 3.4 receives as input two vectors that contain: (i) the +aggregated travel times that are available for a given time window +and (ii) the corresponding road segments. For instance, consider +a road network 𝐺 that is comprised of |𝐸| = 5 road segments [𝑟𝑖], +𝑖 ∈ [1, . . . , 5]. If for a particular time window only the travel times +𝑡𝑡2 and 𝑡𝑡5 of road segments 𝑟2 and 𝑟5 respectively are available, +then the inputs to the deep learning model will be the following: the +vector of the travel times T = [𝑡𝑡2,𝑡𝑡5, ∅, ∅, ∅] ∈ R|𝐸 | and the vector +of the road segments ids R = [𝑟2,𝑟5, ∅, ∅, ∅] ∈ Z|𝐸 |. Then, inside the +deep learning model the ids of the road segments are transmitted +to an embedding layer. This layer transforms the vector R into a +matrix of road segments embeddings E = [𝐸2, 𝐸5, ∅, ∅, ∅] ∈ R|𝐸 |×𝑑. +The embedding representation of the road segments is trainable +and initialized with matrix P, computed using matrix factorization +as it was described above. +3.3 +Attention Mechanism +Here, we extend the "Scaled Dot-Product Attention" that was intro- +duced in [8]. The proposed attention mechanism encodes a varying +number of travel time reports received at a particular time window. +We consider as input here the following: (i) the embeddings’ matrix +of the road segments that have been traversed by the probe vehi- +cles at a particular time window along with (ii) the vector of the +corresponding reported travel times. The overview of the proposed +attention mechanism is illustrated in Figure 3. +Initially, the Query 𝑄𝑖, Key 𝐾𝑖 and Value 𝑉𝑖 matrices are com- +puted using the embeddings E of the available road segments, com- +puted earlier. Therefore, three parameter matrices 𝑊 𝑄 +𝑖 +∈ R𝑑×𝑑, +𝑊 𝐾 +𝑖 +∈ R𝑑×𝑑 and 𝑊 𝑉 +𝑖 +∈ R𝑑×𝑑 are trained using the training in- +stances and are used to compute the matrices 𝑄𝑖 = E𝑊 𝑄 +𝑖 , 𝐾𝑖 = +E𝑊 𝐾 +𝑖 +and 𝑉𝑖 = E𝑊 𝑉 +𝑖 . The index 𝑖 ∈ [1, . . . ,ℎ] of the different +parameter matrices stands for the ℎ parallel attention layers. +The next step is to compute the attention scores using the 𝑄𝑖 +and 𝐾𝑖 matrices. The scores indicate the focus that will be placed at +the travel times of other road segments, that have been reported at +the same time window. Multiple attention heads ℎ𝑒𝑎𝑑𝑖 ∈ R|𝐸 |×|𝐸 | +are computed in parallel according to eq. 1 indicating the attention +at each particular road segment. +ℎ𝑒𝑎𝑑𝑖 = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥( +𝑄𝑖𝐾𝑇 +𝑖 +√ +𝑑 +), 𝑖 ∈ [1, . . . ,ℎ] +(1) +The road segments’ embeddings and travel times are then up- +dated considering the computed attention heads. More specifi- +cally we train the parameter matrices 𝑊 𝑂1 ∈ Rℎ𝑑×𝑑 and 𝑊 𝑂2 ∈ +Rℎ|𝐸 |×|𝐸 | that are multiplied with the concatenated values 𝑉𝑖 and +travel times T respectively. +E′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1𝑉1, . . . ,ℎ𝑒𝑎𝑑ℎ𝑉ℎ)𝑊 𝑂1 +(2) +T ′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1T, . . . ,ℎ𝑒𝑎𝑑ℎT)𝑊 𝑂2 +(3) +3.4 +Traffic Transformer’s Architecture +Here we describe our model’s architecture, which is based on the +original implementation of Transformer model described in [8]. +3.4.1 +Encoder. The encoder considers all travel time reports that +are available at a given time window, encodining the traffic con- +dition of that time window. It is comprised by a set of 𝑁 identical +blocks. The first block receives as input the roads segments embed- +dings and the travel times that are available at a given time window, +following the data preparation procedure described in Section 3.2. +The rest encoder blocks receive as input the output of the previous +block. Figure 4 illustrates the overview of the encoder block. + +SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA +Zygouras et al. +Encoder +Roads + Embeddings +Travel +Times +Encoder +Block +Roads + Ids +Travel +Times +Decoder +Block +N +Encoder +Encoder +Block +Roads + Ids +Travel +Times +Decoder +Block +N +... +... +Decoder +Decoder +Block +Query +Roads Ids +∅ +N +∅ +∅ +t - L - 1 +t +t + 1 +... +... +1st Recurrent Cell +Lth Recurrent Cell +Embedding +Embedding +Embedding +Figure 6: Overview of our model. +Each block first transmits the matrix of the available roads em- +beddings E and the corresponding vector of travel times T at the +attention mechanism. The attention mechanism produces the ma- +trix E′ and the vector T ′. Then, residual connections are employed +at the output of the attention mechanism, normalizing the sum of +the received roads segments embeddings E with the output of the +attention mechanism E′. The output is transmitted to two dense +layers followed by another residual connection. For the travel times +the output of each encoder block is the sum of the received travel +times T and the output of the attention mechanism T ′. +3.4.2 +Decoder. The decoder (Figure 5) is responsible to forecast +the travel times of the next time window, considering the encoder’s +output. The decoder consists of a set of 𝑁 blocks, similarly to the +encoder. Each block receives as input the output of the encoder and +the output of the previous block. In the training phase the first block +receives as input (i) the embeddings of the road segments that are +available in the target time window and (ii) a vector of zeros. For the +testing phase the first block receives as input (i) the embeddings +of all the road segments 𝐸 and (ii) a vector of zeros. Recall that +we are working with the Z-Scores of travel times aggregated per +road segment. Consequently, the vector of zeros corresponds to the +average travel for each road segment. +Each block of the decoder contains two attention mechanisms. +Firstly, the embeddings of the queried road segments and the travel +times are transmitted to the first attention mechanism, followed +by a residual connection. Then a second attention mechanism is +employed, receiving as input the embedding matrix E′ +1 that resulted +from the first attention mechanism along with the embeddings and +the travel times that come from the output of the encoder. The main +difference here is that the matrices 𝑉𝑖 and 𝐾𝑖 are computed from +the output of the encoder and that the considered travel times come +from the encoder. Then, the embedding output E′ +2 of the second +attention mechanism is followed again by a residual connection. +This is followed by two dense layers and a second residual connec- +tion. Finally, the travel times that result from each block is the sum +of the original travel times T that were received as input along +with travel times that result from the first and the second attention +mechanism T ′ +1 and T ′ +2 respectively. +3.4.3 +Recurrent Neural Network. The final module of our proposed +model is a recurrent model that considers the sequence of the last +𝐿 time windows. Each cell of the recurrent network encapsulates +an encoder (consisting of 𝑁 encoder blocks) along with a single +decoder block. Here the decoder block is responsible to aggregate +the information that has been encoded from the previous time win- +dow with the information that has been encoded from the current +time window. Figure 6 illustrates this recurrent architecture. The +encoder and the decoder blocks of the different recurrent cells share +the same weights among the 𝐿 different time windows. +The output of the last recurrent cell is used by the decoder model +in order to make forecasts. The decoder model consists of 𝑁 decoder +blocks that are different from each other and from the decoder +block that lies inside the recurrent cells. The output of the last +decoder block contains the predicted travel times of the queried +road segments for the next time window. This will be the Z-Scores +of the travel times for the road segments that were queried at the +first decoder block. +4 +CONCLUSION +In this paper we presented a novel deep learning framework that +considers the current traffic condition of the road network and is +used to forecast the traffic condition. Our framework can efficiently +encode the travel time reports that are available at a particular +time window via an attention mechanism that considers only the +available travel times reports and the corresponding embeddings +of the road segments. +ACKNOWLEDGMENTS +This research has been financed by the European Union through the +H2020 LAMBDA Project (No. 734242), the EU ICT-48 2020 project +TAILOR (No. 952215) and the Horizon Europe AUTOFAIR Project +(No. 101070568). +REFERENCES +[1] +Xiaomin Fang, Jizhou Huang, Fan Wang, Lingke Zeng, Haijin Liang, and Haifeng +Wang. 2020. Constgat: contextual spatial-temporal graph attention network for +travel time estimation at baidu maps. In Proceedings of the 26th ACM SIGKDD +International Conference on Knowledge Discovery & Data Mining, 2697–2705. +[2] +Kun Fu, Fanlin Meng, Jieping Ye, and Zheng Wang. 2020. Compacteta: a fast +inference system for travel time prediction. In Proceedings of the 26th ACM +SIGKDD International Conference on Knowledge Discovery & Data Mining, 3337– +3345. +[3] +Huiting Hong, Yucheng Lin, Xiaoqing Yang, Zang Li, Kung Fu, Zheng Wang, +Xiaohu Qie, and Jieping Ye. 2020. Heteta: heterogeneous information network +embedding for estimating time of arrival. In Proceedings of the 26th ACM SIGKDD +International Conference on Knowledge Discovery & Data Mining, 2444–2454. +[4] +Guangyin Jin, Min Wang, Jinlei Zhang, Hengyu Sha, and Jincai Huang. 2022. +Stgnn-tte: travel time estimation via spatial–temporal graph neural network. +Future Generation Computer Systems, 126, 70–81. +[5] +Xiucheng Li, Gao Cong, Aixin Sun, and Yun Cheng. 2019. Learning travel time +distributions with deep generative model. In The World Wide Web Conference, +1017–1027. +[6] +Yaguang Li, Kun Fu, Zheng Wang, Cyrus Shahabi, Jieping Ye, and Yan Liu. 2018. +Multi-task representation learning for travel time estimation. In Proceedings of +the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data +Mining, 1695–1704. +[7] +Jochen Schiller and Agnès Voisard. 2004. Location-based services. Elsevier. +[8] +Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, +Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you +need. Advances in neural information processing systems, 30, 5998–6008. +[9] +Nikolaos Zygouras, Nikolaos Panagiotou, Yang Li, Dimitrios Gunopulos, and +Leonidas Guibas. 2019. Htte: a hybrid technique for travel time estimation in +sparse data environments. In Proceedings of the 27th ACM SIGSPATIAL Interna- +tional Conference on Advances in Geographic Information Systems, 99–108. + diff --git a/6NE4T4oBgHgl3EQf1g37/content/tmp_files/load_file.txt b/6NE4T4oBgHgl3EQf1g37/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6c8e417c51d6610c2f701a5ed5327e67babb7ca --- /dev/null +++ b/6NE4T4oBgHgl3EQf1g37/content/tmp_files/load_file.txt @@ -0,0 +1,259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf,len=258 +page_content='A Novel Framework for Handling Sparse Data in Traffic Forecast Nikolaos Zygouras Huawei Amsterdam Research Center Netherlands nikolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='zygouras@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='com Dimitrios Gunopulos National and Kapodistrian University of Athens Greece dg@di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='uoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='gr ABSTRACT The ever increasing amount of GPS-equipped vehicles provides in real-time valuable traffic information for the roads traversed by the moving vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In this way, a set of sparse and time evolving traffic reports is generated for each road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' These time series are a valuable asset in order to forecast the future traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In this paper we present a deep learning framework that encodes the sparse recent traffic information and forecasts the future traffic con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Our framework consists of a recurrent part and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The recurrent part employs an attention mechanism that encodes the traffic reports that are available at a particular time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The decoder is responsible to forecast the future traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' CCS CONCEPTS Information systems → Data stream mining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Location based services;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Geographic information systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' KEYWORDS travel time estimation, traffic forecasting, deep learning, trans- former, GPS trajectories, mining mobility data ACM Reference Format: Nikolaos Zygouras and Dimitrios Gunopulos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' A Novel Framework for Handling Sparse Data in Traffic Forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In The 30th International Conference on Advances in Geographic Information Systems (SIGSPATIAL ’22), November 1–4, 2022, Seattle, WA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ACM, New York, NY, USA, 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='1145/3557915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='3560968 1 INTRODUCTION In recent years, the wide usage of mobile devices and the corre- sponding collection of vast amounts of spatiotemporal data have resulted in the development of various novel Location Based Ser- vices (LBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The LBS are software services that integrate geographic information providing appropriate services and information to the users [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Traffic forecasting and travel time estimation are un- doubtedly two of the widely used LBS and a lot of recent research work has been conducted towards improving their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The importance of such services is indicated by the fact that the vast majority of drivers consults several times a day services that Part of this work was done while N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Zygouras was at the National and Kapodistrian University of Athens, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Copyrights for third-party components of this work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA © 2022 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ACM ISBN 978-1-4503-9529-8/22/11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='1145/3557915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='3560968 Porto Airport Estádio do Dragão Timestamp now ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Historical Data Predictions Travel Times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' r1 r2 Porto Airport Estádio do Dragão r|P | q Figure 1: The travel time estimation problem for a given query path 𝑃𝑞 (blue line) and time of departure 𝑡𝑞 in the city of Porto, that starts at 10:00 from the airport of Porto and ends at the Estádio do Dragão, the entire path is decomposed by a set of |𝑃𝑞| road segments 𝑟1 → 𝑟2 → · · · → 𝑟 |𝑃 | and for each road segment we have a time series of travel time re- ports, received by the available probe vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' perform travel time estimation in order to appropriately choose the fastest route to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Motivated by this, in this paper we propose a novel path based travel time estimation technique that considers the available traffic reports that have been received by the set of the available probe vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each probe vehicle moves in the road network and reports the time that was required to traverse each individual road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In this way, for each road segment of the road network a time series of the reported travel times are generated, illustrated at the right part of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Our technique receives a query path along with a time of departure and estimates the time of arrival considering the current traffic condition of the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Our problem is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' A query path 𝑃𝑞 and a time of departure 𝑡𝑞 are received as input and the task is to estimate the time that is required to traverse the whole path 𝑃𝑞 if the driver departs at 𝑡𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' We propose a novel deep learning framework which is comprised of a recurrent part and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The recurrent part encodes the sparse traffic reports that are available at each time window using an attention mechanism and an embedding representations for each road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The decoder is responsible to forecast the traffic condition of the next time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 2 RELATED WORK In DeepGTT the travel time distribution for any route is learnt by conditioning on the real-time traffic [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Initially, an embedding is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='05292v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='LG] 12 Jan 2023 12:30Aveleda Casteloda Maia Lavra Gondim EN542 EN13 Mioue Silva Escura Vila Novada Barca EN105-2 Telha Moreira A28 SaoPedro Fins 160m NTo7Maig-Este,Vermoim PortoViaNorte) A3Porto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='Braqo Alfen Vermoim Nogueira ZonoIndustrial A41 Maiat deifena A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='41 OLIPORI A3 Perafita A41 /Moia /(AE) Broga/A42 Felgueiras EN14 Milheiros Santa Cruz Gueifaes Ermesinde doBispo Refinaria Balio deMatosinhos EN15-1 VILPL dstoias Aguas Santas A4 Guifoes Sao Mamede deInfesta Pedroucos Baguimdo Senhora da Monte Hora 34m Matosinhos EN12 RioTinto Paranhos EM612 EN12 Pargue Aldoar daCidode Ranalde EN15 Bogvista Fanzeres Nevogilde tecedc For Fan Cedofeita Areias/W12Circunva/acao Lordelo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='do Campanttransavia F-GZHUSUPERBOCKSIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA Zygouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' estimated for each link considering its characteristics, then a non- linear factorization model generates the speed and finally an atten- tion mechanism is used to generate the observed travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Also, in HETETA [3] the road map is translated into a multi-relational network, considering the traffic behavior patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Temporal and graph convolutions are used in order to learn spatiotemporal het- erogeneous information, considering recent, daily and weekly traf- fic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' CompactETA [2] provides an accurate ETA estimation with low latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Graph attention network was employed in order to encode spatial and temporal dependencies of the weighted road network and the sequential information of the route is encoded with positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' A multi-layer perceptron was used for online inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The authors in [6] proposed a multitask represen- tation learning model which predicts the travel time of an origin- destination pair extracting a representation that preserves trips properties and road network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ConSTGAT [1] proposed a spatiotemporal graph neural network exploiting the spatial and temporal information with a 3D-attention mechanism and a model with convolutions over local windows in order to capture route’s contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' STGNN-TTE [4] adopted a spatial–temporal module to capture the real-time traffic condition and a transformer layer to estimate the links’ travel time and the total routes’ travel time synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3 OUR APPROACH 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='1 Problem Definition Road Network is represented as a directed graph 𝐺(𝑉, 𝐸), where the nodes 𝑉 represent the junctions and the edges 𝐸 represent the |𝐸| roads segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' A road segment 𝑟 ∈ 𝐸 is the part of the road net- work between two consecutive junctions without any intermediate junction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Trip 𝑇 is a time ordered sequence of |𝑇 | points 𝑝1 → · · · → 𝑝 |𝑇 |;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' each point 𝑝 contains the geospatial coordinates of the moving object along with the corresponding timestamp 𝜏 that the vehicle was at this particular location 𝑝 = (𝑙𝑜𝑛,𝑙𝑎𝑡,𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Map-matched Trip 𝑇𝐺 is a sequence of |𝑇𝐺 | consecutive points 𝑝′ 1 → · · · → 𝑝′ |𝑇𝐺 | that comes from map matching trip 𝑇 on the road network 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each point 𝑝′ corresponds to a road segment that was traversed by𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each point 𝑝′ of the map matched trip contains a triplet (𝑟,𝑡𝑡,𝜏);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 𝑟 is the traversed road segment, 𝑡𝑡 is the travel time of the road segment 𝑟 and is computed assuming that the vehicle moved with the same speed in the road network between two consecutive GPS points and 𝜏 is the timestamp that the travel time is reported to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Travel time reports 𝐷 is the collection of travel times for the road segments as they are extracted by the trips of all the available probe vehicles that traverse the road network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each travel time report (𝑟,𝑡𝑡,𝑡,𝑇𝑖𝑑) contains the information of the map-matched trips enriched by the id of the trip 𝑇𝑖𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Path 𝑃 is a sequence of |𝑃| consecutive road segments 𝑟1 → · · · → 𝑟 |𝑃 |, where 𝑟𝑖 is the 𝑖th road segment of 𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Below we define formally the traffic forecasting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Traffic Forecasting: Given the available travel times of the last 𝐿 time windows T𝑡−𝐿+1:𝑡 a traffic forecasting model forecasts the travel times of the next 𝐻 time windows T𝑡+1:𝑡+𝐻, where the vector T𝑡 contains the travel times of the 𝐸 road segments at time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The Road Network Trajectories Travel Times Reports D Travel Times ZScores Aggregated Travel Times M Time Window length Roads Embeddings Matrix Factorization Map Matching Extracting Roads Segs Statistics Figure 2: Data preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' input matrix T𝑡−𝐿+1:𝑡 ∈ R|𝐸 |×𝐿 has missing values for the roads that were not traversed by any vehicle at a given time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The forecasted matrix T𝑡+1:𝑡+𝐻 ∈ R|𝐸 |×𝐻 contains forecasts for all the road segments 𝐸 for the next 𝐻 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='2 Data Preparation The first step of the proposed framework is to preprocess the raw data and prepare them appropriately in order to feed them to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The overview of the data preparation approach is illustrated in Figure 2 and described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Map Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Firstly, we map-match the available trips matching them to the road network 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each trip 𝑇 is transformed into a map- matched trip 𝑇𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' This procedure generates the set of the available travel time reports 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' This step is common to both the historical data that are used to train our model and the streaming traffic data that will be used to make forecasts in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Modeling the periodicity of traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In order to model the peri- odicity of traffic we estimate from the historical travel time reports the average travel time 𝑎𝑣𝑔_𝑡𝑡𝑖,ℎ𝑜𝑢𝑟 for each road segment 𝑟𝑖 ∈ 𝐸 and for different hours of day ℎ𝑜𝑢𝑟 ∈ [1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Then, we subtract from each travel time the historical average travel time for that road segment at the given hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In this way, we force the deep learning framework to model, for each different road segment, the deviation from the average travel time for the different hours of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Standardizing Travel Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Since road segments have different lengths and speed limits we selected to standardize the travel time reports, considering the average behaviour of each different road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' More specifically, for each road segment 𝑟𝑖 we compute the historical average travel time 𝜇𝑖 and standard deviation of travel times 𝜎𝑖 and we use these values in order to standardize the travel times per road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' For instance, if 𝑡𝑡5 is a travel time that is re- ported for the road segment 𝑟5 then the corresponding Z-Score will be 𝑡𝑡5−𝜇5 𝜎5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In the rest of the paper we assume that travel times are the Z-Scores of travel times with subtracted the average historical travel time for the different hours of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Aggregating travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The historical travel time reports 𝐷 are grouped together generating a sparse matrix 𝑀 ∈ R|𝐸 |×𝑊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The rows of 𝑀 correspond to the |𝐸| road segments of the road network 𝐺 and the columns correspond to the𝑊 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In this work we use time windows of 15 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' If more than one travel time reports are available for a particular road segment 𝑟𝑖 at the same time window 𝑤𝑗 then 𝑀𝑖𝑗 contains the average travel time of the available travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' A Novel Framework for Handling Sparse Data in Traffic Forecast SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA Roads Embeddings Travel Times Scaled Dot-Product Attention Ki Qi Vi headi MatMul MatMul i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='h Concatenate Concatenate Linear Linear Linear Linear Linear h h Figure 3: Multi-Head Scaled Dot-Product Attention Roads Embeddings Travel Times Roads Embeddings Travel Times Attention Mechanism + Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' + + Linear x2 Roads Embeddings Travel Times Attention Mechanism + Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' + + Conv1D x2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' V K Q Encoder Output N Figure 4: Encoder Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' V Roads Embeddings Travel Times Attention Mechanism + Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' + + Linear x2 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' V K Q Attention Mechanism + Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' + K Q Roads Embeddings Travel Times Encoder Output Figure 5: Decoder Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Extracting Road Segments Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' An embedding repre- sentation 𝐸𝑖 is detected for each road segment 𝑟𝑖 considering its historical travel time reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Here, we follow the process intro- duced by [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' We perform matrix factorization in the sparse matrix 𝑀, learning a matrix P ∈ R|𝐸 |×𝑑 contains a 𝑑-dimensional embed- ding representation of the available road segments Feeding the Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The deep learning model that is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='4 receives as input two vectors that contain: (i) the aggregated travel times that are available for a given time window and (ii) the corresponding road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' For instance, consider a road network 𝐺 that is comprised of |𝐸| = 5 road segments [𝑟𝑖], 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' , 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' If for a particular time window only the travel times 𝑡𝑡2 and 𝑡𝑡5 of road segments 𝑟2 and 𝑟5 respectively are available, then the inputs to the deep learning model will be the following: the vector of the travel times T = [𝑡𝑡2,𝑡𝑡5, ∅, ∅, ∅] ∈ R|𝐸 | and the vector of the road segments ids R = [𝑟2,𝑟5, ∅, ∅, ∅] ∈ Z|𝐸 |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Then, inside the deep learning model the ids of the road segments are transmitted to an embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' This layer transforms the vector R into a matrix of road segments embeddings E = [𝐸2, 𝐸5, ∅, ∅, ∅] ∈ R|𝐸 |×𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The embedding representation of the road segments is trainable and initialized with matrix P, computed using matrix factorization as it was described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='3 Attention Mechanism Here, we extend the "Scaled Dot-Product Attention" that was intro- duced in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The proposed attention mechanism encodes a varying number of travel time reports received at a particular time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' We consider as input here the following: (i) the embeddings’ matrix of the road segments that have been traversed by the probe vehi- cles at a particular time window along with (ii) the vector of the corresponding reported travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The overview of the proposed attention mechanism is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Initially, the Query 𝑄𝑖, Key 𝐾𝑖 and Value 𝑉𝑖 matrices are com- puted using the embeddings E of the available road segments, com- puted earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Therefore, three parameter matrices 𝑊 𝑄 𝑖 ∈ R𝑑×𝑑, 𝑊 𝐾 𝑖 ∈ R𝑑×𝑑 and 𝑊 𝑉 𝑖 ∈ R𝑑×𝑑 are trained using the training in- stances and are used to compute the matrices 𝑄𝑖 = E𝑊 𝑄 𝑖 , 𝐾𝑖 = E𝑊 𝐾 𝑖 and 𝑉𝑖 = E𝑊 𝑉 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The index 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ,ℎ] of the different parameter matrices stands for the ℎ parallel attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The next step is to compute the attention scores using the 𝑄𝑖 and 𝐾𝑖 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The scores indicate the focus that will be placed at the travel times of other road segments, that have been reported at the same time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Multiple attention heads ℎ𝑒𝑎𝑑𝑖 ∈ R|𝐸 |×|𝐸 | are computed in parallel according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 1 indicating the attention at each particular road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ℎ𝑒𝑎𝑑𝑖 = 𝑠𝑜𝑓 𝑡𝑚𝑎𝑥( 𝑄𝑖𝐾𝑇 𝑖 √ 𝑑 ), 𝑖 ∈ [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ,ℎ] (1) The road segments’ embeddings and travel times are then up- dated considering the computed attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' More specifi- cally we train the parameter matrices 𝑊 𝑂1 ∈ Rℎ𝑑×𝑑 and 𝑊 𝑂2 ∈ Rℎ|𝐸 |×|𝐸 | that are multiplied with the concatenated values 𝑉𝑖 and travel times T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' E′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1𝑉1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ,ℎ𝑒𝑎𝑑ℎ𝑉ℎ)𝑊 𝑂1 (2) T ′ = 𝐶𝑜𝑛𝑐𝑎𝑡(ℎ𝑒𝑎𝑑1T, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ,ℎ𝑒𝑎𝑑ℎT)𝑊 𝑂2 (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='4 Traffic Transformer’s Architecture Here we describe our model’s architecture, which is based on the original implementation of Transformer model described in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='1 Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The encoder considers all travel time reports that are available at a given time window, encodining the traffic con- dition of that time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' It is comprised by a set of 𝑁 identical blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The first block receives as input the roads segments embed- dings and the travel times that are available at a given time window, following the data preparation procedure described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The rest encoder blocks receive as input the output of the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Figure 4 illustrates the overview of the encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' SIGSPATIAL ’22, November 1–4, 2022, Seattle, WA, USA Zygouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Encoder Roads Embeddings Travel Times Encoder Block Roads Ids Travel Times Decoder Block N Encoder Encoder Block Roads Ids Travel Times Decoder Block N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Decoder Decoder Block Query Roads Ids ∅ N ∅ ∅ t - L - 1 t t + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 1st Recurrent Cell Lth Recurrent Cell Embedding Embedding Embedding Figure 6: Overview of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each block first transmits the matrix of the available roads em- beddings E and the corresponding vector of travel times T at the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The attention mechanism produces the ma- trix E′ and the vector T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Then, residual connections are employed at the output of the attention mechanism, normalizing the sum of the received roads segments embeddings E with the output of the attention mechanism E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The output is transmitted to two dense layers followed by another residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' For the travel times the output of each encoder block is the sum of the received travel times T and the output of the attention mechanism T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='2 Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The decoder (Figure 5) is responsible to forecast the travel times of the next time window, considering the encoder’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The decoder consists of a set of 𝑁 blocks, similarly to the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each block receives as input the output of the encoder and the output of the previous block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' In the training phase the first block receives as input (i) the embeddings of the road segments that are available in the target time window and (ii) a vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' For the testing phase the first block receives as input (i) the embeddings of all the road segments 𝐸 and (ii) a vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Recall that we are working with the Z-Scores of travel times aggregated per road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Consequently, the vector of zeros corresponds to the average travel for each road segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each block of the decoder contains two attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Firstly, the embeddings of the queried road segments and the travel times are transmitted to the first attention mechanism, followed by a residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Then a second attention mechanism is employed, receiving as input the embedding matrix E′ 1 that resulted from the first attention mechanism along with the embeddings and the travel times that come from the output of the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The main difference here is that the matrices 𝑉𝑖 and 𝐾𝑖 are computed from the output of the encoder and that the considered travel times come from the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Then, the embedding output E′ 2 of the second attention mechanism is followed again by a residual connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' This is followed by two dense layers and a second residual connec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Finally, the travel times that result from each block is the sum of the original travel times T that were received as input along with travel times that result from the first and the second attention mechanism T ′ 1 and T ′ 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content='3 Recurrent Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The final module of our proposed model is a recurrent model that considers the sequence of the last 𝐿 time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Each cell of the recurrent network encapsulates an encoder (consisting of 𝑁 encoder blocks) along with a single decoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Here the decoder block is responsible to aggregate the information that has been encoded from the previous time win- dow with the information that has been encoded from the current time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Figure 6 illustrates this recurrent architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The encoder and the decoder blocks of the different recurrent cells share the same weights among the 𝐿 different time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The output of the last recurrent cell is used by the decoder model in order to make forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The decoder model consists of 𝑁 decoder blocks that are different from each other and from the decoder block that lies inside the recurrent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' The output of the last decoder block contains the predicted travel times of the queried road segments for the next time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' This will be the Z-Scores of the travel times for the road segments that were queried at the first decoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 4 CONCLUSION In this paper we presented a novel deep learning framework that considers the current traffic condition of the road network and is used to forecast the traffic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' Our framework can efficiently encode the travel time reports that are available at a particular time window via an attention mechanism that considers only the available travel times reports and the corresponding embeddings of the road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' ACKNOWLEDGMENTS This research has been financed by the European Union through the H2020 LAMBDA Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 734242), the EU ICT-48 2020 project TAILOR (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 952215) and the Horizon Europe AUTOFAIR Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 101070568).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' REFERENCES [1] Xiaomin Fang, Jizhou Huang, Fan Wang, Lingke Zeng, Haijin Liang, and Haifeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf'} +page_content=' 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170716 diff --git a/89E0T4oBgHgl3EQfwgGc/content/tmp_files/2301.02634v1.pdf.txt b/89E0T4oBgHgl3EQfwgGc/content/tmp_files/2301.02634v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa7cf28b2c4ea68244bf447ee2ad641bc294a52d --- /dev/null +++ b/89E0T4oBgHgl3EQfwgGc/content/tmp_files/2301.02634v1.pdf.txt @@ -0,0 +1,695 @@ +arXiv:2301.02634v1 [math.LO] 6 Jan 2023 +ON DISJOINT STATIONARY SEQUENCES +MAXWELL LEVINE +Abstract. We answer a question of Krueger by obtaining disjoint stationary +sequences on successive cardinals. The main idea is an alternative presentation +of a mixed support iteration, using it even more explicitly as a variant of +Mitchell forcing. We also use a Mahlo cardinal to obtain a model in which +ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2, answering a +question of Gilton. +1. Introduction and Background +In order to develop a more vivid picture of the infinite cardinals, set theorists +study a variety of objects that can potentially exist on these cardinals. The objects +of interest for this paper are called disjoint stationary sequences. These were intro- +duced by Krueger to answer a question of Abraham and Shelah about forcing clubs +through stationary sets. Beginning in joint work with Friedman, Krueger wrote a +series of papers in this area, connecting a wide range of concepts and answering +seemingly unrelated questions of Foreman and Todorˇcevi´c [4, 11, 12, 13, 14, 15]. +Generally, the new arguments hinged on the behavior of two-step iterations of the +form Add(τ) ∗ P. +In order to extend the application of these arguments as widely as possible, +Krueger developed the notion of mixed support forcing [12, 15]. These forcings +are to some extent an analog of the forcing that Mitchell used to obtain the tree +property at double successors of regular cardinals. Their most notable feature is the +appearance of quotients insofar as the forcings took the form M ≃ ¯M ∗ Add(τ) ∗ Q +where ¯M is a partial mixed support iteration. The appearance of Add(τ) after +the initial component, together with the preservation properties of the quotient Q, +allowed Krueger’s new arguments to go through various complicated constructions. +Mixed support iterations have found several applications since [5], particularly in +regard to guessing models [16]. +The main idea in this paper is to use a version of Mitchell forcing to accomplish +the task of a mixed support iteration. Specifically, this version of Mitchell forcing +takes the form M ≃ ¯M ∗ Add(τ) ∗ Q.1 The trick used to obtain this structural +property is reminiscent of the one usd by Cummings et al. in “The Eightfold Way” +to demonstrate that subtle variations in the definitions of Mitchell forcing—up to +merely shifting a L´evy collapse by a single coordinate—can substantially alter the +properties of the forcing extension. The benefit of the forcing used here is that it +comes with a projection analysis of the sort that Abraham used for Mitchell forcing +[1]. Both the forcing itself and its quotients are projections of products of the form +1The extent to which all variations of these forcings are equivalent or not is left as a loose +end. Here we only deal with the case where the two-step iteration Add(τ) ∗ P takes the form +Add(τ) ∗ +˙ +Col(µ, δ). +1 + +2 +MAXWELL LEVINE +A× T where A has a good chain condition and T has a good closure property. This +allows us to obtain preservation properties conveniently, without having to delve +into too many technical details. Abraham in fact used this projection analysis to +extend Mitchell’s result to successive cardinals. This is exactly what we do here +for disjoint stationary sequences, answering the first component of a question of +Krueger [15, Question 12.8]: +Theorem 1. Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a +forcing extension in which there are disjoint stationary sequences on ℵ2 and ℵ3. +We lay out the basic definition and concepts in the following subsections and +then develop the proof in Section 2. We also achieve one of Krueger’s separations +for successive cardinals, which answers a component of another one of his questions +[15, Question 12.9]: +Theorem 2. Suppose λ1 < λ2 are two Mahlo cardinals in V . Then there is a forc- +ing extension in which for µ ∈ {ℵ1, ℵ2}, there are stationarily many N ∈ [H(µ+)]µ +that are internally stationary but not internally club. +The last main result is motivated by work of Gilton and Krueger, who answered +a question from “The Eightfold Way” by obtaining stationary reflection for subsets +of ℵ2 ∩ cof(ω) together with failure of approachability at ℵ2 (i.e. ℵ2 /∈ I[ℵ2]) using +disjoint stationary sequences [5]. This result used the fact that the existence of a +disjoint stationary sequence implies failure of approachability. Gilton asked for the +exact consistency strength of the failure of approachability at ℵ2 together with the +nonexistence of a disjoint stationary set on ℵ2 [7, Question 9.0.15]. (He pointed +out that Cox found this separation using PFA [2].) It is known that the failure +of approachability requires the consistency strength of a Mahlo cardinal, and in +Section 3 we show that a Mahlo cardinal is sufficient for the separation: +Theorem 3. Suppose that λ is Mahlo in V . Then there is a forcing extension in +which ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2. +Disjoint stationary sequences are known to be interpretable in terms of canonical +structure (see Fact 6 below), and the main idea for Theorem 3 is a simple master +condition argument that exploits this connection. +We note that all three of these theorems can be generalized to arbitrarily high +cardinals. +1.1. Basic Definitions. We assume familiarity with the basics of forcing and large +cardinals. We use the following conventions: If P is a forcing poset, then p ≤ q +for p, q ∈ P means that p is stronger than q. We say that P is κ-closed if for all +≤P-decreasing sequences ⟨pξ : ξ < τ⟩ with τ < κ, there is a lower bound p, i.e. +p ≤ pξ for all ξ < τ. We say that P has the κ-chain condition of all antichains +A ⊆ P have cardinality strictly less than κ. +Now we give our main definitions: +Definition 4. Given a regular cardinal µ, a disjoint stationary sequence on µ+ is +a sequence ⟨Sα : α ∈ S⟩ such that: +• S ⊆ µ+ ∩ cof(µ) is stationary, +• Sα is a stationary subset of Pµ(α) for all α ∈ S, +• Sα ∩ Sβ = ∅ if α ̸= β. + +ON DISJOINT STATIONARY SEQUENCES +3 +We write DSS(µ+) to say that there is a disjoint stationary sequence on µ+. +Definition 5. Given a stationary N ∈ [H(Θ)]κ,2 we say: +• N is internally unbounded if ∀x ∈ Pκ(N), ∃M ∈ N, x ⊆ M, +• N is internally stationary if Pκ(N) ∩ N is stationary in Pκ(N), +• N is internally club if Pκ(N) ∩ N is club in Pκ(N), +• N is internally approachable if there is an increasing and continuous con- +tinuous chain ⟨Mξ : ξ < κ⟩ such that |Mξ| < κ and ⟨Mη : η < ξ⟩ ∈ Mξ+1 +for all ξ < κ such that N = � +ξ<κ Mξ. +Although disjoint stationary sequences may seem unrelated the separation of +variants of internal approachability, there are deep connections here, for example: +Fact 6 (Krueger, [15]). If µ is regular and 2µ = µ+, then DSS(µ+) is equivalent to +the existence of a stationary set U ⊆ [H(µ+)]µ such that every N ∈ U is internally +internally unbounded but not internally club. +1.2. Projections and Preservation Lemmas. Technically speaking, our main +goal is to show that certain forcing quotients behave nicely. +We will make an +effort to demonstrate the preservation properties of these quotients directly. These +quotients will be defined in terms of projections: +Definition 7. If P1 and P2 are posets, a projection is an onto map π : P1 → P2 +such that: +• p ≤ q implies that π(p) ≤ π(q), +• if r ≤ π(p), then there is some q ≤ p such that π(q) ≤ r. +A projection is trivial if π(p) = π(q) implies that p and q are compatible. +Trivial projections are basically ismorphisms: +Fact 8. If π : P1 → P2 is a trivial projection, then P1 ≃ P2. +For our purposes, we are interested in the preservation of stationary sets. The +chain condition gives us preservation fairly straightforwardly. The following fact is +implicit in parts of the literature, and a version of it can be found in this paper in +the form of Proposition 26. +Fact 9. If P has the µ-chain condition and S ⊂ Pµ(X) is stationary, then P forces +that S is stationary in P V +µ (X). +However, we must place demands on our stationary sets in order for them to be +preserved by closed forcings. +Definition 10. A stationary set S ⊂ Pµ(H(Θ)) is internally approachable of length +τ if for all N ∈ S with N ≺ H(Θ), there is a continuous chain of elementary +submodels ⟨Mi : i < τ⟩ such that N = � +i<τ Mi and for all i < τ, ⟨Mi : i < j⟩ ∈ +Mj+1. In this case we write S ⊆ IA(τ). +Fact 11. If S ⊂ Pµ(H(Θ)) ∩ IA(τ) is an internally approachable stationary set, +τ < µ, and P is µ-closed, then P forces that S is stationary in Pµ(H(Θ)V ). +2See Jech for details on stationary sets [10]. + +4 +MAXWELL LEVINE +1.3. Costationarity of the Ground Model. The notion of ground model co- +stationarity is a key ingredient in arguments pertaining to disjoint stationary se- +quences. It will specifically give us the disjointness, since we will be picking sta- +tionary sets that are not added by initial segments of these forcings. +Gitik obtained the classical result: +Fact 12 (Gitik [8]). If V ⊂ W are models of ZFC with the same ordinals, W \ V +contains a real, and κ is a regular cardinal in W such that (κ+)W ≤ λ, then +P W +κ (λ) \ V is stationary. +Because we will need Fact 11, we will actually use Krueger’s refinement of Gitik’s +theorem: +Fact 13 (Krueger [15]). Suppose V ⊂ W are models of ZFC with the same ordinals, +W \ V contains a real, µ is a regular cardinal in W, and X ∈ V is such that +(µ+)W ⊆ X, and that in W, Θ is a regular cardinal such that X ⊂ H(Θ). Then in +W the set {N ∈ Pµ(H(Θ)) ∩ IA(ω) : N ∩ X /∈ V } is stationary. +2. The New Mitchell Forcing +2.1. Defining the Forcing. In this subsection we will illustrate the basic idea of +this paper by using our new take on Mitchell forcing to prove a known result: +Theorem 14 (Krueger [15]). If λ is a Mahlo cardinal and µ < λ are regular +cardinals, there is a forcing extension in which 2ω = µ+ = λ and there is a disjoint +stationary sequence on λ. +Specifically, we will define a forcing M+(τ, µ, λ) such that the model W in +Theorem 14 can be realized as an extension by M+(ω, µ, λ). +For standard technical reasons, we define a poset ismorphic to Add(τ, λ): +Definition 15. Given a regular τ and a set of ordinals Y , we let Add∗(τ, Y ) be +the poset consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1} +where | dom p| < τ. We let p ≤Add∗(τ,Y ) q if and only if p ⊇ q. +Note: In later subsections we will conflate Add(τ, λ) and Add∗(τ, λ) to simplify +notation. +Definition 16. Let λ be inaccessible and let τ < µ < λ be regular cardinals such +that τ <τ = τ. We define a forcing M+(τ, µ, λ) that consists of pairs (p, q) such +that: +(1) p ∈ Add∗(τ, λ), +(2) q is a function such that: +(a) dom q ⊂ {δ < λ : δ is inaccessible}, +(b) | dom q| < µ, +(c) ∀δ ∈ dom(q), p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ∈ +˙ +Col(µ, δ)”. +We let (p, q) ≤ (p′, q′) if and only if: +(i) p ≤Add∗(τ,λ) p′, +(ii) dom q ⊇ dom q′, +(iii) for all δ ∈ dom q′, p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ≤ ˙ +Col(µ,δ) q′(δ)” +First we go through the more routine properties that one would expect of this +forcing. + +ON DISJOINT STATIONARY SEQUENCES +5 +Proposition 17. M+(τ, µ, λ) is τ-closed and λ-Knaster. +Proof. Closure uses the facts that Add∗(τ, λ) is τ-closed and ⊩Add(τ,δ+1) “ ˙ +Col(µ, δ) +is µ-closed” for all δ. Knasterness uses a standard application of the Delta System +Lemma. +□ +Crucially, we get a nice termspace: +Definition 18. Let T = T(M+(τ, µ, λ)) be the poset consisting of conditions q +such that: +(1) dom q ⊂ λ ∩ {δ < λ : δ is inaccessible}, +(2) | dom q| < µ, +(3) ∀δ ∈ dom q, ⊩Add∗(τ,δ+1) q(δ) ∈ +˙ +Col(µ, δ)”. +Most importantly, we let q ≤ q′ if and only if: +(i) dom q ⊇ dom q′, +(ii) for all δ ∈ dom q, ⊩Add∗(τ,δ+1) “q(δ) ≤ q′(δ)”. +Proposition 19. There is a projection Add∗(τ, λ)×T(M+(τ, µ, λ)) ։ M+(τ, µ, λ). +Proof. We let π be the projection with the definition π(p, q) = (p, q). This is au- +tomatically order-preserving because the ordering ≤Add∗(τ,λ)×T is coarser than the +ordering ≤M+(τ,µ,λ). For obtaining the density condition, suppose (r, s) ≤M+(τ,µ,λ) +(p0, q0). We want to find some (p1, q1) such that (p1, q1) ≤Add∗(τ,λ)×T (p0, q0) and +(p1, q1) ≤M+(τ,µ,λ) (r, s). To do this, we first let p1 = r, and then we define q1 +with dom q1 = dom r such that at each coordinate δ ∈ dom q1, we use standard +arguments on names to show that we can get both p0 ↾ ((δ + 1) × τ) ⊩Add∗(τ,λ) +“q1(δ) ≤ s(δ)” as well as 1Add∗(τ,λ) ⊩Add∗(τ,λ) “q1(δ) ≤ q0(δ)”. +□ +Proposition 20. T is µ-closed. +Proof. This as an application of the Mixing Principle. Given a ≤T-decreasing se- +quence ⟨qi : i < τ⟩ with τ < µ we let d = � +i<τ dom qi. Then we define a lower +bound ¯q with domain d such that for all δ ∈ d, q(δ) is a canonically-defined name +for a lower bound of the qi(δ)’s (where i is large enough that δ ∈ dom qi). +□ +Then we get the standard consequences of the termspace analysis: +Proposition 21. The following are true in any extension by M+(τ, µ, λ): +(1) V -cardinals up to and including µ are cardinals. +(2) For all α < λ, |α| = µ. +(3) λ = µ+. +(4) 2τ = λ. +Proof. (1) follows from the projection analysis and the fact that T is µ-closed and +Add∗(τ, λ) is τ +-cc, and from τ-closure of M+(τ, µ, λ). (2) follows from the fact +that for all inaccessible δ < λ, M+(τ, µ, λ) projects onto Col(µ, δ). (3) follows from +(1) and (2) plus λ-Knasterness. (4) follows from the fact that M+(µ, λ) projects +onto Add∗(τ, λ), so it forces that 2τ ≥ λ. Since the poset has size λ, it also forces +that 2τ ≤ λ. +□ +The following lemma is the crux of the new idea. + +6 +MAXWELL LEVINE +Lemma 22. If δ0 < λ is inaccessible, then there is a forcing equivalence +M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ) ∗ Q +where M+(τ, µ, δ0) ∗ Add(τ) forces that Q is a projection of a product of a µ-closed +forcing and a τ +-cc forcing. +Proof. More precisely, we will show that there is a forcing equivalence M+(τ, µ, λ) ≃ +M+(τ, µ, δ0) ∗ Add(τ) ∗ (P × R) where the following hold in the extension by +M+(τ, µ, δ0) ∗ Add(τ): +• R is a projection of a product of a µ-closed forcing and Add∗(τ, λ), and +• V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”. +The statement of the lemma can then be obtained by merging P with the closed +component of the product that projects onto R. +First we describe P and R. To do this, we fix some notation. Given Y ⊆ λ, we let +πY +Add denote the projection (p, q) → p↾(Y × τ) from M+(τ, µ, λ) onto Add∗(τ, Y ). +For any poset P, we employ the convention that Γ(P) denotes a canonical name for +a P-generic. If X ⊂ P, then we use the notation ↑ X := {q ∈ P : ∃p ∈ X, p ≤ q}. +We will let +P := Col(µ, δ0)V [(↑(πδ0 +Add”Γ(M+(τ,µ,δ0))))×Γ(Add(τ))] +if we are working in an extension by M+(τ, µ, δ0) ∗ Add(τ). (In other words, the +poset P will be the version of Col(µ, δ0) as interpreted in the extension of V by +Add∗(τ, δ + 1) where the initial coordinates come from M+(τ, µ, δ0) and the last +coordinate comes from the additional copy of Add(τ).) +Still working in an extension by M+(τ, µ, δ0) ∗ Add(τ), the poset R consists of +pairs (p, q) such that the following hold: +(1) p ∈ Add∗(τ, (δ0, λ)), +(2) q is a function such that +(a) dom q ⊂ {δ ∈ (δ0, λ) : δ is inaccessible}, +(b) | dom q| < µ, +(c) ∀δ ∈ dom(q), p↾((δ0, (δ + 1)) × τ) ⊩Add∗(τ,(δ0,δ+1)) “q(δ) ∈ +˙ +Col(µ, δ)”. +The ordering is the one analogous to that of M+(τ, µ, λ). An easy adaptation of +the arguments for the projection analysis for M+(τ, µ, λ) will then give a projection +analysis for R. +The rest of the proof of the lemma consists of verifying the more substantial +claims. +Claim 23. M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R). +Proof. We identify M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R) with the dense subset of con- +ditions ((r, s), t, u, (r, ˙s′)) such that ˙s′ is forced to have a specific domain in V . The +fact that this subset is dense follows from the fact that M+(τ, µ, λ) ∗ Add(τ) has +the µ-covering property. +We will argue that there is a trivial projection defined by +π : (p, q) �→ ((p↾(δ0 × τ), q↾δ0) +� +�� +� +M+(µ,δ0) +, p↾({δ0} × τ) +� +�� +� +Add(τ) +, q∗(δ0) +� �� � +P +, (¯p, ¯q) +� �� � +R +) +such that +• ¯p := p↾((δ0, λ) × τ); + +ON DISJOINT STATIONARY SEQUENCES +7 +• q∗(δ0) is obtained by changing q(δ0) from an Add∗(τ, δ0 + 1)-name to an +Add(τ) as interpreted in the extension by the relevant generic, namely +(↑ (πδ0 +Add”Γ(M+(τ, µ, δ0)))); +• ¯q has domain (δ0, λ), and for each δ ∈ (δ0, λ), ¯q(δ) has changes analogous +to the changes made to q∗(δ0). +It is clear that π is order-preserving. We also want to show that if +((r, s), t, u, (r′, ˙s′)) ≤M+(τ,µ,δ0)∗Add(τ)∗(P×R) π(p0, q0) +then there is some (p1, q1) ≤M+(µ,λ) (p0, q0) such that π(p1, q1) ≤ ((r, s), t, u, (r′, s′)). +This can be done by taking: +• p1 = r ∪ ˜t ∪ r′ where ˜t writes t as as a partial function {δ} × τ → {0, 1}, +• q1 = s ∪ ˜u ∪ ˜s′ where ˜u reinterprets u as a Add∗(δ0 + 1)-name and for each +δ ∈ dom( ˙s′), ˜s′ reinterprets ˙s′(δ) as a Add∗(δ + 1)-name. +Last, we argue that π(p0, q0) = π(p1, q1) implies that (p0, q0) and (p1, q1) are +compatible. Suppose that (p0, q0) and (p1, q1) are incompatible. If p0 and p1 are +incompatible as elements of Add∗(τ, λ), then one of pi ↾ (δ0 × τ), pi ↾ ({δ0} × τ), +and pi ↾((δ0, λ) × τ) must be distinct for i = 0 and i = 1. Otherwise, there is some +p′ ≤ p0, p1 and some δ ∈ dom q0∩dom q1 inaccessible such that p′ ⊩ “q0(δ) ⊥ q1(δ)”, +which implies that q0(δ) ̸= q1(δ). Therefore, one of qi ↾ δ0, qi(δ0), or qi ↾ (δ0, λ) is +distinct for i ∈ {0, 1}. +□ +Claim 24. V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”. +Proof. In fact, our argument will also show that V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P = +Col(µ, δ0)”. We fix some arbitrary generics: +• G is M+(τ, µ, δ0)-generic over V , +• r is Add(τ)-generic over V [G], +• H is the Add∗(τ, δ0)-generic induced from G by πδ0 +Add, +• K is the generic for the quotient of M+(τ, µ, δ0) by Add∗(τ, δ0), i.e. the +generic such that V [H][K] = V [G], +• T is the generic for the termspace forcing T(M+(τ, µ, δ0)), so that V [G] ⊂ +V [T ][H]. +It is enough to argue that V [G][r] |= “P is µ-closed” knowing that V [H][r] |= “P +is µ-closed”. Because adjoining G does not change the definition of Add(τ), and +because K is defined in terms of the subsets of τ adjoined by the filter H, we have +V [G][r] = V [H][K][r] = V [H][r][K]. Therefore, it is enough to show that K does +not add <µ-sequences over V [H][r], so that V [H][r]’s version of Col(µ, δ0) remains +µ-closed in V [G][r]. We have +V [H][r] ⊂ V [H][r][K] = V [H][K][r] = V [G][r] ⊂ V [T ][H][r] = V [H][r][T ], +and Easton’s Lemma implies that T does not add new <µ-sequences over V [H][r], +so therefore K does not add new <µ-sequences over V [H][r] since it is an interme- +diate factor of the extension. +□ +This completes the proof of the lemma. +□ +Now we have an application for the case where τ = ω. +Proposition 25. If λ is Mahlo then V [M+(ω, µ, λ)] |= DSS(λ). +This basically repeats Krueger’s argument for [15, Theorem 9.1]. + +8 +MAXWELL LEVINE +Proof. Let G be M+(ω, µ, λ)-generic over V . The set of V -inaccessibles in λ will +form the stationary set S ⊂ µ+ ∩ cof(µ) carrying the disjoint stationary sequence +in the extension by M+(ω, µ, λ). For every such δ ∈ S, let ¯G be the generic on +M+(ω, µ, δ) induced by G and let r be the Add(ω)-generic induced by G via π{δ} +Add. +We use Fact 13 to obtain a stationary set S∗ +δ ⊂ Pµ(H(δ))V [ ¯ +G][r] such that for all +N ∈ S∗ +δ, N ∩ δ /∈ V [ ¯G] and such that S∗ +δ is also internally approachable by a ω- +sequence. Therefore we can apply Lemma 22 with Fact 11 and then Fact 9 to find +that S∗ +δ is stationary in V [G]. We then let Sδ = {N ∩ δ : N ∈ S∗ +δ}, and we see that +⟨Sδ : δ ∈ S⟩ is a disjoint stationary sequence. +□ +2.2. Proving the Main Theorems. Now we will apply the new version of Mitchell +forcing to answer Krueger’s questions. Theorem 1 follows quickly: +Proof of Theorem 1. Begin with a ground model V in which λ1 < λ2 and the λ’s +are Mahlo. +Let M1 = M+(ω, ℵ1, λ1). +(Any λ1-sized forcing that turns λ1 into +ℵ2 and adds a disjoint stationary sequence on ℵ2 would work, so we could also +use a more standard mixed support iteration.) Then let ˙M2 be an M1-name for +M+(ω, λ1, λ2). We argue that if G1 is M1-generic over V and G2 is ˙M2[G1]-generic +over V [G1], then V [G1][G2] |= “DSS(λ1)∧DSS(λ2)”. We get DSS(λ2) from the fact +that λ2 remains Mahlo in V [G1] together with Proposition 25, so we only need to +argue that the disjoint stationary sequence ⃗S := ⟨Sα : α ∈ S⟩ ∈ V [G1] remains a +disjoint stationary sequence in V [G1][G2]. +Working in V [G1], preservation of ⃗S follows from the projection analysis: Let H1 +and H2 be chosen so that H1 is T := T(M2)-generic over V [G1], H2 is Add(ω, λ2)V [G1]- +generic over V [G1][H1], and V [G1][G2] ⊆ V [G1][H1][H2]. Since T is λ1-closed, it +preserves stationarity of S and the Sα’s, and Add(ω, λ2)V [G1] still has the countable +chain condition in V [G1][H1]. It follows that the stationarity of S is preserved in +V [G1][H1][H2], as well as the stationarity of the Sα’s (by Fact 9). Therefore ⃗S is a +disjoint stationary sequence on λ1 in V [G1][G2]. +□ +It will take a bit more work to show that Theorem 2 holds in the same model +given for Theorem 1. Note that we cannot just apply Fact 6 because 2ω = ℵ3 in +the model for Theorem 1, plus it is consistent that there can be a stationary set +which is internally unbounded but not internally stationary [13]. +We will give some facts on preservation of the distinction between stationary +sets that are internally stationary but not internally club: +Proposition 26. Suppose P is ν-closed and S ⊆ Pδ(X) is a stationary set such +that |X|<δ ≤ ν and δ ≤ ν. Then ⊩P “S is stationary in Pδ(X)”. +Proof. Let +˙C be a P-name for a club in Pδ(X). +Let ⃗x = ⟨xξ : ξ ≤ ¯ν⟩ be an +enumeration of Pδ(X) (where ¯ν ≤ ν). We construct a sequence ⃗z = ⟨zξ : ξ ≤ ¯ν⟩ ⊆ +Pδ(X) and a ≤P-descending sequence ⟨pξ : ξ ≤ ¯ν⟩ such that for all ξ, pξ ⊩ “xξ ⊆ +zξ ∈ ˙C”. Let D be the set of unions � +i<¯δ zξi for all increasing chains ⟨zξi : i < ¯δ⟩ ⊂ +⃗z (where ¯δ < δ). Since D is a club in Pδ(X) defined in V , there is some w ∈ D ∩ S. +Let ⟨zξi : i < ¯δ⟩ be an ⊆-increasing chain with ¯δ < δ such that � +i<¯δ zξi = w and +let ξ∗ < ¯ν be such that ξ∗ > supi<¯δ ξi. Then pξ∗ ⊩ “w ∈ ˙C ∩ S”. +□ +Proposition 27. Let P1 have the δ-chain condition, let P2 be ν-closed, and let X +be a set such that |X|δ ≤ ν with δ+ ≤ ν. If S ⊆ [X]δ is stationary and internally + +ON DISJOINT STATIONARY SEQUENCES +9 +stationary but not internally club, then P1 × P2 forces that S is stationary and +internally stationary but not internally club. +Proof. First, S remains stationary in the extension by P2 by Proposition 26, and it +remains stationary in the further extension by P1 by the fact that P1 still has the +δ-chain condition together with Fact 9. If N ∈ S, then N ∩ Pδ(N) is stationary, so +its stationarity is preserved by the same reasoning, using the fact that we still have +the appropriate chain condition. The fact that N is not internally club is preserved +in the extension by P2 because of ν-closure and the fact that δ ≤ ν, and then it +is preserved in the further extension by P1 because the proof of Fact 9 shows that +added clubs contain ground model clubs. +□ +We use a concept from Harrington and Shelah to handle Mahlo cardinals: +Definition 28. [9] Let N be a model of some fragment of ZFC. We say that M ≺ N +is rich if the following hold: +(1) λ ∈ M; +(2) ¯λ := M ∩ λ ∈ λ; +(3) ¯λ is an inaccessible cardinal in N; +(4) The size of M is ¯λ; +(5) M is closed under < ¯λ-sequences and ¯λ < λ. +Lemma 29. If λ is Mahlo, then M+(ω, µ, λ) forces that there are stationarily many +Z ∈ [µ+]µ which are internally stationary but not internally club. +This follows Krueger’s proof of [15, Theorem 10.1], making necessary changes +for Mahlo cardinals, and including enough details to show that we can get the +necessary preservation of stationarity simply from the projection analysis. We do +not need guessing functions (which are used in Krueger’s argument) because we are +only obtaining one instance of separation per large cardinal. +Proof of Lemma 29. Denote M := M+(ω, µ, λ) and let ˙C be an M-name for a club +in ([H(µ+)]µ)V [M]. We want to find an M-name ˙Z for an element of ([H(µ+)]µ)V [M]∩ +˙C that is internally stationary but not internally club. Let ˙F be an M-name for +a function (H(µ+)V [M])<ω → H(µ+)V [M] with the property that all of its closure +points are in ˙C. Let Θ be as large as needed for the following discussion and let N +be the structure (H(Θ), ∈, <Θ, M, ˙F, λ, µ) where <Θ is a well-ordering of H(Θ). +Since λ is Mahlo, we can find some M ≺ N with µ ⊂ M that is a rich submodel of +cardinality ¯λ. Now set G to be M-generic over V . Note that H(λ)V [G] = H(λ)[G] +because M has the λ-chain condition and M ⊂ H(λ). We will argue that Z := +M[G] ∩ H(λ)[G] is what we are looking for. +Claim 30. Z ∈ C := ˙C[G]. +Proof. We have ¯λ ≤ |Z| ≤ |M| ≤ ¯λ and ¯λ has cardinality µ in N[G], so Z ∈ +[H(λ)V [G]]µ. If a1, . . . , an ∈ Z, there are M-names ˙b1, . . . , ˙bn ∈ M ∩ H(λ) such +that ai = ˙bi[G] for all 1 ≤ i ≤ n. +By elementarity, M contains the <Θ-least +maximal antichain A ⊂ M of conditions deciding ˙F(˙b1, . . . , ˙bn). Since |A| < λ, +|A| ∈ M∩λ = ¯λ, so it will follow that A ⊂ M. Therefore if p ∈ G∩A, then p ∈ M in +particular, so p ⊩ ˙F(˙b1, . . . , ˙bn) = ˙b∗ for some ˙b∗ ∈ M∩H(λ) where we automatically +get ˙b∗ ∈ H(¯λ), and therefore F(a1, . . . , an) = a∗ := ˙b∗[G] ∈ M[G] ∩ H(λ)[G] = Z +(where of course F := ˙F[G]). +□ + +10 +MAXWELL LEVINE +For the rest of the proof let ¯G := πM(G) where πM is the Mostowski collapse +relative to M. Since πM(M) = M+(ω, µ, ¯λ), there is an extension πM : M[G] ∼= +πM(M)[ ¯G]. We also denote h := πM(H(λ)[G] ∩ M[G]). Note that h<¯λ ⊂ h by the +facts that M is rich and πM(M) has the ¯λ-chain condition. +Claim 31. Z is internally stationary. +Proof. First, we argue that S := Pµ(h)N[ ¯ +G] is stationary in N[G]. By Lemma 22, +the quotient M/ ¯G is a projection of a forcing of the form A1 ∗ ( ˙T × A2) where A1 +has the countable chain condition, ˙T is an A1-name for a µ-closed forcing, and A2 +also has the countable chain condition. Let K1, KT , and K2 be respective generics +such that V [G] ⊆ V [ ¯G][K1][KT ][K2]. Working in N[ ¯G], note that S′ ∩ IA(ω) is +stationary, and therefore has its stationarity preserved in V [ ¯G][K1] by Fact 9. +We must also show that the stationarity of S′ will be preserved by countably +closed forcings over N[ ¯G][K1]. Suppose ⟨Mn : n < ω⟩ witnesses internal approach- +ability of some N ∈ S′ in V [ ¯G] with respect to the structure H(λ+)V [ ¯ +G], and let +Mω := � +n<ω Mn. Then we can see that ⟨Mn[K1] : n < ω⟩ is a chain of elementary +submodels of H(λ)[ ¯G][K1] = H(λ)V [ ¯ +G][K1]. We also have Mn[K1] ∩ V [ ¯G] = M and +Mω[K1] ∩ V [ ¯G] = Mω ∈ S′ with Mω[K1] ≺ H(λ)V [ ¯ +G][K1]. If we choose the Mn’s to +be elementary substructures of H(λ+)V [ ¯ +G](∈, <∗, ˙C, . . .) where <∗ is a well-ordering +and ˙C is a A1 ∗ ˙T-name for a club, then an argument almost exactly like the one +showing that internal approachability is preserved (i.e. the proof of Fact 11) will +show that S′ is stationary in N[ ¯G][K1][KT ]. +Then the extension of N[ ¯G][K1][KT][K2] over N[ ¯G][K1][KT ] preserves the sta- +tionarity of S′ by another application of Fact 9, so we get stationarity in N[G]. +Now that we have established preservation of stationarity of S′, we can finish +the argument. Since |h| = µ in N[G], we can write h = � +i<µ xi where ⟨xi : i < µ⟩ +is a continuous and ⊂-increasing chain of elements of Pµ(h). The chain is a club +in h, so there is a stationary X ⊆ µ such that {xi : i ∈ X} ⊆ T . For all i < µ, +the fact that |xi| < µ implies that xi ∈ h, and so xi = πM(yi) for some yi ∈ Z. +Therefore ⟨yi : i < µ⟩ is ⊂-increasing and continuous with union Z, and in particular +⟨yi : i ∈ X⟩ is stationary in Z. +□ +Claim 32. Z is not internally club. +Proof. Suppose for contradiction that Z is internally club and hence that there is +a ⊂-increasing and continuous chain ⟨Zi : i < µ⟩ ∈ N[G] with |Zi| < µ for all i < µ +and � +i<µ Zi = Z. So for all i < µ, Zi ⊂ Z, and so ⟨πM[Zi] : i < µ⟩ is an ⊂- +increasing and continuous chain with union h. If we let Wi := πM[Zi] for all i < µ, +then the fact that |Wi| < µ implies that Wi = πM(Zi). Therefore ⟨Wi : i < µ⟩ is a +continuous and ⊂-increasing chain of sets in Pµ(h) with union h. +Next we define a set U ∈ N[ ¯G][r] (where r is the generic induced by G from +π{¯λ} +Add) as +{A ∈ Pµ(H(χ)) ∩ IA(ω) : A ∩ h /∈ N[ ¯G]}. +We have a real in N[ ¯G][r] \ N[ ¯G] and (µ+)N[ ¯ +G][r] = λ ⊂ H(λ). Hence we apply +Fact 13 to see that U is stationary in N[ ¯G][r], and it remains stationary in N[G] by +the preservation properties of the quotient (i.e. Lemma 22 combined with Fact 11 +and Fact 9). Therefore in N[G], {A ∩ h : A ∈ U} is stationary in Pµ(h). Since +⟨Wi : i < µ⟩ is club in h, there is some i < µ such that Wi = A ∩ h for some A ∈ U. + +ON DISJOINT STATIONARY SEQUENCES +11 +But by definition, A ∩ h /∈ N[ ¯G], and subsets of Wi of cardinality < ¯λ are in N[ ¯G], +so this is a contradiction. +□ +This completes the proof of the lemma. +□ +Proof of Theorem 2. Let M1 be any λ1-sized forcing that turns λ1 into ℵ2 and adds +stationarily many N ∈ [H(ℵ2)]ℵ1 that are internally stationary but not internally +club. Let ˙M2 be an M1-name for M+(ω, λ1, λ2), let G1 be M1-generic over V , and +let G2 be ˙M2[G1]-generic over V [G1]. Then we can see that the theorem holds +in V [G1][G2]: the distinction between internally stationary and internally club on +[H(ℵ2)]ℵ1 is preserved in V [G1][G2] by Proposition 27, and we get a distinction +between internally stationary and internally club for [H(ℵ3)]ℵ2 by Lemma 29. +□ +3. A Club Forcing and a Guessing Sequence +3.1. A review of the tools. The main idea of the proof of Theorem 3 is to force +a club through the complement of a canonical stationary set, which is described as +follows: +Fact 33 (Krueger,[15]). Suppose µ is an uncountable regular cardinal and µ<µ ≤ +µ+. Let x = ⟨xα : α < µ+⟩ enumerate [µ+]<µ and let +S(x) := {α ∈ µ+ ∩ cof(µ) : Pµ(α) \ ⟨xβ : β < α⟩ is stationary}. +Then DSS(µ+) holds if and only if S(x) is stationary. +The natural thing to do is to define the following: +Definition 34. Let µ be an uncountable regular cardinal such that µ<µ = µ+ +and let x and S(x) be defined as in Fact 33. Then let P(x) be the set of closed +bounded subsets p of µ+ such that p ∩ S(x) = ∅. We let p′ ≤ p if and only if +p′ ∩ (max p + 1) = p. +We will also crucially need a characterization of diamonds. This following ap- +pears in joint work with Gilton and Stejskalov´a [6]. +Fact 35. The following are equivalent: +(1) λ is Mahlo and ♦λ(Reg) (where of course Reg = {τ < λ : τ regular}) holds. +(2) There is a function ℓ : λ → Vλ such that for every transitive structure N +satisfying a rich fragment of ZFC that is closed under λ+-sequences in V , +the following holds: For every A ∈ N with A ∈ H(λ+) and any a ⊂ H with +|a| < λ, there is a rich M ≺ N with a ∪ {ℓ} ⊂ M such that ℓ(¯λ) = πM(A) +(where ¯λ = M ∩ λ and πM is the Mostowski collapse).3 +We can always use such an ℓ assuming the consistency of a Mahlo cardinal: If λ +is Mahlo in a model V , then it is Mahlo in G¨odel’s class L where ♦λ(S) holds for +all regular λ and stationary S ⊂ λ. +We use a poset that appears in Gilton’s thesis [7] and is discussed in the same +paper with the guessing sequence [6]. We denote this poset MG +ℓ (κ, λ) and black-box +its basic properties: +Fact 36. [3, 7] The following hold for MG +ℓ (κ, λ): +3The original is stated with a different quantification—for all rich structures, there exists a +function, not the other way around. However, the proof works with the quantification used here. + +12 +MAXWELL LEVINE +• MG +ℓ (κ, λ) has the λ-chain condition; +• MG +ℓ (κ, λ) is κ-closed; +• If ℓ(δ) = P for some κ+-closed forcing, then we have the forcing equivalence: +MG +ℓ (κ, λ) ≃ MG +ℓ (κ, δ) ∗ (P × Add(κ, δ⊕)) ∗ Nδ⊕ +where: +– α⊕ takes the least inaccessible larger than α, and +– Nδ⊕ is a projection of a product of a square-κ+-cc and a κ+-closed +forcing. +3.2. The proof. Now we prove Theorem 3. Fix κ and λ as in the statement of +the theorem and let µ = κ+. We can assume that ♦λ(Reg) holds, so let ℓ witness +Fact 35 and let M = MG +ℓ (κ, λ). We have V [M] |= µ<µ ≤ µ+, so we fix an M-name +˙x of [µ+]<µ in V [M] as well as a sequence of names ⟨ ˙xα : α < µ+⟩ that canonically +represent the elements listed by ˙x. Then let ˙P be an M-name for P( ˙x). Let G be +M-generic over V and let H be P := ˙P[G]-generic over V [G]. Then the model in +which the theorem is realized is V [G][H]. +Note: If M ≺ N is rich and πM is the Mostowski collapse relative to M, we will +typically denote πM(a) as ¯a. +The following lemma is the crux of the proof: +Lemma 37. Let M ≺ N be a rich model chosen to witness Fact 35 in the sense +of having the properties that M ∩ λ = ¯λ and ℓ(¯λ) = πM( ˙P(˙x)). Suppose ¯G0 ∗ ¯H0 is +¯M ∗ ¯P-generic over V . +Then there is a G0 ∗ H0 which is M ∗ P( ˙x)-generic over V and a rich M ≺ N +such that: +(1) if j : ¯M → M ⊂ N is the inverse of the Mostowski collapse, then there is a +lift j : ¯M[ ¯G0][ ¯H0] → N[G0][H0]; +(2) ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0]; +(3) N[G0] is an extension of N[ ¯G0][ ¯H0] by Add(κ, (M ∩ λ)⊕) ∗ N(M∩λ)⊕. +Proof. We will lift the elementary embedding j : ¯M → N to j : ¯M[ ¯G0][ ¯H0] → +N[G0][H0]. We therefore fix the notation ¯λ = M ∩ λ, and we have an ¯M-generic +¯G0, so we let P = ˙P( ˙x)[G0]. +To perform the lift, we need to show that we can absorb the generic ¯H0. We +can see that N[ ¯G0] |= “πM(P) is ¯λ-closed”, which follows from the fact that ¯M has +the ¯λ-chain condition. By the guessing property of ℓ we have a forcing equivalence +M/G0 ≃ (¯P × Add(κ, ¯λ⊕)) ∗ ˙N¯λ⊕, giving us (3). +The first stage of the lift j : ¯M[ ¯G0] → N[G0] works by choosing a generic G′ over +M/ ¯G0 such that G′ projects to ¯H0. Then we let G0 = ¯G0 × G′ and we see that +j” ¯G0 ⊆ G0. +To lift the embedding further, we use a master condition argument. Specifically, +we want to show that ∪ ¯H0 ∪ {¯λ} is a condition in P. This follows because ¯λ /∈ S(x) +(as evaluated in N[G0]) because ¯M[ ¯G0]<¯λ ⊂ ¯M[ ¯G0] and therefore Pµ(¯λ) \ ⟨xβ : β < +¯λ⟩ will be empty, so of course it will be nonstationary. Hence we choose H0 to be a +generic containing ∪ ¯H0 ∪ {¯λ} . It then follows that ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0], giving +us (2). +□ +Proposition 38. ˙P[G] is λ-distributive over V [G]. + +ON DISJOINT STATIONARY SEQUENCES +13 +Proof. Suppose there were (m, ˙p) ∈ M ∗ ˙P forcing that some ˙f collapses λ over +V . Then a suitably-chosen N := (H(Θ), ∈, <Θ, M ∗ ˙P, (m, ˙p), ˙f, . . .) would contain +the <Θ-least such example, and so we can find a rich M ≺ N witnessing Fact 35 +with (m, ˙p) ∈ M and such that ℓ(¯λ) = πM( ˙P). Then (2) from Lemma 37 obtains a +contradiction. +□ +Proposition 39. V [G][H] |= ¬DSS(µ+). +Proof. Since P is λ-distributive over V [G], x remains an enumeration of [µ+]<µ in +V [M][P]. Moreover, P forces that S(x) is nonstationary in V [M][P], so we can apply +Fact 33. +□ +Proposition 40. V [G][H] |= ¬AP(µ+). +Proof. This is exactly as in Lemma 5.9 [6], where we imitate the argument of “The +Eightfold Way” and use property (3) of the lift, except that here P stands for a +P(x) rather than the iteration Pα used in [6]. The main point is that if we are +using an embedding j : ¯M[ ¯G][ ¯H] → N[G][H], then the extension by G ∗ H over +the extension by ¯G ∗ ¯H has the correct branch preservation properties (as given by +the distributivity of ˙P[G] and the closure and square-chain condition of the posets +projecting onto N¯λ⊕). +□ +Now we are finished with the proof of Theorem 3. +4. Further directions +We propose some other considerations along the lines of the question: Why did +we have to do more work to get Theorem 2 after obtaining Theorem 1? Or rather, +is the assumption 2µ = µ+ necessary for Fact 6? +Question 1. Is it consistent for µ regular that exactly one of DSS(µ+) and “inter- +nally club and internally unbounded are distinct for [H(µ+)]µ” holds? +On a similar note, the assumption that 2µ = |H(µ+)| is also used in a folklore +result that assuming 2µ = µ+, the distinction between internally unbounded and +internally approachable for [µ+]µ requires a Mahlo cardinal. +Question 2. What is the exact equiconsistency strength of the separation of inter- +nally approachable and internally unbounded for [H(µ+)]µ for regular µ? +References +[1] Uri Abraham. Aronszajn trees on ℵ2 and ℵ3. Ann. Pure Appl. Logic, 24(3):213–230, 1983. +[2] Sean D. Cox. Forcing axioms, approachability, and stationary set reflection. J. Symb. Log., +86(2):499–530, 2021. +[3] James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot, and Dima Sinapova. +The eightfold way. Journal of Symbolic Logic, 83(1):349–371, 2018. +[4] Sy-David Friedman and John Krueger. Thin stationary sets and disjoint club sequences. +Trans. Amer. Math. Soc., 359(5):2407–2420, 2007. +[5] Thomas Gilton and John Krueger. A note on the eightfold way. Proc. Amer. Math. Soc., +148(3):1283–1293, 2020. +[6] Thomas Gilton, Maxwell Levine, and ˇS´arka Stejskalov´a. Trees and stationary reflection at +double successors of regular cardinals. Journal of Symbolic Logic. To appear. +[7] Thomas Daniells Gilton. On the Infinitary Combinatorics of Small Cardinals and the Car- +dinality of the Continuum. ProQuest LLC, Ann Arbor, MI, 2019. Thesis (Ph.D.)–University +of California, Los Angeles. + +14 +MAXWELL LEVINE +[8] Moti Gitik. Nonsplitting subset of Pκ(κ+). J. Symbolic Logic, 50(4):881–894 (1986), 1985. +[9] Leo Harrington and Saharon Shelah. Some exact equiconsistency results in set theory. Notre +Dame Journal of Formal Logic, 26(2):178–188, 1985. +[10] Thomas Jech. Set Theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, the +third millennium, revised and expanded edition, 2003. +[11] John Krueger. Internally club and approachable. Adv. Math., 213(2):734–740, 2007. +[12] John Krueger. A general Mitchell style iteration. MLQ Math. Log. Q., 54(6):641–651, 2008. +[13] John Krueger. Internal approachability and reflection. J. Math. Log., 8(1):23–39, 2008. +[14] John Krueger. Internally club and approachable for larger structures. Fund. Math., +201(2):115–129, 2008. +[15] John Krueger. Some applications of mixed support iterations. Ann. Pure Appl. Logic, 158(1- +2):40–57, 2009. +[16] Matteo Viale. Guessing models and generalized Laver diamond. Ann. Pure Appl. Logic, +163(11):1660–1678, 2012. + diff --git a/89E0T4oBgHgl3EQfwgGc/content/tmp_files/load_file.txt b/89E0T4oBgHgl3EQfwgGc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86a86ef174141af820bfd64e28ece58f761a2b92 --- /dev/null +++ b/89E0T4oBgHgl3EQfwgGc/content/tmp_files/load_file.txt @@ -0,0 +1,495 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf,len=494 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='02634v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='LO] 6 Jan 2023 ON DISJOINT STATIONARY SEQUENCES MAXWELL LEVINE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We answer a question of Krueger by obtaining disjoint stationary sequences on successive cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The main idea is an alternative presentation of a mixed support iteration, using it even more explicitly as a variant of Mitchell forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We also use a Mahlo cardinal to obtain a model in which ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2, answering a question of Gilton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Introduction and Background In order to develop a more vivid picture of the infinite cardinals, set theorists study a variety of objects that can potentially exist on these cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The objects of interest for this paper are called disjoint stationary sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' These were intro- duced by Krueger to answer a question of Abraham and Shelah about forcing clubs through stationary sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Beginning in joint work with Friedman, Krueger wrote a series of papers in this area, connecting a wide range of concepts and answering seemingly unrelated questions of Foreman and Todorˇcevi´c [4, 11, 12, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Generally, the new arguments hinged on the behavior of two-step iterations of the form Add(τ) ∗ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' In order to extend the application of these arguments as widely as possible, Krueger developed the notion of mixed support forcing [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' These forcings are to some extent an analog of the forcing that Mitchell used to obtain the tree property at double successors of regular cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Their most notable feature is the appearance of quotients insofar as the forcings took the form M ≃ ¯M ∗ Add(τ) ∗ Q where ¯M is a partial mixed support iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The appearance of Add(τ) after the initial component, together with the preservation properties of the quotient Q, allowed Krueger’s new arguments to go through various complicated constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Mixed support iterations have found several applications since [5], particularly in regard to guessing models [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The main idea in this paper is to use a version of Mitchell forcing to accomplish the task of a mixed support iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Specifically, this version of Mitchell forcing takes the form M ≃ ¯M ∗ Add(τ) ∗ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1 The trick used to obtain this structural property is reminiscent of the one usd by Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' in “The Eightfold Way” to demonstrate that subtle variations in the definitions of Mitchell forcing—up to merely shifting a L´evy collapse by a single coordinate—can substantially alter the properties of the forcing extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The benefit of the forcing used here is that it comes with a projection analysis of the sort that Abraham used for Mitchell forcing [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Both the forcing itself and its quotients are projections of products of the form 1The extent to which all variations of these forcings are equivalent or not is left as a loose end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Here we only deal with the case where the two-step iteration Add(τ) ∗ P takes the form Add(τ) ∗ ˙ Col(µ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 1 2 MAXWELL LEVINE A× T where A has a good chain condition and T has a good closure property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This allows us to obtain preservation properties conveniently, without having to delve into too many technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Abraham in fact used this projection analysis to extend Mitchell’s result to successive cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This is exactly what we do here for disjoint stationary sequences, answering the first component of a question of Krueger [15, Question 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='8]: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose λ1 < λ2 are two Mahlo cardinals in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then there is a forcing extension in which there are disjoint stationary sequences on ℵ2 and ℵ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We lay out the basic definition and concepts in the following subsections and then develop the proof in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We also achieve one of Krueger’s separations for successive cardinals, which answers a component of another one of his questions [15, Question 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='9]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose λ1 < λ2 are two Mahlo cardinals in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then there is a forc- ing extension in which for µ ∈ {ℵ1, ℵ2}, there are stationarily many N ∈ [H(µ+)]µ that are internally stationary but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The last main result is motivated by work of Gilton and Krueger, who answered a question from “The Eightfold Way” by obtaining stationary reflection for subsets of ℵ2 ∩ cof(ω) together with failure of approachability at ℵ2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ℵ2 /∈ I[ℵ2]) using disjoint stationary sequences [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This result used the fact that the existence of a disjoint stationary sequence implies failure of approachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Gilton asked for the exact consistency strength of the failure of approachability at ℵ2 together with the nonexistence of a disjoint stationary set on ℵ2 [7, Question 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (He pointed out that Cox found this separation using PFA [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=') It is known that the failure of approachability requires the consistency strength of a Mahlo cardinal, and in Section 3 we show that a Mahlo cardinal is sufficient for the separation: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose that λ is Mahlo in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then there is a forcing extension in which ℵ2 /∈ I[ℵ2] and there is no disjoint stationary sequence on ℵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Disjoint stationary sequences are known to be interpretable in terms of canonical structure (see Fact 6 below), and the main idea for Theorem 3 is a simple master condition argument that exploits this connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We note that all three of these theorems can be generalized to arbitrarily high cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Basic Definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We assume familiarity with the basics of forcing and large cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We use the following conventions: If P is a forcing poset, then p ≤ q for p, q ∈ P means that p is stronger than q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We say that P is κ-closed if for all ≤P-decreasing sequences ⟨pξ : ξ < τ⟩ with τ < κ, there is a lower bound p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' p ≤ pξ for all ξ < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We say that P has the κ-chain condition of all antichains A ⊆ P have cardinality strictly less than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Now we give our main definitions: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Given a regular cardinal µ, a disjoint stationary sequence on µ+ is a sequence ⟨Sα : α ∈ S⟩ such that: S ⊆ µ+ ∩ cof(µ) is stationary, Sα is a stationary subset of Pµ(α) for all α ∈ S, Sα ∩ Sβ = ∅ if α ̸= β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ON DISJOINT STATIONARY SEQUENCES 3 We write DSS(µ+) to say that there is a disjoint stationary sequence on µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Given a stationary N ∈ [H(Θ)]κ,2 we say: N is internally unbounded if ∀x ∈ Pκ(N), ∃M ∈ N, x ⊆ M, N is internally stationary if Pκ(N) ∩ N is stationary in Pκ(N), N is internally club if Pκ(N) ∩ N is club in Pκ(N), N is internally approachable if there is an increasing and continuous con- tinuous chain ⟨Mξ : ξ < κ⟩ such that |Mξ| < κ and ⟨Mη : η < ξ⟩ ∈ Mξ+1 for all ξ < κ such that N = � ξ<κ Mξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Although disjoint stationary sequences may seem unrelated the separation of variants of internal approachability, there are deep connections here, for example: Fact 6 (Krueger, [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If µ is regular and 2µ = µ+, then DSS(µ+) is equivalent to the existence of a stationary set U ⊆ [H(µ+)]µ such that every N ∈ U is internally internally unbounded but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Projections and Preservation Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Technically speaking, our main goal is to show that certain forcing quotients behave nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will make an effort to demonstrate the preservation properties of these quotients directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' These quotients will be defined in terms of projections: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If P1 and P2 are posets, a projection is an onto map π : P1 → P2 such that: p ≤ q implies that π(p) ≤ π(q), if r ≤ π(p), then there is some q ≤ p such that π(q) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' A projection is trivial if π(p) = π(q) implies that p and q are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Trivial projections are basically ismorphisms: Fact 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If π : P1 → P2 is a trivial projection, then P1 ≃ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For our purposes, we are interested in the preservation of stationary sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The chain condition gives us preservation fairly straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The following fact is implicit in parts of the literature, and a version of it can be found in this paper in the form of Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Fact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If P has the µ-chain condition and S ⊂ Pµ(X) is stationary, then P forces that S is stationary in P V µ (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' However, we must place demands on our stationary sets in order for them to be preserved by closed forcings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' A stationary set S ⊂ Pµ(H(Θ)) is internally approachable of length τ if for all N ∈ S with N ≺ H(Θ), there is a continuous chain of elementary submodels ⟨Mi : i < τ⟩ such that N = � i<τ Mi and for all i < τ, ⟨Mi : i < j⟩ ∈ Mj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' In this case we write S ⊆ IA(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Fact 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If S ⊂ Pµ(H(Θ)) ∩ IA(τ) is an internally approachable stationary set, τ < µ, and P is µ-closed, then P forces that S is stationary in Pµ(H(Θ)V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 2See Jech for details on stationary sets [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 4 MAXWELL LEVINE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Costationarity of the Ground Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The notion of ground model co- stationarity is a key ingredient in arguments pertaining to disjoint stationary se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' It will specifically give us the disjointness, since we will be picking sta- tionary sets that are not added by initial segments of these forcings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Gitik obtained the classical result: Fact 12 (Gitik [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If V ⊂ W are models of ZFC with the same ordinals, W \\ V contains a real, and κ is a regular cardinal in W such that (κ+)W ≤ λ, then P W κ (λ) \\ V is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Because we will need Fact 11, we will actually use Krueger’s refinement of Gitik’s theorem: Fact 13 (Krueger [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose V ⊂ W are models of ZFC with the same ordinals, W \\ V contains a real, µ is a regular cardinal in W, and X ∈ V is such that (µ+)W ⊆ X, and that in W, Θ is a regular cardinal such that X ⊂ H(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then in W the set {N ∈ Pµ(H(Θ)) ∩ IA(ω) : N ∩ X /∈ V } is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The New Mitchell Forcing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Defining the Forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' In this subsection we will illustrate the basic idea of this paper by using our new take on Mitchell forcing to prove a known result: Theorem 14 (Krueger [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If λ is a Mahlo cardinal and µ < λ are regular cardinals, there is a forcing extension in which 2ω = µ+ = λ and there is a disjoint stationary sequence on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Specifically, we will define a forcing M+(τ, µ, λ) such that the model W in Theorem 14 can be realized as an extension by M+(ω, µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For standard technical reasons, we define a poset ismorphic to Add(τ, λ): Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Given a regular τ and a set of ordinals Y , we let Add∗(τ, Y ) be the poset consisting of partial functions p : {δ ∈ Y : δ is inaccessible} × τ → {0, 1} where | dom p| < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We let p ≤Add∗(τ,Y ) q if and only if p ⊇ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Note: In later subsections we will conflate Add(τ, λ) and Add∗(τ, λ) to simplify notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let λ be inaccessible and let τ < µ < λ be regular cardinals such that τ <τ = τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We define a forcing M+(τ, µ, λ) that consists of pairs (p, q) such that: (1) p ∈ Add∗(τ, λ), (2) q is a function such that: (a) dom q ⊂ {δ < λ : δ is inaccessible}, (b) | dom q| < µ, (c) ∀δ ∈ dom(q), p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ∈ ˙ Col(µ, δ)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We let (p, q) ≤ (p′, q′) if and only if: (i) p ≤Add∗(τ,λ) p′, (ii) dom q ⊇ dom q′, (iii) for all δ ∈ dom q′, p↾((δ + 1) × τ) ⊩Add∗(τ,δ+1) “q(δ) ≤ ˙ Col(µ,δ) q′(δ)” First we go through the more routine properties that one would expect of this forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ON DISJOINT STATIONARY SEQUENCES 5 Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' M+(τ, µ, λ) is τ-closed and λ-Knaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Closure uses the facts that Add∗(τ, λ) is τ-closed and ⊩Add(τ,δ+1) “ ˙ Col(µ, δ) is µ-closed” for all δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Knasterness uses a standard application of the Delta System Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Crucially, we get a nice termspace: Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let T = T(M+(τ, µ, λ)) be the poset consisting of conditions q such that: (1) dom q ⊂ λ ∩ {δ < λ : δ is inaccessible}, (2) | dom q| < µ, (3) ∀δ ∈ dom q, ⊩Add∗(τ,δ+1) q(δ) ∈ ˙ Col(µ, δ)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Most importantly, we let q ≤ q′ if and only if: (i) dom q ⊇ dom q′, (ii) for all δ ∈ dom q, ⊩Add∗(τ,δ+1) “q(δ) ≤ q′(δ)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proposition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' There is a projection Add∗(τ, λ)×T(M+(τ, µ, λ)) ։ M+(τ, µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We let π be the projection with the definition π(p, q) = (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This is au- tomatically order-preserving because the ordering ≤Add∗(τ,λ)×T is coarser than the ordering ≤M+(τ,µ,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For obtaining the density condition, suppose (r, s) ≤M+(τ,µ,λ) (p0, q0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We want to find some (p1, q1) such that (p1, q1) ≤Add∗(τ,λ)×T (p0, q0) and (p1, q1) ≤M+(τ,µ,λ) (r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' To do this, we first let p1 = r, and then we define q1 with dom q1 = dom r such that at each coordinate δ ∈ dom q1, we use standard arguments on names to show that we can get both p0 ↾ ((δ + 1) × τ) ⊩Add∗(τ,λ) “q1(δ) ≤ s(δ)” as well as 1Add∗(τ,λ) ⊩Add∗(τ,λ) “q1(δ) ≤ q0(δ)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' T is µ-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This as an application of the Mixing Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Given a ≤T-decreasing se- quence ⟨qi : i < τ⟩ with τ < µ we let d = � i<τ dom qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then we define a lower bound ¯q with domain d such that for all δ ∈ d, q(δ) is a canonically-defined name for a lower bound of the qi(δ)’s (where i is large enough that δ ∈ dom qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Then we get the standard consequences of the termspace analysis: Proposition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The following are true in any extension by M+(τ, µ, λ): (1) V -cardinals up to and including µ are cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (2) For all α < λ, |α| = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (3) λ = µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (4) 2τ = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (1) follows from the projection analysis and the fact that T is µ-closed and Add∗(τ, λ) is τ +-cc, and from τ-closure of M+(τ, µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (2) follows from the fact that for all inaccessible δ < λ, M+(τ, µ, λ) projects onto Col(µ, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (3) follows from (1) and (2) plus λ-Knasterness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (4) follows from the fact that M+(µ, λ) projects onto Add∗(τ, λ), so it forces that 2τ ≥ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since the poset has size λ, it also forces that 2τ ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ The following lemma is the crux of the new idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 6 MAXWELL LEVINE Lemma 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If δ0 < λ is inaccessible, then there is a forcing equivalence M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ) ∗ Q where M+(τ, µ, δ0) ∗ Add(τ) forces that Q is a projection of a product of a µ-closed forcing and a τ +-cc forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' More precisely, we will show that there is a forcing equivalence M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ) ∗ (P × R) where the following hold in the extension by M+(τ, µ, δ0) ∗ Add(τ): R is a projection of a product of a µ-closed forcing and Add∗(τ, λ), and V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The statement of the lemma can then be obtained by merging P with the closed component of the product that projects onto R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' First we describe P and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' To do this, we fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Given Y ⊆ λ, we let πY Add denote the projection (p, q) → p↾(Y × τ) from M+(τ, µ, λ) onto Add∗(τ, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For any poset P, we employ the convention that Γ(P) denotes a canonical name for a P-generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If X ⊂ P, then we use the notation ↑ X := {q ∈ P : ∃p ∈ X, p ≤ q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will let P := Col(µ, δ0)V [(↑(πδ0 Add”Γ(M+(τ,µ,δ0))))×Γ(Add(τ))] if we are working in an extension by M+(τ, µ, δ0) ∗ Add(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (In other words, the poset P will be the version of Col(µ, δ0) as interpreted in the extension of V by Add∗(τ, δ + 1) where the initial coordinates come from M+(τ, µ, δ0) and the last coordinate comes from the additional copy of Add(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=') Still working in an extension by M+(τ, µ, δ0) ∗ Add(τ), the poset R consists of pairs (p, q) such that the following hold: (1) p ∈ Add∗(τ, (δ0, λ)), (2) q is a function such that (a) dom q ⊂ {δ ∈ (δ0, λ) : δ is inaccessible}, (b) | dom q| < µ, (c) ∀δ ∈ dom(q), p↾((δ0, (δ + 1)) × τ) ⊩Add∗(τ,(δ0,δ+1)) “q(δ) ∈ ˙ Col(µ, δ)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The ordering is the one analogous to that of M+(τ, µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' An easy adaptation of the arguments for the projection analysis for M+(τ, µ, λ) will then give a projection analysis for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The rest of the proof of the lemma consists of verifying the more substantial claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Claim 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' M+(τ, µ, λ) ≃ M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We identify M+(τ, µ, δ0) ∗ Add(τ, 1) ∗ (P × R) with the dense subset of con- ditions ((r, s), t, u, (r, ˙s′)) such that ˙s′ is forced to have a specific domain in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The fact that this subset is dense follows from the fact that M+(τ, µ, λ) ∗ Add(τ) has the µ-covering property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will argue that there is a trivial projection defined by π : (p, q) �→ ((p↾(δ0 × τ), q↾δ0) � �� � M+(µ,δ0) , p↾({δ0} × τ) � �� � Add(τ) , q∗(δ0) � �� � P , (¯p, ¯q) � �� � R ) such that ¯p := p↾((δ0, λ) × τ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ON DISJOINT STATIONARY SEQUENCES 7 q∗(δ0) is obtained by changing q(δ0) from an Add∗(τ, δ0 + 1)-name to an Add(τ) as interpreted in the extension by the relevant generic, namely (↑ (πδ0 Add”Γ(M+(τ, µ, δ0))));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ¯q has domain (δ0, λ), and for each δ ∈ (δ0, λ), ¯q(δ) has changes analogous to the changes made to q∗(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' It is clear that π is order-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We also want to show that if ((r, s), t, u, (r′, ˙s′)) ≤M+(τ,µ,δ0)∗Add(τ)∗(P×R) π(p0, q0) then there is some (p1, q1) ≤M+(µ,λ) (p0, q0) such that π(p1, q1) ≤ ((r, s), t, u, (r′, s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This can be done by taking: p1 = r ∪ ˜t ∪ r′ where ˜t writes t as as a partial function {δ} × τ → {0, 1}, q1 = s ∪ ˜u ∪ ˜s′ where ˜u reinterprets u as a Add∗(δ0 + 1)-name and for each δ ∈ dom( ˙s′), ˜s′ reinterprets ˙s′(δ) as a Add∗(δ + 1)-name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Last, we argue that π(p0, q0) = π(p1, q1) implies that (p0, q0) and (p1, q1) are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose that (p0, q0) and (p1, q1) are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If p0 and p1 are incompatible as elements of Add∗(τ, λ), then one of pi ↾ (δ0 × τ), pi ↾ ({δ0} × τ), and pi ↾((δ0, λ) × τ) must be distinct for i = 0 and i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Otherwise, there is some p′ ≤ p0, p1 and some δ ∈ dom q0∩dom q1 inaccessible such that p′ ⊩ “q0(δ) ⊥ q1(δ)”, which implies that q0(δ) ̸= q1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore, one of qi ↾ δ0, qi(δ0), or qi ↾ (δ0, λ) is distinct for i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Claim 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P is µ-closed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' In fact, our argument will also show that V [M+(τ, µ, δ0)][Add(τ, 1)] |= “P = Col(µ, δ0)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We fix some arbitrary generics: G is M+(τ, µ, δ0)-generic over V , r is Add(τ)-generic over V [G], H is the Add∗(τ, δ0)-generic induced from G by πδ0 Add, K is the generic for the quotient of M+(τ, µ, δ0) by Add∗(τ, δ0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' the generic such that V [H][K] = V [G], T is the generic for the termspace forcing T(M+(τ, µ, δ0)), so that V [G] ⊂ V [T ][H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' It is enough to argue that V [G][r] |= “P is µ-closed” knowing that V [H][r] |= “P is µ-closed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Because adjoining G does not change the definition of Add(τ), and because K is defined in terms of the subsets of τ adjoined by the filter H, we have V [G][r] = V [H][K][r] = V [H][r][K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore, it is enough to show that K does not add <µ-sequences over V [H][r], so that V [H][r]’s version of Col(µ, δ0) remains µ-closed in V [G][r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We have V [H][r] ⊂ V [H][r][K] = V [H][K][r] = V [G][r] ⊂ V [T ][H][r] = V [H][r][T ], and Easton’s Lemma implies that T does not add new <µ-sequences over V [H][r], so therefore K does not add new <µ-sequences over V [H][r] since it is an interme- diate factor of the extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Now we have an application for the case where τ = ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If λ is Mahlo then V [M+(ω, µ, λ)] |= DSS(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This basically repeats Krueger’s argument for [15, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 8 MAXWELL LEVINE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let G be M+(ω, µ, λ)-generic over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The set of V -inaccessibles in λ will form the stationary set S ⊂ µ+ ∩ cof(µ) carrying the disjoint stationary sequence in the extension by M+(ω, µ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For every such δ ∈ S, let ¯G be the generic on M+(ω, µ, δ) induced by G and let r be the Add(ω)-generic induced by G via π{δ} Add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We use Fact 13 to obtain a stationary set S∗ δ ⊂ Pµ(H(δ))V [ ¯ G][r] such that for all N ∈ S∗ δ, N ∩ δ /∈ V [ ¯G] and such that S∗ δ is also internally approachable by a ω- sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore we can apply Lemma 22 with Fact 11 and then Fact 9 to find that S∗ δ is stationary in V [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We then let Sδ = {N ∩ δ : N ∈ S∗ δ}, and we see that ⟨Sδ : δ ∈ S⟩ is a disjoint stationary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proving the Main Theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Now we will apply the new version of Mitchell forcing to answer Krueger’s questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Theorem 1 follows quickly: Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Begin with a ground model V in which λ1 < λ2 and the λ’s are Mahlo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let M1 = M+(ω, ℵ1, λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (Any λ1-sized forcing that turns λ1 into ℵ2 and adds a disjoint stationary sequence on ℵ2 would work, so we could also use a more standard mixed support iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=') Then let ˙M2 be an M1-name for M+(ω, λ1, λ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We argue that if G1 is M1-generic over V and G2 is ˙M2[G1]-generic over V [G1], then V [G1][G2] |= “DSS(λ1)∧DSS(λ2)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We get DSS(λ2) from the fact that λ2 remains Mahlo in V [G1] together with Proposition 25, so we only need to argue that the disjoint stationary sequence ⃗S := ⟨Sα : α ∈ S⟩ ∈ V [G1] remains a disjoint stationary sequence in V [G1][G2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Working in V [G1], preservation of ⃗S follows from the projection analysis: Let H1 and H2 be chosen so that H1 is T := T(M2)-generic over V [G1], H2 is Add(ω, λ2)V [G1]- generic over V [G1][H1], and V [G1][G2] ⊆ V [G1][H1][H2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since T is λ1-closed, it preserves stationarity of S and the Sα’s, and Add(ω, λ2)V [G1] still has the countable chain condition in V [G1][H1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' It follows that the stationarity of S is preserved in V [G1][H1][H2], as well as the stationarity of the Sα’s (by Fact 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore ⃗S is a disjoint stationary sequence on λ1 in V [G1][G2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ It will take a bit more work to show that Theorem 2 holds in the same model given for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Note that we cannot just apply Fact 6 because 2ω = ℵ3 in the model for Theorem 1, plus it is consistent that there can be a stationary set which is internally unbounded but not internally stationary [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will give some facts on preservation of the distinction between stationary sets that are internally stationary but not internally club: Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose P is ν-closed and S ⊆ Pδ(X) is a stationary set such that |X|<δ ≤ ν and δ ≤ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then ⊩P “S is stationary in Pδ(X)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let ˙C be a P-name for a club in Pδ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let ⃗x = ⟨xξ : ξ ≤ ¯ν⟩ be an enumeration of Pδ(X) (where ¯ν ≤ ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We construct a sequence ⃗z = ⟨zξ : ξ ≤ ¯ν⟩ ⊆ Pδ(X) and a ≤P-descending sequence ⟨pξ : ξ ≤ ¯ν⟩ such that for all ξ, pξ ⊩ “xξ ⊆ zξ ∈ ˙C”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let D be the set of unions � i<¯δ zξi for all increasing chains ⟨zξi : i < ¯δ⟩ ⊂ ⃗z (where ¯δ < δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since D is a club in Pδ(X) defined in V , there is some w ∈ D ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let ⟨zξi : i < ¯δ⟩ be an ⊆-increasing chain with ¯δ < δ such that � i<¯δ zξi = w and let ξ∗ < ¯ν be such that ξ∗ > supi<¯δ ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then pξ∗ ⊩ “w ∈ ˙C ∩ S”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proposition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let P1 have the δ-chain condition, let P2 be ν-closed, and let X be a set such that |X|δ ≤ ν with δ+ ≤ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If S ⊆ [X]δ is stationary and internally ON DISJOINT STATIONARY SEQUENCES 9 stationary but not internally club, then P1 × P2 forces that S is stationary and internally stationary but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' First, S remains stationary in the extension by P2 by Proposition 26, and it remains stationary in the further extension by P1 by the fact that P1 still has the δ-chain condition together with Fact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If N ∈ S, then N ∩ Pδ(N) is stationary, so its stationarity is preserved by the same reasoning, using the fact that we still have the appropriate chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The fact that N is not internally club is preserved in the extension by P2 because of ν-closure and the fact that δ ≤ ν, and then it is preserved in the further extension by P1 because the proof of Fact 9 shows that added clubs contain ground model clubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ We use a concept from Harrington and Shelah to handle Mahlo cardinals: Definition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [9] Let N be a model of some fragment of ZFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We say that M ≺ N is rich if the following hold: (1) λ ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (2) ¯λ := M ∩ λ ∈ λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (3) ¯λ is an inaccessible cardinal in N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (4) The size of M is ¯λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (5) M is closed under < ¯λ-sequences and ¯λ < λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If λ is Mahlo, then M+(ω, µ, λ) forces that there are stationarily many Z ∈ [µ+]µ which are internally stationary but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This follows Krueger’s proof of [15, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1], making necessary changes for Mahlo cardinals, and including enough details to show that we can get the necessary preservation of stationarity simply from the projection analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We do not need guessing functions (which are used in Krueger’s argument) because we are only obtaining one instance of separation per large cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof of Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Denote M := M+(ω, µ, λ) and let ˙C be an M-name for a club in ([H(µ+)]µ)V [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We want to find an M-name ˙Z for an element of ([H(µ+)]µ)V [M]∩ ˙C that is internally stationary but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let ˙F be an M-name for a function (H(µ+)V [M])<ω → H(µ+)V [M] with the property that all of its closure points are in ˙C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let Θ be as large as needed for the following discussion and let N be the structure (H(Θ), ∈, <Θ, M, ˙F, λ, µ) where <Θ is a well-ordering of H(Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since λ is Mahlo, we can find some M ≺ N with µ ⊂ M that is a rich submodel of cardinality ¯λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Now set G to be M-generic over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Note that H(λ)V [G] = H(λ)[G] because M has the λ-chain condition and M ⊂ H(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will argue that Z := M[G] ∩ H(λ)[G] is what we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Claim 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Z ∈ C := ˙C[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We have ¯λ ≤ |Z| ≤ |M| ≤ ¯λ and ¯λ has cardinality µ in N[G], so Z ∈ [H(λ)V [G]]µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' , an ∈ Z, there are M-names ˙b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' , ˙bn ∈ M ∩ H(λ) such that ai = ˙bi[G] for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' By elementarity, M contains the <Θ-least maximal antichain A ⊂ M of conditions deciding ˙F(˙b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' , ˙bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since |A| < λ, |A| ∈ M∩λ = ¯λ, so it will follow that A ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore if p ∈ G∩A, then p ∈ M in particular, so p ⊩ ˙F(˙b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' , ˙bn) = ˙b∗ for some ˙b∗ ∈ M∩H(λ) where we automatically get ˙b∗ ∈ H(¯λ), and therefore F(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' , an) = a∗ := ˙b∗[G] ∈ M[G] ∩ H(λ)[G] = Z (where of course F := ˙F[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ 10 MAXWELL LEVINE For the rest of the proof let ¯G := πM(G) where πM is the Mostowski collapse relative to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since πM(M) = M+(ω, µ, ¯λ), there is an extension πM : M[G] ∼= πM(M)[ ¯G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We also denote h := πM(H(λ)[G] ∩ M[G]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Note that h<¯λ ⊂ h by the facts that M is rich and πM(M) has the ¯λ-chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Claim 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Z is internally stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' First, we argue that S := Pµ(h)N[ ¯ G] is stationary in N[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' By Lemma 22, the quotient M/ ¯G is a projection of a forcing of the form A1 ∗ ( ˙T × A2) where A1 has the countable chain condition, ˙T is an A1-name for a µ-closed forcing, and A2 also has the countable chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let K1, KT , and K2 be respective generics such that V [G] ⊆ V [ ¯G][K1][KT ][K2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Working in N[ ¯G], note that S′ ∩ IA(ω) is stationary, and therefore has its stationarity preserved in V [ ¯G][K1] by Fact 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We must also show that the stationarity of S′ will be preserved by countably closed forcings over N[ ¯G][K1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose ⟨Mn : n < ω⟩ witnesses internal approach- ability of some N ∈ S′ in V [ ¯G] with respect to the structure H(λ+)V [ ¯ G], and let Mω := � n<ω Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then we can see that ⟨Mn[K1] : n < ω⟩ is a chain of elementary submodels of H(λ)[ ¯G][K1] = H(λ)V [ ¯ G][K1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We also have Mn[K1] ∩ V [ ¯G] = M and Mω[K1] ∩ V [ ¯G] = Mω ∈ S′ with Mω[K1] ≺ H(λ)V [ ¯ G][K1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If we choose the Mn’s to be elementary substructures of H(λ+)V [ ¯ G](∈, <∗, ˙C, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=') where <∗ is a well-ordering and ˙C is a A1 ∗ ˙T-name for a club, then an argument almost exactly like the one showing that internal approachability is preserved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' the proof of Fact 11) will show that S′ is stationary in N[ ¯G][K1][KT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then the extension of N[ ¯G][K1][KT][K2] over N[ ¯G][K1][KT ] preserves the sta- tionarity of S′ by another application of Fact 9, so we get stationarity in N[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Now that we have established preservation of stationarity of S′, we can finish the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since |h| = µ in N[G], we can write h = � i<µ xi where ⟨xi : i < µ⟩ is a continuous and ⊂-increasing chain of elements of Pµ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The chain is a club in h, so there is a stationary X ⊆ µ such that {xi : i ∈ X} ⊆ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' For all i < µ, the fact that |xi| < µ implies that xi ∈ h, and so xi = πM(yi) for some yi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore ⟨yi : i < µ⟩ is ⊂-increasing and continuous with union Z, and in particular ⟨yi : i ∈ X⟩ is stationary in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Claim 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Z is not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose for contradiction that Z is internally club and hence that there is a ⊂-increasing and continuous chain ⟨Zi : i < µ⟩ ∈ N[G] with |Zi| < µ for all i < µ and � i<µ Zi = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' So for all i < µ, Zi ⊂ Z, and so ⟨πM[Zi] : i < µ⟩ is an ⊂- increasing and continuous chain with union h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If we let Wi := πM[Zi] for all i < µ, then the fact that |Wi| < µ implies that Wi = πM(Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore ⟨Wi : i < µ⟩ is a continuous and ⊂-increasing chain of sets in Pµ(h) with union h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Next we define a set U ∈ N[ ¯G][r] (where r is the generic induced by G from π{¯λ} Add) as {A ∈ Pµ(H(χ)) ∩ IA(ω) : A ∩ h /∈ N[ ¯G]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We have a real in N[ ¯G][r] \\ N[ ¯G] and (µ+)N[ ¯ G][r] = λ ⊂ H(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Hence we apply Fact 13 to see that U is stationary in N[ ¯G][r], and it remains stationary in N[G] by the preservation properties of the quotient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Lemma 22 combined with Fact 11 and Fact 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Therefore in N[G], {A ∩ h : A ∈ U} is stationary in Pµ(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since ⟨Wi : i < µ⟩ is club in h, there is some i < µ such that Wi = A ∩ h for some A ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ON DISJOINT STATIONARY SEQUENCES 11 But by definition, A ∩ h /∈ N[ ¯G], and subsets of Wi of cardinality < ¯λ are in N[ ¯G], so this is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let M1 be any λ1-sized forcing that turns λ1 into ℵ2 and adds stationarily many N ∈ [H(ℵ2)]ℵ1 that are internally stationary but not internally club.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let ˙M2 be an M1-name for M+(ω, λ1, λ2), let G1 be M1-generic over V , and let G2 be ˙M2[G1]-generic over V [G1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then we can see that the theorem holds in V [G1][G2]: the distinction between internally stationary and internally club on [H(ℵ2)]ℵ1 is preserved in V [G1][G2] by Proposition 27, and we get a distinction between internally stationary and internally club for [H(ℵ3)]ℵ2 by Lemma 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' A Club Forcing and a Guessing Sequence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' A review of the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The main idea of the proof of Theorem 3 is to force a club through the complement of a canonical stationary set, which is described as follows: Fact 33 (Krueger,[15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose µ is an uncountable regular cardinal and µ<µ ≤ µ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let x = ⟨xα : α < µ+⟩ enumerate [µ+]<µ and let S(x) := {α ∈ µ+ ∩ cof(µ) : Pµ(α) \\ ⟨xβ : β < α⟩ is stationary}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then DSS(µ+) holds if and only if S(x) is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The natural thing to do is to define the following: Definition 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let µ be an uncountable regular cardinal such that µ<µ = µ+ and let x and S(x) be defined as in Fact 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then let P(x) be the set of closed bounded subsets p of µ+ such that p ∩ S(x) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We let p′ ≤ p if and only if p′ ∩ (max p + 1) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will also crucially need a characterization of diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This following ap- pears in joint work with Gilton and Stejskalov´a [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Fact 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The following are equivalent: (1) λ is Mahlo and ♦λ(Reg) (where of course Reg = {τ < λ : τ regular}) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (2) There is a function ℓ : λ → Vλ such that for every transitive structure N satisfying a rich fragment of ZFC that is closed under λ+-sequences in V , the following holds: For every A ∈ N with A ∈ H(λ+) and any a ⊂ H with |a| < λ, there is a rich M ≺ N with a ∪ {ℓ} ⊂ M such that ℓ(¯λ) = πM(A) (where ¯λ = M ∩ λ and πM is the Mostowski collapse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='3 We can always use such an ℓ assuming the consistency of a Mahlo cardinal: If λ is Mahlo in a model V , then it is Mahlo in G¨odel’s class L where ♦λ(S) holds for all regular λ and stationary S ⊂ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We use a poset that appears in Gilton’s thesis [7] and is discussed in the same paper with the guessing sequence [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We denote this poset MG ℓ (κ, λ) and black-box its basic properties: Fact 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [3, 7] The following hold for MG ℓ (κ, λ): 3The original is stated with a different quantification—for all rich structures, there exists a function, not the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' However, the proof works with the quantification used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 12 MAXWELL LEVINE MG ℓ (κ, λ) has the λ-chain condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' MG ℓ (κ, λ) is κ-closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' If ℓ(δ) = P for some κ+-closed forcing, then we have the forcing equivalence: MG ℓ (κ, λ) ≃ MG ℓ (κ, δ) ∗ (P × Add(κ, δ⊕)) ∗ Nδ⊕ where: – α⊕ takes the least inaccessible larger than α, and – Nδ⊕ is a projection of a product of a square-κ+-cc and a κ+-closed forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Now we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Fix κ and λ as in the statement of the theorem and let µ = κ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We can assume that ♦λ(Reg) holds, so let ℓ witness Fact 35 and let M = MG ℓ (κ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We have V [M] |= µ<µ ≤ µ+, so we fix an M-name ˙x of [µ+]<µ in V [M] as well as a sequence of names ⟨ ˙xα : α < µ+⟩ that canonically represent the elements listed by ˙x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then let ˙P be an M-name for P( ˙x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let G be M-generic over V and let H be P := ˙P[G]-generic over V [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then the model in which the theorem is realized is V [G][H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Note: If M ≺ N is rich and πM is the Mostowski collapse relative to M, we will typically denote πM(a) as ¯a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The following lemma is the crux of the proof: Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Let M ≺ N be a rich model chosen to witness Fact 35 in the sense of having the properties that M ∩ λ = ¯λ and ℓ(¯λ) = πM( ˙P(˙x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose ¯G0 ∗ ¯H0 is ¯M ∗ ¯P-generic over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then there is a G0 ∗ H0 which is M ∗ P( ˙x)-generic over V and a rich M ≺ N such that: (1) if j : ¯M → M ⊂ N is the inverse of the Mostowski collapse, then there is a lift j : ¯M[ ¯G0][ ¯H0] → N[G0][H0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (2) ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' (3) N[G0] is an extension of N[ ¯G0][ ¯H0] by Add(κ, (M ∩ λ)⊕) ∗ N(M∩λ)⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We will lift the elementary embedding j : ¯M → N to j : ¯M[ ¯G0][ ¯H0] → N[G0][H0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We therefore fix the notation ¯λ = M ∩ λ, and we have an ¯M-generic ¯G0, so we let P = ˙P( ˙x)[G0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' To perform the lift, we need to show that we can absorb the generic ¯H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' We can see that N[ ¯G0] |= “πM(P) is ¯λ-closed”, which follows from the fact that ¯M has the ¯λ-chain condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' By the guessing property of ℓ we have a forcing equivalence M/G0 ≃ (¯P × Add(κ, ¯λ⊕)) ∗ ˙N¯λ⊕, giving us (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The first stage of the lift j : ¯M[ ¯G0] → N[G0] works by choosing a generic G′ over M/ ¯G0 such that G′ projects to ¯H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then we let G0 = ¯G0 × G′ and we see that j” ¯G0 ⊆ G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' To lift the embedding further, we use a master condition argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Specifically, we want to show that ∪ ¯H0 ∪ {¯λ} is a condition in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This follows because ¯λ /∈ S(x) (as evaluated in N[G0]) because ¯M[ ¯G0]<¯λ ⊂ ¯M[ ¯G0] and therefore Pµ(¯λ) \\ ⟨xβ : β < ¯λ⟩ will be empty, so of course it will be nonstationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Hence we choose H0 to be a generic containing ∪ ¯H0 ∪ {¯λ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' It then follows that ¯M[ ¯G0][ ¯H0]<¯λ ⊆ ¯M[ ¯G0], giving us (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proposition 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ˙P[G] is λ-distributive over V [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' ON DISJOINT STATIONARY SEQUENCES 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Suppose there were (m, ˙p) ∈ M ∗ ˙P forcing that some ˙f collapses λ over V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then a suitably-chosen N := (H(Θ), ∈, <Θ, M ∗ ˙P, (m, ˙p), ˙f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=') would contain the <Θ-least such example, and so we can find a rich M ≺ N witnessing Fact 35 with (m, ˙p) ∈ M and such that ℓ(¯λ) = πM( ˙P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Then (2) from Lemma 37 obtains a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proposition 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' V [G][H] |= ¬DSS(µ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Since P is λ-distributive over V [G], x remains an enumeration of [µ+]<µ in V [M][P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Moreover, P forces that S(x) is nonstationary in V [M][P], so we can apply Fact 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Proposition 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' V [G][H] |= ¬AP(µ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' This is exactly as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content='9 [6], where we imitate the argument of “The Eightfold Way” and use property (3) of the lift, except that here P stands for a P(x) rather than the iteration Pα used in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The main point is that if we are using an embedding j : ¯M[ ¯G][ ¯H] → N[G][H], then the extension by G ∗ H over the extension by ¯G ∗ ¯H has the correct branch preservation properties (as given by the distributivity of ˙P[G] and the closure and square-chain condition of the posets projecting onto N¯λ⊕).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' □ Now we are finished with the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Further directions We propose some other considerations along the lines of the question: Why did we have to do more work to get Theorem 2 after obtaining Theorem 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Or rather, is the assumption 2µ = µ+ necessary for Fact 6?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Is it consistent for µ regular that exactly one of DSS(µ+) and “inter- nally club and internally unbounded are distinct for [H(µ+)]µ” holds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' On a similar note, the assumption that 2µ = |H(µ+)| is also used in a folklore result that assuming 2µ = µ+, the distinction between internally unbounded and internally approachable for [µ+]µ requires a Mahlo cardinal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' What is the exact equiconsistency strength of the separation of inter- nally approachable and internally unbounded for [H(µ+)]µ for regular µ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' References [1] Uri Abraham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Aronszajn trees on ℵ2 and ℵ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Logic, 24(3):213–230, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [2] Sean D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [3] James Cummings, Sy-David Friedman, Menachem Magidor, Assaf Rinot, and Dima Sinapova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' The eightfold way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Journal of Symbolic Logic, 83(1):349–371, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [4] Sy-David Friedman and John Krueger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Thin stationary sets and disjoint club sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=', 359(5):2407–2420, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [5] Thomas Gilton and John Krueger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Trees and stationary reflection at double successors of regular cardinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' Journal of Symbolic Logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' [7] Thomas Daniells Gilton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} +page_content=' On the Infinitary Combinatorics of Small Cardinals and the Car- dinality of the Continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf'} 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100644 index 0000000000000000000000000000000000000000..11d57a43dcbd3cdddfe0daff48e20b5d977843ee --- /dev/null +++ b/9tFJT4oBgHgl3EQfoyxM/content/tmp_files/2301.11597v1.pdf.txt @@ -0,0 +1,1172 @@ +1 +MissBeamNet: Learning Missing Doppler Velocity +Log Beam Measurements +Mor Yona and Itzik Klein +Abstract—One of the primary means of sea exploration is +autonomous underwater vehicles (AUVs). To perform these tasks, +AUVs must navigate the rough challenging sea environment. +AUVs usually employ an inertial navigation system (INS), aided +by a Doppler velocity log (DVL), to provide the required +navigation accuracy. The DVL transmits four acoustic beams to +the seafloor, and by measuring changes in the frequency of the +returning beams, the DVL can estimate the AUV velocity vector. +However, in practical scenarios, not all the beams are successfully +reflected. When only three beams are available, the accuracy of +the velocity vector is degraded. When fewer than three beams +are reflected, the DVL cannot estimate the AUV velocity vector. +This paper presents a data-driven approach, MissBeamNet, to +regress the missing beams in partial DVL beam measurement +cases. To that end, a deep neural network (DNN) model is +designed to process the available beams along with past DVL +measurements to regress the missing beams. The AUV velocity +vector is estimated using the available measured and regressed +beams. To validate the proposed approach, sea experiments were +made with the ”Snapir” AUV, resulting in an 11 hours dataset of +DVL measurements. Our results show that the proposed system +can accurately estimate velocity vectors in situations of missing +beam measurements. Our dataset and codebase implementing +the described framework is available at our GitHub repository. +Index Terms—Autonomous Underwater Vehicles, Navigation, +Doppler Velocity Log, Deep-Learning +I. INTRODUCTION +The demand for autonomous underwater vehicles (AUV) +is significantly growing [1], [2], [3], [4]. AUVs are used in +a variety of applications, such as seafloor exploration and +mapping [5], pipeline inspection [6], [7], and underwater mine +detection [8]. An accurate navigation system is necessary +for the AUV to navigate challenging sea conditions and +successfully perform the required tasks. From a navigational +perspective, the commonly used global navigation satellite +system (GNSS) is unavailable underwater. Furthermore, un- +derwater currents and the ever-changing landscape make it +difficult to use simultaneous localization and mapping (SLAM) +[9]. Consequently, most AUVs employ an inertial navigation +system (INS) aided by a Doppler velocity log (DVL). The INS +provides a complete navigation solution comprising position, +velocity, and orientation using three-axis accelerometers and +three-axis gyroscopes. However, due to inertial measurement +errors, the pure inertial solution will drift over time [10]. +The DVL provides an accurate estimate of the AUV velocity +vector, which is used to aid the INS and obtain an accurate +navigation solution. The fusion between INS and DVL is well +addressed in the literature under normal DVL operating condi- +tions. For example, a rotational dynamic model was shown to +improve the INS/DVL fusion performance [11]. Furthermore, +an adaptive Kalman filter aimed at finding the optimal window +length for each measurement has been suggested [12]. In +order to improve the extended Kalman filter, an innovative +unscented Kalman filter was developed for AUV navigation +[13]. Recently, a dedicated neural network was proposed to +cope with current estimation during INS/DVL fusion [14]. +The DVL emits four acoustic beams to the seafloor and mea- +sures the changes in the reflected beams’ frequency. Using the +frequency shift, the beam’s velocity is calculated. The AUV +velocity vector can be estimated when at least three beams +are reflected back. In real-life scenarios, however, beams may +not reflect back to the DVL for several reasons, such as if the +AUV passes over a deep trench in one of the directions, an +underwater sand wave changes the seafloor surface, or when +the AUV operates in extreme roll and pitch angles. In such +scenarios, the DVL cannot estimate the AUV velocity vector, +and the INS/DVL loosely coupled approach cannot be applied. +Since the tightly coupled approach uses any of the available +beams, it can be implemented for the fusion process. Yet, for +practical considerations, the loosely coupled method is usually +implemented [15], [16]. To cope with situations of partial +beam measurement, a model-based extended loosely coupled +approach was suggested [17]. +The use of data-driven approaches in navigation and their +benefits over model-based approaches were recently summa- +rized in [18]. A novel method of improving the accuracy of +the estimated DVL velocity in underwater navigation using a +neural network structure was suggested [19]. Furthermore, a +deep learning network that utilizes attitude and heading data +in order to improve navigation accuracy and fault tolerance +was developed [20]. +This paper presents, a learning framework, MissBeamNet, to +regress the missing DVL beams and enable AUV velocity vec- +tor estimation. To that end, we leveraged our initial research +to regress only a single beam [21]. The contributions of this +research are: +‚ A modular framework capable of regressing one, two, or +three missing beams. +‚ A robust long short-term memory network architecture +able to accurately regress the missing beams. +‚ Inclusion of depth measurements to improve beam re- +gression accuracy. +‚ A GitHub repository containing our code and dataset as a +benchmark dataset and solution and to encourage further +research in the field. +Here, we provide a thorough analysis of the missing beam sce- +narios. In addition, we compare our results to two model-based +approaches: 1) an average of the missing beam to estimate the +current one (baseline) and 2) the virtual beam approach [17]. +arXiv:2301.11597v1 [cs.RO] 27 Jan 2023 + +2 +All analyses were made on a dataset consisting of 11 hours +of DVL recordings made by the Snapir AUV [23] during its +mission in the Mediterranean Sea. We further demonstrate +the superiority of MissBeamNet over current model-based +approaches and its ability to estimate the AUV velocity vector +in situations of missing DVL beam measurements. +The remainder of the paper is organized as follows: Section II +describes the AUV sensors and the model-based partial DVL +approaches. Section III presents our MissBeamNet framework, +while Section IV gives our sea experiment results. Finally, our +conclusions are presented in Section V. +II. PROBLEM FORMULATION +This section briefly describes the AUV sensors used in this +research and presents the baseline model-based approaches to +coping with missing beam measurements. +A. AUV Sensors +1) DVL: The DVL transmits four acoustic beams to the +seafloor, which reach the seafloor and bounce back to the DVL +transducers. The DVL measures the change in frequency in +each direction. Based on [24], the relative velocity of each +beam is calculated by: +Vrel “ pFD ` bF,D ` nF,Dq1000 ¨ Cp1 ` SFcq +2fs +(1) +where FD is the Doppler frequency shift, bF,D and nF,D are +the bias and noise of the Doppler frequency shift, respectively, +SFc is the scale factor error, C is the speed of sound, and fs +is the transmitted acoustic frequency. The DVL transducers +send acoustic beams in four directions. The standard DVL +configuration is the ”Janus Doppler configuration”.In this +configuration, the transducers are in an ”X” shape, and the +direction of each beam is described by the following equation: +bi “ +» +—– +cos ˜ψi sin ˜θ +sin ˜ψi sin ˜θ +cos ˜θ +fi +ffifl +(2) +where ˜θ is the (fixed) pitch angle and ˜ψi is the yaw angle +defined for each beam i as: +˜ψi “ pi ´ 1q90 deg `45 deg, i “ 1, 2, 3, 4. +(3) +The estimated DVL velocity in the platform frame is: +˜vp +t{p “ pAT Aq´1AT y +(4) +where ˜vp +t{p is the velocity vector, A is the direction matrix +defined as: +A “ +» +———– +bT +1 +bT +2 +bT +3 +bT +4 +fi +ffiffiffifl +(5) +and y is the measured beams vector +y “ +“ +˜y1 +˜y2 +˜y3 +˜y4 +‰T . +(6) +2) Pressure sensor: A pressure sensor measures the pres- +sure of a fluid or gas. In underwater navigation, a pressure +sensor can be used to measure the water pressure at different +depths, which can be used to determine the depth of the sub- +merged vehicle. The underlying physical equation to estimate +the AUV depth is [26]: +p “ ρ ¨ g ¨ h ` ρp0 +(7) +where p [Kpa] is the measured pressure, p0 is the pressure +in +the +atmosphere +equalling +101.3[Kpa], +ρ +is +water +density[kg{m3], +g +is +the +gravity +magnitude, +assumed +here constant and equal to 9.81 [m{s2], and h [m] is the +depth of the AUV. +B. Model-based approaches for missing beams +1) Average: An average in a time window refers to the +average value of a measurement over a specific period of time +(the ’time window’). This can be useful for smoothing out +noisy or erratic measurements, and reducing the effects of +random errors. In the context of measurement synthesizing the +average is a standard method that uses the average between the +measurements in the previous time window, to assume the cur- +rent measurement. For a time window with N measurements, +the average is : +AV pxq “ 1 +N +N +ÿ +k“1 +xk +(8) +The size of the time window is chosen based on the charac- +teristics of the sensor and system. For example, a small time +window may be used for measurements that change rapidly, +while a larger time window may be more suitable for relatively +stable measurements. +2) Virtual Beam: The last velocity vector measurement can +be utilized to predict the current velocity vector [17]. This +method replaces the missing DVL beam measurement with +the previously available measurement. For example, if beam +#1 is absent, solving (4) with the known velocity vector at +k ´ 1 gives an estimate of its velocity: +y1,k « bT +1 rˆvx,k´1 +ˆvy,k´1 +ˆvz,k´1s +(9) +where k is the time index and ˆvj,k´1 is the estimated velocity +component from the previous step for j “ x, y, z. This +approximated beam velocity is then used, together with the +measured beams, in (4) to predict the current velocity vector. +III. MISSBEAMNET FRAMEWORK +We propose a deep learning framework, MissBeamNet, as a +mechanism to handle missing DVL beam measurements (1,2, +or 3 beams) and allow the estimation of the AUV velocity +vector. The MissBeamNet framework utilizes n past DVL +beam measurements and the currently available beams as input +to an end-to-end neural network, which regresses the missing +beams. Then, the regressed and currently available measured +beams are plugged into the model-based least squares (LS) +estimator to estimate the AUV velocity vector. Figure 1 + +3 +describes our MissBeamNet framework. +Our proposed MissBeamNet can cope with the following +scenarios: +‚ If three beams are available, MissBeamNet will regress +one missing beam. +‚ If two beams are available, MissBeamNet will regress +two missing beams. +‚ If one beam is available, MissBeamNet will regress three +missing beams. +Note, that MissBeamNet was not designed to handle complete +DVL outages, as it requires at least one available beam. For +total outages, other solutions exist [27], [28]. We consider two +Fig. 1. +MissBeamNet framework utilizing past DVL beam measurements to +regress the missing beams. +types of neural networks as our baseline network architectures. +The first is based on a one-dimension convolution neural +network (CNN), while the other is based on long-short-term +memory (LSTM) cells. Both networks have been proven to +work with time-series data, such as those considered in our +scenario. +A. Baseline Network Architectures +1) Convolutional Neural Network: In CNN layers, there is +a sparse interaction between the input and output, as appose to +fully connected layers, where all the input parameters directly +interact with the output. The convolution operator is a linear +operator that involves multiplying an input with a kernel +containing learned parameters. The kernel slides through the +input, and the result is the sum of all the multiplications: +yt “ +pÿ +k“1 +xt`kwk +(10) +where t is the timestamp, p is the kernel length, w is the +learned kernel parameter, and x,y are the input and output, re- +spectively. The fact that CNN shares parameters by passing the +same kernels through all the input makes CNN architectures +very popular in situations with large inputs. Figure 2 describes +our baseline CNN architecture, including network parameters, +for a scenario of two missing beams. The network is a multi- +head network where past DVL measurements are the input to +the first head, and current DVL measurements are the input to +the second head. The same structure and parameters are used +when one or three beams are missing. The selected activation +function between the layers is Relu and the stride and padding +are set to one. +2) Long Short-Term Memory Network: LSTM is an ad- +vanced version of a recurrent neural network (RNN) and solves +its shortcomings. RNNs are capable of handling temporal +data by using information from prior inputs. However, if +the sequence is long, the RNN may face a problem known +as vanishing/exploding gradients [29]. For example, when a +Fig. 2. +Baseline CNN architecture with an example of two missing beams. +gradient is small, it may continue to decrease until the model +is no longer learning. The LSTM addresses these problems +using three types of gates: The forget gate, the input gate, and +the output gate. +The role of the forget gate is to forget unwanted information +from the previous output and current input: +ft “ σpxtU f ` ht´1W f ` bfq +(11) +where xt is the input, ht´1 is the output of the previous LSTM +cell, W f and bf are the weights and biases of the forget gate, +respectively. In (11) sigmoid function is employed to bring the +parameter it wants to forget closer to zero. The output of the +forget gate is then multiplied by the previous cell state. The +role of the input gate is to update the cell state Ct´1, by first +calculating the input gate it: +it “ σpxtU i ` ht´1W i ` biq +(12) +where U i and wi are the gate weights and bi is the bias. +Second, calculating the estimated cell state ˜Ct: +˜Ct “ tanhpxtU g ` ht´1W g ` bgq +(13) +where U g and wg are the gate weights and bg is the bias. The +results of (11),(12), and (13) are used for the current cell state +calculations: +Ct “ ft ¨ Ct´1 ` it ¨ ˜Ct +(14) +As the name implies, the output gate ot determines which +parameters are important as the output and next hidden state. +ot “ σpxtU o ` ht´1W o ` boq +(15) +where U t and wt are the gate weights and bt is the bias. The +output gate results are then multiplied by a tanh layer of the +cell state to calculate the current output and hidden state +ht “ ot ¨ tanhpCtq +(16) +Figure 3 describes our LSTM baseline network structure. +Previous beam measurements are used as input to the LSTM +layers. After the LSTM features extraction, the features are +concatenated with available beam measurements into a fully +connected layer, which performs the final process resulting in +the output of the regressed missing beams. Note that, like our +baseline CNN network, this is a multi-head network where past +DVL measurements are inputs to the first head, and current +DVL measurements are inputs to the second head. Figure 4 +describes the LSTM architecture parameters in the scenario of +two missing beams. The activation function between the layers +is Relu. The same structure and parameters are also used when +one or three beams are missing. + +Current beams +E- +LSEstimator +★Velocityvector +Past n DVL beam measurements, +Neural network +Regressed beamsInput: +1DCNN1 +1DCNN2 +1DCNN3 +1DCNN 4 +1DCNN5 +1DCNN6 +1DCNN7 +4X6 +16@2X1 +32@3X1 +64@3X1 +64@3X1 +128@3X1 +128@3X1 +256@3X1 +Flatten layer +Fully connected 1 +Dropoutlayer +Fully connected 2 +1280 +probability = 0.3 +640 +Fully connected 3 +Output: +2 missing +10 +beams +Current beams +24 +Fig. 3. +Baseline LSTM structure. +Fig. 4. +Baseline LSTM architecture with an example of two missing beams. +B. Training Process +The training process of deep neural networks requires defin- +ing a loss function. The common loss functions for regression +problems are mean absolute error (MAE) or mean squared +error (MSE). In this paper, we use MSE loss defined by: +MSE “ 1 +n +n +ÿ +i“1 +pyactual ´ ypredictedq2 +(17) +where n is the number of samples, yactual is the target, and +ypredicted is the model output. Generally, the MSE loss func- +tion will try to adjust the model to better handle outliers than +MAE due to the MSE squared error. However, in our scenarios, +an AUV operates in varying sea conditions, therefore we adopt +the MSE loss. During training, the loss function is calculated +after each forward propagation in order to use the method of +gradient descent and set the DNN initial weight and biases on +values that will provide the desired result. Forward propagation +is the process of the data going through all the layers of +the architecture, like (10) for CNN and (11)-(16) for LSTM +networks. After completing the forward propagation process, +the back propagation process updates the weights and biases +of all the layers with a gradient descent principle +θ “ θ ´ η∇θJpθq +(18) +where θ is the vector of weights and biases, Jpθq is the loss +function with the DNN weights and biases set to θ, ∇θ is the +gradient operator, and η is the learning rate. +The learning rate is a crucial hyperparameter, which dictates +how fast the weights and biases change after each training +batch. If the selected learning rate is too low, it might converge +in a local minimum, and if it is too high, the model might not +converge at a minimum. Our selected optimizer for all tested +architectures is an adaptive moment estimation (ADAM) [30]. +IV. ANALYSIS RESULTS +A. Dataset Description +To examine the proposed approach, data from sea experi- +ments were employed. All experiments were conducted in the +Mediterranean Sea by the ”Snapir” (ECA A18D), a 5.5[m] +long AUV capable of reaching 3000[m] depth. It is equipped +with the Teledyne RDI Work Horse navigator DVL[31], which +has a four-beams Janus convex configuration with a sample +rate of 1[Hz]. To train the deep neural network, first, all invalid +Fig. 5. +The ”Snapir” being pulled out of the water after a successful mission. +DVL data was removed (some of the invalid readings occurred +when actual beams were missing). Then, the data was divided +into routes that the AUV performed. Approximately 60% of +the missions were used as the training dataset and the rest as +the test dataset. The training dataset comprised 23,243 samples +corresponding to 387 minutes of recording, and the test dataset +had 276 minutes of recording (16,618 samples). +Figure 6 shows an experiment with challenging dynamics +which is part of the test dataset. +Fig. 6. +Experiment example from the test dataset. +The total of 663 minutes of recordings consists of two +parts: 300 minutes from our initial data collection cam- +paign [22] and 363 minutes in the current campaign. +The complete dataset is publicly available at our GitHub +https://github.com/ansfl/MissBeamNet. + +output +fully connected 2 +fully connected 1 +OOOOOOO +flatten layer +co +LSTM +LSTM +LSTM +LSTM +LSTM +LSTM +ho +cell +cell +cell +cell +cell +cell +Beams +Beams t +Beams t +Beams t. +Beams t. +Beams t, +Beam 1 +current +beams +Beam 2 +5 +Beam 3 +5 +Beam 4Input: +LSTM +Fully connected 1 +4X6 +hidden state = 500 +3000 +Fully connected 1 +2 missing +Output +6 +beams +Current beams: +20 +9- +10 +15 +-20 +-25 +30 +35 +0 +1000 +2000 +3000 +4000 +5000 +Time [s]5 +B. Performance Metric +Performance metrics compare different models/methods and +choose the one with the best performance. Throughout the +research, we used the performance metric of root mean squared +error (RMSE), which is widely used to evaluate models on +regression tasks. RMSE is calculated by taking the root of the +average of squared differences between the predicted values +and the target values +RMSE “ +cřn +i“1pyactual ´ ypredictedq2 +n +(19) +The RMSE results are in the same units as the original data, +making it easy to interpret. +C. Baseline Architectures Comparison +To compare our two baseline architectures described in +Section +III-B, we consider a scenario with two missing +beams, namely, beams #1 and #2, and assume six past beam +measurements are used. In addition to these two baseline +architectures, we examine the possibility of using only +past beam measurements instead of the baseline multi-head +approach. These two architectures are denoted as CNN A +and LSTM A. The results of the test dataset in terms of +RMSE are presented in Figure 7. The results shows that +Fig. 7. +RMSE results for network architecture comparison. +using the baseline architectures (multi-head) obtained better +performance than working with all the inputs in a single +head. In addition, the performance of the baseline LSTM +showed an improvement of 27 % over the baseline CNN. +D. Number of Past Beam Measurement Influence +The number of past measurements utilized by the network +is defined as the window-size length. The length of the +optimal window size is crucial for model performance. The +window size regularizes the model performance between the +long and short movement patterns. If the selected window +size is too short, the model might miss the pattern of the +AUV movement, and if it is too long, the model might not +react well enough to a movement that just started. Figure 8 +shows the RMSE of the baseline LSTM model with different +window-size lengths (between 3-10) when beams #1 and +#2 are being regressed. The results suggest that the optimal +window-size length on our dataset is six measurements. +Fig. 8. +RMSE as a function of the window size for the baseline LSTM +network. +E. Additional Input Information +To improve the model performance even further, additional +inputs are considered. +1) Depth Sensor: The last depth sensor reading. +2) AUV Velocity Vector: Domain knowledge is used to +transform the raw data (in this case, the beams) into +meaningful features using feature engineering. Feature +engineering is prevalent in classical machine learning +methods, but less in deep neural networks. The assump- +tion when using a neural network is thet model will +learn the essential relations between features indepen- +dently. The beams and the velocity vector are related, +as the latter is estimated using the former. That is, +the model is not receiving new information. Yet, in +the proposed LSTM-based model, there are only two +fully connected layers, and therefore feature engineering +may help achieve better accuracy or shorten the network +convergence time. +Figure 9 describes the performance of each input with our +baseline LSTM architecture, including additional inputs of 1) +depth, 2) velocity vector, and 3) depth and velocity vector. +The tested case is when beams #1 and #2 are missing, and +beams #3 and #4 are inserted as a two-phase input to our +baseline LSTM network. All of the additional inputs improved +Fig. 9. +Velocity RMSE as a function of different input selection for our +baseline LSTM network. +the performance of the baseline LSTM, and the best approach +was obtained using all three input types - beam measurements +(baseline), depth sensors, and the velocity vector. In this +instance, there was a 16% improvement compared to the +LSTM baseline. +F. Missing Beams Analysis +There are 14 combinations of missing beams: four combina- +tions of one missing beam, six of two missing beams, and four +of three missing beams. In the proposed approach a different +network needs to be trained for each of those combinations. +Training for all networks used the same hyper-parameters: + +0.14 +0.12 +0.10 +MSE +0.06 +0.04 +0.02 +0.00 +LSTMA +Baseline LSTM +CNN-A +Baseline CNN +Architecture0.100 +0.095 +[m/s] +0.090 +E +0.085 +0.080 +0.075 +3 +4 +5 +6 +7 +8 +9 +10 +# of past measurements0.0750 +0.0725 +0.0700 +兰0.0675 +S +MSE +20.0650 +0.0625 +0.0600 +0.0575 +Baseline LSTM +Baseline LSTM + +Baseline LSTM + +Baseline LSTM + +altitude +velocity vector +altitude + velocity vector +Architecture6 +MSE loss function with a learning rate of 0.00005, batch size +of 1 sequence, and 150 epochs. In the following sections, we +present the performance of our MissBeamNet approach com- +pared to the average (baseline) and virtual beam approaches. +For this analysis, we employed our baseline LSTM network +described in Section 3.1.2. Based on the results of Section +4.4, we use six past DVL beam measurements and, based on +Section 4.5, both the depth sensor reading and velocity vector +were are added as additional inputs. The results were obtained +on the testing dataset. +1) One missing beam: When one beam is missing, the +least squared approach (4) can be used to obtain the estimated +AUV velocity vector. Table I presents the results of estimating +the missing beam, the speed error obtained when using the +estimated fourth beam together with the measured three, and +the improvement of our MissBeamNet approach over the two +model-based approaches. +Both model-based and MissBeamNet methods were supe- +rior tp the three beams solution, indicating that regressing the +fourth beam is critical to improving the AUV speed estima- +tion accuracy. Specifically, MissBeamNet, improved the speed +accuracy by over 90%. In addition, MissBeamNet performed +significantly better than the model-based approaches, with a +40%-68% improvement. Taking the mean of performance of +all four scenarios MissBeamNet improved the model-based +approaches by over 49.8 %. +2) Two Missing Beams: When considering two missing +beams, six different combinations exist. In such scenarios, +the AUV velocity cannot be estimated. Following the same +procedure as the previous one missing beam scenarios, Table +II presents the results of two missing beams. The results show +a significant difference between the speed error in each com- +bination, even in the model-based approaches, emphasizing +the problem’s complexity. Yet, in all cases, MissBeamNet +was more accurate than the model-based approaches, with a +minimum improvement of 20% that reached almost 50%. The +average improvement over the baseline model-based approach +was 28.7% compared to 49.8% when only one beam was miss- +ing. This is attributed to the model receiving less information +from two current beams compared to three when only one is +missing. +3) Three Missing Beams: Table III presents the results +for the four scenarios in which three beams are missing. As +expected, the speed error when three beams are missing is +higher than in the two or one missing beams scenarios. Yet, +MissBeamNet use improved results by at least 21% over the +model-based approaches. For three missing beams, the average +improvement was 24% compared to 28.7% when two beams +were missing, only 4.7% less, indicating that even with only +one beam at hand, the AUV velocity can be estimated. +4) Hyperparameter Tuning: One of the main challenges in +deep learning research is to find the best combination of hy- +perparameters for the proposed architecture. Each architecture +has several parameters that can influence model performance, +including the number of layers, the number of parameters in +each layer, the type of cost function, the learning rate, and +batch size. To demonstrate the potential of hyperparameter +tuning, we evaluated three different hyperparameters. The +first was the learning rate, which affects how much each +batch changes the weights and biases III-B. The second hyper +parameter was the number of hidden parameters in the LSTM +layer ht III-A2, and the third hyperparameter is the number +of parameters in the LSTM output. To test the importance of +hyperparameter tuning, each parameter was set with a few +available options, and a seed was set (equal initialization +in each run). We focused on a one missing beam scenario, +which has four options - missing beam #1, #2, #3, or #4. +For each case, 15 randomly selected combinations of the +three hyperparameters were examined. Table IV presents the +potential of hyperparameter tuning. It is important to note that +out of the 15 tested hyperparameter combinations, only a few +were better than the results before tuning. Yet, they were able +to improve the missing beam estimation and, consequently, +reduce the speed error and increase the rate of improvement +compared to the two model-based approaches. +V. CONCLUSIONS +Here, we presented MissBeamNet, a deep learning-based +framework developed to compensate for partial DVL measure- +ment scenarios (1, 2, or 3 missing beams). To that end, an +LSTM-based dedicated DNN was derived. We demonstrated +that the best input to the network is past DVL measurements, +past depth sensor measurements, previous velocity vectors, +and the currently available measured beams. Once the missing +beams are regressed, they are combined with the available +beams and plugged into the classical model-based approach +to estimate the AUV velocity vector. +To evaluate MissBeamNet, sea experiments with the Uni- +versity of Haifa’s ”Snapir” AUV were conducted. The data +included several trajectories collected for different purposes +and under various sea conditions. We provide a thorough +analysis of all 14 missing beam combinations and explore +several means to enhance our baseline architecture. The results +show that MissBeamNet allows estimating the missing DVL +beams and, consequently, the AUV velocity vector. Addi- +tionally, MissBeamNet significantly improves the accuracy of +the velocity vector in all examined scenarios compared to +the model-based approaches. The improvement of all three +missing beam combinations was above 20 % over the model- +based approaches. For two missing beams, performance was +generally better compared to three missing beams since the +model uses one additional measured beam. Finally, we show +that hyperparameters-tuned models improve the accuracy of +MissBeamNet by more than 40%. +REFERENCES +1 Q. Luo, Y. Shao, J. Li, X. Yan and C. Liu, A multi-AUV cooperative +navigation method based on the augmented adaptive embedded cuba- +ture Kalman filter algorithm., Neural Comput and Applic vol. 34, pp. +18975–18992 2022. +2 M. Mohammadi, M.M. Arefi, N. Vafamand and O. 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Hunt, Autonomous Underwater Vehicles +(AUVs): their past, present and future contributions to the advancement +of marine geoscience Marine Geology., vol. 352, pp. 451-468. 2014. + +7 +TABLE I +ONE MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET +Case +Approach +Beam 1 +Beam 2 +Beam 3 +Beam 4 +Avg. +results +Missing beam [m/s] +Average (baseline) +0.110 +0.101 +0.101 +0.111 +0.106 +Virtual beam +0.139 +0.109 +0.110 +0.129 +0.121 +MissBeamNet (ours) +0.065 +0.048 +0.034 +0.067 +0.053 +Speed error [m/s] +Average (baseline) +0.066 +0.061 +0.061 +0.067 +0.064 +Virtual beam +0.079 +0.065 +0.066 +0.077 +0.072 +Three beams +0.450 +0.437 +0.438 +0.449 +0.443 +MissBeamNet (ours) +0.039 +0.029 +0.021 +0.040 +0.032 +MissBeamNet improvement % +Average (baseline) +40.9 +52.4 +65.6 +40.3 +49.8 +Virtual beam +50.6 +55.3 +68.1 +48.0 +55.5 +Three beams +91.3 +93.3 +95.2 +91.1 +92.7 +TABLE II +TWO MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET +Case +Approach +Beam +1,2 +Beam +1,3 +Beam +1,4 +Beam +2,3 +Beam +2,4 +Beam +3,4 +Avg. +result +Missing beams [m/s] +Average (baseline) +0.106 +0.106 +0.111 +0.101 +0.107 +0.106 +0.106 +Virtual beam +0.121 +0.121 +0.131 +0.110 +0.119 +0.120 +0.120 +MissBeamNet (ours) +0.062 +0.052 +0.085 +0.076 +0.057 +0.066 +0.066 +Speed error [m/s] +Average (baseline) +0.092 +0.077 +0.096 +0.088 +0.079 +0.092 +0.087 +Virtual beam +0.106 +0.069 +0.114 +0.096 +0.065 +0.105 +0.092 +MissBeamNet (ours) +0.055 +0.057 +0.075 +0.066 +0.061 +0.058 +0.062 +MissBeamNet improvement % +Average (baseline) +40.2 +25.9 +21.9 +25.0 +22.8 +36.9 +28.7 +Virtual beam +48.1 +17.4 +34.2 +31.25 +6.15 +44.7 +30.3 +TABLE III +THREE MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET +Case +Approach +Beam +1,2,3 +Beam +2,3,4 +Beam +1,2,4 +Beam +1,3,4 +Avg. +results +Missing beams [m/s] +Average (baseline) +0.104 +0.108 +0.107 +0.105 +0.106 +Virtual beam +0.118 +0.124 +0.124 +0.116 +0.120 +MissBeamNet (ours) +0.071 +0.073 +0.077 +0.071 +0.073 +Speed error [m/s] +Average (baseline) +0.102 +0.108 +0.106 +0.103 +0.105 +Virtual beam +0.102 +0.109 +0.111 +0.099 +0.105 +MissBeamNet (ours) +0.077 +0.081 +0.083 +0.078 +0.079 +MissBeamNet improvement % +Average (baseline) +24.5 +25.0 +21.7 +24.3 +23.9 +Virtual beam +24.5 +25.7 +25.2 +21.2 +24.1 +TABLE IV +HYPERPARAMETERS TUNING +Case +Approach +Beam 1 +Beam 2 +Beam 3 +Beam 4 +Missing beam [m/s] +Before tuning +0.065 +0.048 +0.034 +0.067 +After tuning +0.011 +0.017 +0.02 +0.012 +Speed error [m/s] +Before tuning +0.039 +0.029 +0.021 +0.040 +After tuning +0.007 +0.010 +0.012 +0.007 +Learning rate +Before tuning +5e-05 +5e-05 +5e-05 +5e-05 +After tuning +1e-04 +1e-04 +5e-05 +1e-05 +hidden LSTM +parameters ht +Before tuning +500 +500 +500 +500 +After tuning +250 +750 +750 +100 +LSTM output +parameters +Before tuning +7 +7 +7 +7 +After tuning +7 +5 +7 +5 +MissBeamNet Tuning +improvement % +Average (baseline) +89.4 +83.6 +80.3 +89.5 +Virtual beam +91.1 +84.6 +81.8 +90.9 +Three beams +98.4 +97.7 +97.2 +98.4 +Before tuning +82.2 +67.5 +42.8 +82.5 +4 E. 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Bengio, On the difficulty of training re- +current neural networks Proceedings of the 30th International Conference +on Machine Learning, pp.1310-1318, 2013. +30 P. Diederik and B. Jimmy Adam: A Method for Stochastic Optimization, +3rd International Conference for Learning Representations, 2015. +31 Teledyne marine manual for the Teledyne RDI Work Horse navigator +DVL. + diff --git a/9tFJT4oBgHgl3EQfoyxM/content/tmp_files/load_file.txt b/9tFJT4oBgHgl3EQfoyxM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5864a257375d3ead4528826b1a776aa5ee106370 --- /dev/null +++ b/9tFJT4oBgHgl3EQfoyxM/content/tmp_files/load_file.txt @@ -0,0 +1,709 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf,len=708 +page_content='1 MissBeamNet: Learning Missing Doppler Velocity Log Beam Measurements Mor Yona and Itzik Klein Abstract—One of the primary means of sea exploration is autonomous underwater vehicles (AUVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To perform these tasks, AUVs must navigate the rough challenging sea environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' AUVs usually employ an inertial navigation system (INS), aided by a Doppler velocity log (DVL), to provide the required navigation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The DVL transmits four acoustic beams to the seafloor, and by measuring changes in the frequency of the returning beams, the DVL can estimate the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' However, in practical scenarios, not all the beams are successfully reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' When only three beams are available, the accuracy of the velocity vector is degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' When fewer than three beams are reflected, the DVL cannot estimate the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This paper presents a data-driven approach, MissBeamNet, to regress the missing beams in partial DVL beam measurement cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To that end, a deep neural network (DNN) model is designed to process the available beams along with past DVL measurements to regress the missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The AUV velocity vector is estimated using the available measured and regressed beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To validate the proposed approach, sea experiments were made with the ”Snapir” AUV, resulting in an 11 hours dataset of DVL measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Our results show that the proposed system can accurately estimate velocity vectors in situations of missing beam measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Our dataset and codebase implementing the described framework is available at our GitHub repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Index Terms—Autonomous Underwater Vehicles, Navigation, Doppler Velocity Log, Deep-Learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' INTRODUCTION The demand for autonomous underwater vehicles (AUV) is significantly growing [1], [2], [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' AUVs are used in a variety of applications, such as seafloor exploration and mapping [5], pipeline inspection [6], [7], and underwater mine detection [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' An accurate navigation system is necessary for the AUV to navigate challenging sea conditions and successfully perform the required tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' From a navigational perspective, the commonly used global navigation satellite system (GNSS) is unavailable underwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Furthermore, un- derwater currents and the ever-changing landscape make it difficult to use simultaneous localization and mapping (SLAM) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Consequently, most AUVs employ an inertial navigation system (INS) aided by a Doppler velocity log (DVL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The INS provides a complete navigation solution comprising position, velocity, and orientation using three-axis accelerometers and three-axis gyroscopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' However, due to inertial measurement errors, the pure inertial solution will drift over time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The DVL provides an accurate estimate of the AUV velocity vector, which is used to aid the INS and obtain an accurate navigation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The fusion between INS and DVL is well addressed in the literature under normal DVL operating condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For example, a rotational dynamic model was shown to improve the INS/DVL fusion performance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Furthermore, an adaptive Kalman filter aimed at finding the optimal window length for each measurement has been suggested [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In order to improve the extended Kalman filter, an innovative unscented Kalman filter was developed for AUV navigation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Recently, a dedicated neural network was proposed to cope with current estimation during INS/DVL fusion [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The DVL emits four acoustic beams to the seafloor and mea- sures the changes in the reflected beams’ frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Using the frequency shift, the beam’s velocity is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The AUV velocity vector can be estimated when at least three beams are reflected back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In real-life scenarios, however, beams may not reflect back to the DVL for several reasons, such as if the AUV passes over a deep trench in one of the directions, an underwater sand wave changes the seafloor surface, or when the AUV operates in extreme roll and pitch angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In such scenarios, the DVL cannot estimate the AUV velocity vector, and the INS/DVL loosely coupled approach cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Since the tightly coupled approach uses any of the available beams, it can be implemented for the fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yet, for practical considerations, the loosely coupled method is usually implemented [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To cope with situations of partial beam measurement, a model-based extended loosely coupled approach was suggested [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The use of data-driven approaches in navigation and their benefits over model-based approaches were recently summa- rized in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' A novel method of improving the accuracy of the estimated DVL velocity in underwater navigation using a neural network structure was suggested [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Furthermore, a deep learning network that utilizes attitude and heading data in order to improve navigation accuracy and fault tolerance was developed [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This paper presents, a learning framework, MissBeamNet, to regress the missing DVL beams and enable AUV velocity vec- tor estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To that end, we leveraged our initial research to regress only a single beam [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The contributions of this research are: ‚ A modular framework capable of regressing one, two, or three missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ‚ A robust long short-term memory network architecture able to accurately regress the missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ‚ Inclusion of depth measurements to improve beam re- gression accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ‚ A GitHub repository containing our code and dataset as a benchmark dataset and solution and to encourage further research in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Here, we provide a thorough analysis of the missing beam sce- narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In addition, we compare our results to two model-based approaches: 1) an average of the missing beam to estimate the current one (baseline) and 2) the virtual beam approach [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='11597v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='RO] 27 Jan 2023 2 All analyses were made on a dataset consisting of 11 hours of DVL recordings made by the Snapir AUV [23] during its mission in the Mediterranean Sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' We further demonstrate the superiority of MissBeamNet over current model-based approaches and its ability to estimate the AUV velocity vector in situations of missing DVL beam measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section II describes the AUV sensors and the model-based partial DVL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Section III presents our MissBeamNet framework, while Section IV gives our sea experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Finally, our conclusions are presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' PROBLEM FORMULATION This section briefly describes the AUV sensors used in this research and presents the baseline model-based approaches to coping with missing beam measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' AUV Sensors 1) DVL: The DVL transmits four acoustic beams to the seafloor, which reach the seafloor and bounce back to the DVL transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The DVL measures the change in frequency in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Based on [24], the relative velocity of each beam is calculated by: Vrel “ pFD ` bF,D ` nF,Dq1000 ¨ Cp1 ` SFcq 2fs (1) where FD is the Doppler frequency shift, bF,D and nF,D are the bias and noise of the Doppler frequency shift, respectively, SFc is the scale factor error, C is the speed of sound, and fs is the transmitted acoustic frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The DVL transducers send acoustic beams in four directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The standard DVL configuration is the ”Janus Doppler configuration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='In this configuration, the transducers are in an ”X” shape, and the direction of each beam is described by the following equation: bi “ » —– cos ˜ψi sin ˜θ sin ˜ψi sin ˜θ cos ˜θ fi ffifl (2) where ˜θ is the (fixed) pitch angle and ˜ψi is the yaw angle defined for each beam i as: ˜ψi “ pi ´ 1q90 deg `45 deg, i “ 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' (3) The estimated DVL velocity in the platform frame is: ˜vp t{p “ pAT Aq´1AT y (4) where ˜vp t{p is the velocity vector, A is the direction matrix defined as: A “ » ———– bT 1 bT 2 bT 3 bT 4 fi ffiffiffifl (5) and y is the measured beams vector y “ “ ˜y1 ˜y2 ˜y3 ˜y4 ‰T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' (6) 2) Pressure sensor: A pressure sensor measures the pres- sure of a fluid or gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In underwater navigation, a pressure sensor can be used to measure the water pressure at different depths, which can be used to determine the depth of the sub- merged vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The underlying physical equation to estimate the AUV depth is [26]: p “ ρ ¨ g ¨ h ` ρp0 (7) where p [Kpa] is the measured pressure, p0 is the pressure in the atmosphere equalling 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3[Kpa], ρ is water density[kg{m3], g is the gravity magnitude, assumed here constant and equal to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='81 [m{s2], and h [m] is the depth of the AUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Model-based approaches for missing beams 1) Average: An average in a time window refers to the average value of a measurement over a specific period of time (the ’time window’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This can be useful for smoothing out noisy or erratic measurements, and reducing the effects of random errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In the context of measurement synthesizing the average is a standard method that uses the average between the measurements in the previous time window, to assume the cur- rent measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For a time window with N measurements, the average is : AV pxq “ 1 N N ÿ k“1 xk (8) The size of the time window is chosen based on the charac- teristics of the sensor and system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For example, a small time window may be used for measurements that change rapidly, while a larger time window may be more suitable for relatively stable measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2) Virtual Beam: The last velocity vector measurement can be utilized to predict the current velocity vector [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This method replaces the missing DVL beam measurement with the previously available measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For example, if beam #1 is absent, solving (4) with the known velocity vector at k ´ 1 gives an estimate of its velocity: y1,k « bT 1 rˆvx,k´1 ˆvy,k´1 ˆvz,k´1s (9) where k is the time index and ˆvj,k´1 is the estimated velocity component from the previous step for j “ x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This approximated beam velocity is then used, together with the measured beams, in (4) to predict the current velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' MISSBEAMNET FRAMEWORK We propose a deep learning framework, MissBeamNet, as a mechanism to handle missing DVL beam measurements (1,2, or 3 beams) and allow the estimation of the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The MissBeamNet framework utilizes n past DVL beam measurements and the currently available beams as input to an end-to-end neural network, which regresses the missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Then, the regressed and currently available measured beams are plugged into the model-based least squares (LS) estimator to estimate the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 1 3 describes our MissBeamNet framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Our proposed MissBeamNet can cope with the following scenarios: ‚ If three beams are available, MissBeamNet will regress one missing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ‚ If two beams are available, MissBeamNet will regress two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ‚ If one beam is available, MissBeamNet will regress three missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Note, that MissBeamNet was not designed to handle complete DVL outages, as it requires at least one available beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For total outages, other solutions exist [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' We consider two Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' MissBeamNet framework utilizing past DVL beam measurements to regress the missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' types of neural networks as our baseline network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The first is based on a one-dimension convolution neural network (CNN), while the other is based on long-short-term memory (LSTM) cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Both networks have been proven to work with time-series data, such as those considered in our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Baseline Network Architectures 1) Convolutional Neural Network: In CNN layers, there is a sparse interaction between the input and output, as appose to fully connected layers, where all the input parameters directly interact with the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The convolution operator is a linear operator that involves multiplying an input with a kernel containing learned parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The kernel slides through the input, and the result is the sum of all the multiplications: yt “ pÿ k“1 xt`kwk (10) where t is the timestamp, p is the kernel length, w is the learned kernel parameter, and x,y are the input and output, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The fact that CNN shares parameters by passing the same kernels through all the input makes CNN architectures very popular in situations with large inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 2 describes our baseline CNN architecture, including network parameters, for a scenario of two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The network is a multi- head network where past DVL measurements are the input to the first head, and current DVL measurements are the input to the second head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The same structure and parameters are used when one or three beams are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The selected activation function between the layers is Relu and the stride and padding are set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2) Long Short-Term Memory Network: LSTM is an ad- vanced version of a recurrent neural network (RNN) and solves its shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' RNNs are capable of handling temporal data by using information from prior inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' However, if the sequence is long, the RNN may face a problem known as vanishing/exploding gradients [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For example, when a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Baseline CNN architecture with an example of two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' gradient is small, it may continue to decrease until the model is no longer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The LSTM addresses these problems using three types of gates: The forget gate, the input gate, and the output gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The role of the forget gate is to forget unwanted information from the previous output and current input: ft “ σpxtU f ` ht´1W f ` bfq (11) where xt is the input, ht´1 is the output of the previous LSTM cell, W f and bf are the weights and biases of the forget gate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In (11) sigmoid function is employed to bring the parameter it wants to forget closer to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The output of the forget gate is then multiplied by the previous cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The role of the input gate is to update the cell state Ct´1, by first calculating the input gate it: it “ σpxtU i ` ht´1W i ` biq (12) where U i and wi are the gate weights and bi is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Second, calculating the estimated cell state ˜Ct: ˜Ct “ tanhpxtU g ` ht´1W g ` bgq (13) where U g and wg are the gate weights and bg is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results of (11),(12), and (13) are used for the current cell state calculations: Ct “ ft ¨ Ct´1 ` it ¨ ˜Ct (14) As the name implies, the output gate ot determines which parameters are important as the output and next hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ot “ σpxtU o ` ht´1W o ` boq (15) where U t and wt are the gate weights and bt is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The output gate results are then multiplied by a tanh layer of the cell state to calculate the current output and hidden state ht “ ot ¨ tanhpCtq (16) Figure 3 describes our LSTM baseline network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Previous beam measurements are used as input to the LSTM layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' After the LSTM features extraction, the features are concatenated with available beam measurements into a fully connected layer, which performs the final process resulting in the output of the regressed missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Note that, like our baseline CNN network, this is a multi-head network where past DVL measurements are inputs to the first head, and current DVL measurements are inputs to the second head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 4 describes the LSTM architecture parameters in the scenario of two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The activation function between the layers is Relu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The same structure and parameters are also used when one or three beams are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Current beams E- LSEstimator ★Velocityvector Past n DVL beam measurements, Neural network Regressed beamsInput: 1DCNN1 1DCNN2 1DCNN3 1DCNN 4 1DCNN5 1DCNN6 1DCNN7 4X6 16@2X1 32@3X1 64@3X1 64@3X1 128@3X1 128@3X1 256@3X1 Flatten layer Fully connected 1 Dropoutlayer Fully connected 2 1280 probability = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 640 Fully connected 3 Output: 2 missing 10 beams Current beams 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Baseline LSTM structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Baseline LSTM architecture with an example of two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Training Process The training process of deep neural networks requires defin- ing a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The common loss functions for regression problems are mean absolute error (MAE) or mean squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In this paper, we use MSE loss defined by: MSE “ 1 n n ÿ i“1 pyactual ´ ypredictedq2 (17) where n is the number of samples, yactual is the target, and ypredicted is the model output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Generally, the MSE loss func- tion will try to adjust the model to better handle outliers than MAE due to the MSE squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' However, in our scenarios, an AUV operates in varying sea conditions, therefore we adopt the MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' During training, the loss function is calculated after each forward propagation in order to use the method of gradient descent and set the DNN initial weight and biases on values that will provide the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Forward propagation is the process of the data going through all the layers of the architecture, like (10) for CNN and (11)-(16) for LSTM networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' After completing the forward propagation process, the back propagation process updates the weights and biases of all the layers with a gradient descent principle θ “ θ ´ η∇θJpθq (18) where θ is the vector of weights and biases, Jpθq is the loss function with the DNN weights and biases set to θ, ∇θ is the gradient operator, and η is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The learning rate is a crucial hyperparameter, which dictates how fast the weights and biases change after each training batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' If the selected learning rate is too low, it might converge in a local minimum, and if it is too high, the model might not converge at a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Our selected optimizer for all tested architectures is an adaptive moment estimation (ADAM) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' ANALYSIS RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Dataset Description To examine the proposed approach, data from sea experi- ments were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' All experiments were conducted in the Mediterranean Sea by the ”Snapir” (ECA A18D), a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5[m] long AUV capable of reaching 3000[m] depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' It is equipped with the Teledyne RDI Work Horse navigator DVL[31], which has a four-beams Janus convex configuration with a sample rate of 1[Hz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To train the deep neural network, first, all invalid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The ”Snapir” being pulled out of the water after a successful mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' DVL data was removed (some of the invalid readings occurred when actual beams were missing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Then, the data was divided into routes that the AUV performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Approximately 60% of the missions were used as the training dataset and the rest as the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The training dataset comprised 23,243 samples corresponding to 387 minutes of recording, and the test dataset had 276 minutes of recording (16,618 samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 6 shows an experiment with challenging dynamics which is part of the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Experiment example from the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The total of 663 minutes of recordings consists of two parts: 300 minutes from our initial data collection cam- paign [22] and 363 minutes in the current campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The complete dataset is publicly available at our GitHub https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='com/ansfl/MissBeamNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' output fully connected 2 fully connected 1 OOOOOOO flatten layer co LSTM LSTM LSTM LSTM LSTM LSTM ho cell cell cell cell cell cell Beams Beams t Beams t Beams t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Beams t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Beams t, Beam 1 current beams Beam 2 5 Beam 3 5 Beam 4Input: LSTM Fully connected 1 4X6 hidden state = 500 3000 Fully connected 1 2 missing Output 6 beams Current beams: 20 9- 10 15 20 25 30 35 0 1000 2000 3000 4000 5000 Time [s]5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Performance Metric Performance metrics compare different models/methods and choose the one with the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Throughout the research, we used the performance metric of root mean squared error (RMSE), which is widely used to evaluate models on regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' RMSE is calculated by taking the root of the average of squared differences between the predicted values and the target values RMSE “ cřn i“1pyactual ´ ypredictedq2 n (19) The RMSE results are in the same units as the original data, making it easy to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Baseline Architectures Comparison To compare our two baseline architectures described in Section III-B, we consider a scenario with two missing beams, namely, beams #1 and #2, and assume six past beam measurements are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In addition to these two baseline architectures, we examine the possibility of using only past beam measurements instead of the baseline multi-head approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' These two architectures are denoted as CNN A and LSTM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results of the test dataset in terms of RMSE are presented in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results shows that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' RMSE results for network architecture comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' using the baseline architectures (multi-head) obtained better performance than working with all the inputs in a single head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In addition, the performance of the baseline LSTM showed an improvement of 27 % over the baseline CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Number of Past Beam Measurement Influence The number of past measurements utilized by the network is defined as the window-size length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The length of the optimal window size is crucial for model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The window size regularizes the model performance between the long and short movement patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' If the selected window size is too short, the model might miss the pattern of the AUV movement, and if it is too long, the model might not react well enough to a movement that just started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 8 shows the RMSE of the baseline LSTM model with different window-size lengths (between 3-10) when beams #1 and #2 are being regressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results suggest that the optimal window-size length on our dataset is six measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' RMSE as a function of the window size for the baseline LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Additional Input Information To improve the model performance even further, additional inputs are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 1) Depth Sensor: The last depth sensor reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2) AUV Velocity Vector: Domain knowledge is used to transform the raw data (in this case, the beams) into meaningful features using feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Feature engineering is prevalent in classical machine learning methods, but less in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The assump- tion when using a neural network is thet model will learn the essential relations between features indepen- dently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The beams and the velocity vector are related, as the latter is estimated using the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' That is, the model is not receiving new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yet, in the proposed LSTM-based model, there are only two fully connected layers, and therefore feature engineering may help achieve better accuracy or shorten the network convergence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Figure 9 describes the performance of each input with our baseline LSTM architecture, including additional inputs of 1) depth, 2) velocity vector, and 3) depth and velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The tested case is when beams #1 and #2 are missing, and beams #3 and #4 are inserted as a two-phase input to our baseline LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' All of the additional inputs improved Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Velocity RMSE as a function of different input selection for our baseline LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' the performance of the baseline LSTM, and the best approach was obtained using all three input types - beam measurements (baseline), depth sensors, and the velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In this instance, there was a 16% improvement compared to the LSTM baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Missing Beams Analysis There are 14 combinations of missing beams: four combina- tions of one missing beam, six of two missing beams, and four of three missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In the proposed approach a different network needs to be trained for each of those combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Training for all networks used the same hyper-parameters: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='10 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='00 LSTMA Baseline LSTM CNN-A Baseline CNN Architecture0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='095 [m/s] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='090 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='075 3 4 5 6 7 8 9 10 # of past measurements0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0700 兰0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0675 S MSE 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0575 Baseline LSTM Baseline LSTM + Baseline LSTM + Baseline LSTM + altitude velocity vector altitude + velocity vector Architecture6 MSE loss function with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='00005, batch size of 1 sequence, and 150 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In the following sections, we present the performance of our MissBeamNet approach com- pared to the average (baseline) and virtual beam approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For this analysis, we employed our baseline LSTM network described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Based on the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='4, we use six past DVL beam measurements and, based on Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5, both the depth sensor reading and velocity vector were are added as additional inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results were obtained on the testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 1) One missing beam: When one beam is missing, the least squared approach (4) can be used to obtain the estimated AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Table I presents the results of estimating the missing beam, the speed error obtained when using the estimated fourth beam together with the measured three, and the improvement of our MissBeamNet approach over the two model-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Both model-based and MissBeamNet methods were supe- rior tp the three beams solution, indicating that regressing the fourth beam is critical to improving the AUV speed estima- tion accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Specifically, MissBeamNet, improved the speed accuracy by over 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In addition, MissBeamNet performed significantly better than the model-based approaches, with a 40%-68% improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Taking the mean of performance of all four scenarios MissBeamNet improved the model-based approaches by over 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='8 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2) Two Missing Beams: When considering two missing beams, six different combinations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' In such scenarios, the AUV velocity cannot be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Following the same procedure as the previous one missing beam scenarios, Table II presents the results of two missing beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results show a significant difference between the speed error in each com- bination, even in the model-based approaches, emphasizing the problem’s complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yet, in all cases, MissBeamNet was more accurate than the model-based approaches, with a minimum improvement of 20% that reached almost 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The average improvement over the baseline model-based approach was 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7% compared to 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='8% when only one beam was miss- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' This is attributed to the model receiving less information from two current beams compared to three when only one is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 3) Three Missing Beams: Table III presents the results for the four scenarios in which three beams are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' As expected, the speed error when three beams are missing is higher than in the two or one missing beams scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yet, MissBeamNet use improved results by at least 21% over the model-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For three missing beams, the average improvement was 24% compared to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7% when two beams were missing, only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7% less, indicating that even with only one beam at hand, the AUV velocity can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 4) Hyperparameter Tuning: One of the main challenges in deep learning research is to find the best combination of hy- perparameters for the proposed architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Each architecture has several parameters that can influence model performance, including the number of layers, the number of parameters in each layer, the type of cost function, the learning rate, and batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To demonstrate the potential of hyperparameter tuning, we evaluated three different hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The first was the learning rate, which affects how much each batch changes the weights and biases III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The second hyper parameter was the number of hidden parameters in the LSTM layer ht III-A2, and the third hyperparameter is the number of parameters in the LSTM output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To test the importance of hyperparameter tuning, each parameter was set with a few available options, and a seed was set (equal initialization in each run).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' We focused on a one missing beam scenario, which has four options - missing beam #1, #2, #3, or #4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For each case, 15 randomly selected combinations of the three hyperparameters were examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Table IV presents the potential of hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' It is important to note that out of the 15 tested hyperparameter combinations, only a few were better than the results before tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yet, they were able to improve the missing beam estimation and, consequently, reduce the speed error and increase the rate of improvement compared to the two model-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' CONCLUSIONS Here, we presented MissBeamNet, a deep learning-based framework developed to compensate for partial DVL measure- ment scenarios (1, 2, or 3 missing beams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To that end, an LSTM-based dedicated DNN was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' We demonstrated that the best input to the network is past DVL measurements, past depth sensor measurements, previous velocity vectors, and the currently available measured beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Once the missing beams are regressed, they are combined with the available beams and plugged into the classical model-based approach to estimate the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' To evaluate MissBeamNet, sea experiments with the Uni- versity of Haifa’s ”Snapir” AUV were conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The data included several trajectories collected for different purposes and under various sea conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' We provide a thorough analysis of all 14 missing beam combinations and explore several means to enhance our baseline architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The results show that MissBeamNet allows estimating the missing DVL beams and, consequently, the AUV velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Addi- tionally, MissBeamNet significantly improves the accuracy of the velocity vector in all examined scenarios compared to the model-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' The improvement of all three missing beam combinations was above 20 % over the model- based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' For two missing beams, performance was generally better compared to three missing beams since the model uses one additional measured beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Finally, we show that hyperparameters-tuned models improve the accuracy of MissBeamNet by more than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' REFERENCES 1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Shao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Yan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Liu, A multi-AUV 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Ruhl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Morris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Peakall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Parsons, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Sumner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Darby, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Dorrell and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' Hunt, Autonomous Underwater Vehicles (AUVs): their past, present and future contributions to the advancement of marine geoscience Marine Geology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 352, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 451-468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' 7 TABLE I ONE MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET Case Approach Beam 1 Beam 2 Beam 3 Beam 4 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' results Missing beam [m/s] Average (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='106 Virtual beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='121 MissBeamNet (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='053 Speed error [m/s] Average (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='064 Virtual beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='072 Three beams 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='443 MissBeamNet (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='032 MissBeamNet improvement % Average (baseline) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='8 Virtual beam 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='6 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5 Three beams 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7 TABLE II TWO MISSING BEAM SCENARIO RESULTS ON THE TEST DATASET Case Approach Beam 1,2 Beam 1,3 Beam 1,4 Beam 2,3 Beam 2,4 Beam 3,4 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' result Missing beams [m/s] Average (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='106 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='120 MissBeamNet (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='066 Speed error [m/s] Average (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} 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+page_content='055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='062 MissBeamNet improvement % Average (baseline) 40.' metadata={'source': 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THE TEST DATASET Case Approach Beam 1,2,3 Beam 2,3,4 Beam 1,2,4 Beam 1,3,4 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content=' results Missing beams [m/s] Average (baseline) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='106 Virtual beam 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='105 Virtual beam 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='105 MissBeamNet (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='079 MissBeamNet improvement % Average (baseline) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='9 Virtual beam 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='1 TABLE IV HYPERPARAMETERS TUNING Case Approach Beam 1 Beam 2 Beam 3 Beam 4 Missing beam [m/s] Before tuning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='067 After tuning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='012 Speed error [m/s] Before tuning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='040 After tuning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='007 Learning rate Before tuning 5e-05 5e-05 5e-05 5e-05 After tuning 1e-04 1e-04 5e-05 1e-05 hidden LSTM parameters ht Before tuning 500 500 500 500 After tuning 250 750 750 100 LSTM output parameters Before tuning 7 7 7 7 After tuning 7 5 7 5 MissBeamNet Tuning improvement % Average (baseline) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='6 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5 Virtual beam 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='9 Three beams 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='4 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='7 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='4 Before tuning 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} +page_content='8 82.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf'} diff --git a/9tFST4oBgHgl3EQfbTje/content/tmp_files/2301.13799v1.pdf.txt b/9tFST4oBgHgl3EQfbTje/content/tmp_files/2301.13799v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6c9e5b63376316c1e497e906b2a138fdf301bdd --- /dev/null +++ b/9tFST4oBgHgl3EQfbTje/content/tmp_files/2301.13799v1.pdf.txt @@ -0,0 +1,2754 @@ +Highlights +Partitioning Distributed Compute Jobs with Reinforcement Learn- +ing and Graph Neural Networks +Christopher W. F. Parsonson, Zacharaya Shabka, Alessandro Ottino, Georgios +Zervas +• Demonstrate that deciding how much to partition distributed jobs is a +key factor in determining overall system throughput. +• Demonstrate that optimising for only the job completion time leads +to high blocking rates and poor throughput in dynamic job arrival +scenarios. +• Introduce a new partitioning algorithm which leverages reinforcement +learning, a graph neural network, and a novel formulation of the user- +defined job completion time specification to automatically learn to +partition jobs such that the blocking rate is minimised and user require- +ments are met. +• Demonstrate the proposed algorithm out-performing baselines on a state- +of-the-art optical network architecture running five real deep learning +computation graphs. +arXiv:2301.13799v1 [cs.LG] 31 Jan 2023 + +Partitioning Distributed Compute Jobs with +Reinforcement Learning and Graph Neural Networks +Christopher W. F. Parsonson1,∗, Zacharaya Shabka1, Alessandro Ottino1, +Georgios Zervas1 +∗Corresponding author: zciccwf@ucl.ac.uk +1UCL + +Partitioning Distributed Compute Jobs with +Reinforcement Learning and Graph Neural Networks +Christopher W. F. Parsonson1,∗, Zacharaya Shabka1, Alessandro Ottino1, +Georgios Zervas1 +Abstract +From natural language processing to genome sequencing, large-scale ma- +chine learning models are bringing advances to a broad range of fields. Many of +these models are too large to be trained on a single machine, and instead must +be distributed across multiple devices. This has motivated the research of new +compute and network systems capable of handling such tasks. In particular, +recent work has focused on developing management schemes which decide +how to allocate distributed resources such that some overall objective, such +as minimising the job completion time (JCT), is optimised. However, such +studies omit explicit consideration of how much a job should be distributed, +usually assuming that maximum distribution is desirable. In this work, we +show that maximum parallelisation is sub-optimal in relation to user-critical +metrics such as throughput and blocking rate. To address this, we propose +PAC-ML (partitioning for asynchronous computing with machine learning). +PAC-ML leverages a graph neural network and reinforcement learning to +learn how much to partition computation graphs such that the number of +jobs which meet arbitrary user-defined JCT requirements is maximised. In +experiments with five real deep learning computation graphs on a recently +proposed optical architecture across four user-defined JCT requirement distri- +butions, we demonstrate PAC-ML achieving up to 56.2% lower blocking rates +in dynamic job arrival settings than the canonical maximum parallelisation +strategy used by most prior works. +Keywords: +Deep Learning, Reinforcement Learning, Graph Neural Networks, +∗Corresponding author: zciccwf@ucl.ac.uk +1UCL +Preprint submitted to Journal of Parallel and Distributed Computing +February 1, 2023 + +Distributed Asynchronous Computing, Job Partitioning, Optical Networks +1. Introduction +The last decade has seen an exponential increase in the amount of compute +demanded by big data jobs such as artificial intelligence (AI) and genome +processing, with resource requirements doubling every 3.4 months since 2012; +50× faster than Moore’s Law (OpenAI, 2018). This trend is showing no +sign of slowing down. The fundamental relationship between neural net- +work accuracy and scale (Kaplan et al., 2020) provides a strong incentive +for practitioners seeking performance improvement to further increase their +resource requirements. Moreover, brain-scale AI will require at least as many +parameters as the ≈1 000 trillion synapses present in the human brain (Furber, +2016); several orders of magnitude more than the largest models used today. +The compute time and memory requirements of state-of-the-art big data +applications already far outstrip the capabilities of any single hardware +device. For example, one of the current largest deep neural networks (DNNs), +Megatron-Turing natural language generation (MT-NLG) (Smith et al., 2022), +contains 530 billion parameters. +These parameters alone occupy ≈1 000 +GB, exceeding the capacity of the largest A100 GPU by over an order of +magnitude, and the parameter loss gradients tracked during training occupy +several times more. Even if the model could be fitted onto a single device, +the training time would be ≈900 years2. To address these compute time and +memory demands, rather than using a single device, big data jobs must be +distributed and parallelised across a cluster of machines. For example, the +Selene supercomputing cluster (NVIDIA, 2020) consists of 358 400 A100 GPU +tensor cores, bringing the MT-NLG training time from 900 years down to the +order of days3. +However, parallelising jobs across ever-more machines brings its own +challenges. With any parallelisation strategy, at some point the output of each +‘worker’ (a single device processing at least part of a job) must be collected and +synchronised to get the overall result of the parallelised computation. This +synchronisation requires communication between the workers. As the number +2Assuming it takes 8 V100 GPUs 36 years to train a 175 billion parameter model +(NVIDIA, 2022) and extrapolating. +3Assuming a linear parallelisation speedup and 0 communication overhead. +2 + +Figure 1: How the network overhead of six distributed deep learning jobs (encompassing +object tracking, recommendation, natural language processing, and image recognition) +increases with the number of workers used in Meta’s GPU cluster (Wang et al., 2022). +of workers used to execute a job is increased, the per-worker computation +demands decrease, but the overall communication overhead between workers +grows (see Figure 1). This shifts the performance bottleneck away from +the workers themselves and into the network connecting them, and brings +additional challenges with managing varying traffic characteristics for different +job types and parallelisation strategies (Wang et al., 2022; Parsonson et al., +2022a; Benjamin et al., 2021, 2022). +To address the communication bottleneck in distributed computing, recent +works have sought to develop optical clusters (Benjamin et al., 2020; Ballani +et al., 2020; Khani et al., 2021; Wang et al., 2022; Ottino et al., 2022); machines +interconnected by optical switches (Parsonson et al., 2020; Gerard et al., 2020, +2021). Compared to their electronic counterparts, optically switched networks +offer orders of magnitude improvements in scalability, bandwidth, latency, +and power consumption (Ballani et al., 2020; Zervas et al., 2018; Mishra et al., +2021) (see Section 3). +Optical clusters are typically operated under the optical circuit switched +(OCS) paradigm due to its non-blocking circuit configurations with high +capacity and scalability (Raja et al., 2021). OCS networks are fundamentally +different from the electronic packet switched (EPS) architectures used by +most current clusters, resulting in entirely new communication patterns and +resource demand characteristics. Consequently, new compute and network +resource management schemes are needed in order to optimally allocate jobs +and maximise performance. +3 + +Job 1 +Job 3 +Job 5 +Job 2 +Job 4 +Job 6 +60 +40: +20 - +101 +102 +# WorkersOf the many resource management tasks which must be performed in a +compute cluster, job partitioning (how to split a job up across how many +devices) is key to overall performance. More partitioning can lead to lower +compute times. However, it may also increase network overhead and occupancy +of cluster resources, possibly leading to future jobs being blocked upon arrival +and consequently lower overall cluster throughput. Prior works such as SiP- +ML (Khani et al., 2021) have introduced simple partitioning heuristics for +optical networks which have notably improved cluster performance. However, +they have not been designed under the more realistic setting of dynamic +and stochastic job arrivals, have not considered the state of the cluster in a +‘network-aware’ manner when making partitioning decisions, and have been +crafted to optimise for the sub-optimal objective of minimising job completion +time (JCT). +In this work, we first argue that simply minimising the JCT is a naive +objective because it brazenly encourages more parallelisation of a job re- +quest without considering the effect this has on the ability of a cluster to +service subsequent jobs. We then introduce a new more subtle formulation +of the optimisation metric, the user-defined blocking rate, which more aptly +encompasses the desires of cluster users. Next, we propose a simple modi- +fication of the quantised SiP-ML partitioner which, rather than maximally +parallelising all jobs, minimally parallelises them such that they meet the +user-defined maximum acceptable completion time. Then, we propose a +novel network-aware partitioning strategy (see Figure 5 and Section 5) called +PAC-ML (partitioning for asynchronous computing with machine learning) +which utilises reinforcement learning (RL) and a graph neural network (GNN) +to flexibly meet the demands of the user in an arbitrary manner given the +current state of the cluster network. Finally, we demonstrate our method in +simulation on the recently propsed RAMP optical architecture (Ottino et al., +2022), achieving up to 56.2% lower blocking rates than the best heuristic +baseline. We show that different user-defined demand environments require +different partitioning strategies for optimal results, and that a key advantage +of PAC-ML is that it is able to discover performant strategies automatically +without the need for handcrafted heuristics or environment-specific tuning. +2. Related Work +Recent years have seen a surge of interest in developing methods to +distribute machine learning (ML) tasks across multiple devices (Ben-Nun +4 + +and Hoefler, 2019; Mayer and Jacobsen, 2020). One approach has been to +optimise the physical plane of the distributed cluster such as its compute +and network devices and architectures (Parsonson et al., 2020; Khani et al., +2021; Wang et al., 2022; Ottino et al., 2022). +In this work, we instead +focus on optimising the virtual plane, which determines how physical layer +resources are allocated to execute a job. We divide the virtual plane into +three sub-components: Job (1) partitioning (how many devices to use); (2) +placement (which devices to use); and (3) scheduling (in which order to use +the devices). Many prior virtual plane works have considered (2) and (3) (how +to distribute), whereas we focus on (1) (how much to distribute). However, +in this section we comment on recent progress across all these fields, since we +leverage this progress throughout our work. +ML for discrete optimisation. Many combinatorial optimisation (CO) +problems turn out to be NP-hard, rendering exhaustive search techniques +intractable for practical application (Bengio et al., 2021). Consequently, +practitioners rely on either approximate algorithms, which give restricted +performance guarantees and poor scalability (Williamson and Shmoys, 2011), +or heuristics, which have limited solution efficacy (Halim and Ismail, 2019). +Since the first application of neural networks to CO by Hopfield and Tank +(1985), the last decade has seen a resurgence in ML-for-CO (Bello* et al., 2017; +Dai et al., 2017; Barrett et al., 2019; Gasse et al., 2019; Barrett et al., 2022; +Parsonson et al., 2022b). The advantages of ML-for-CO over approximation +algorithms and heuristics include handling complex problems at scale, learning +either without external input and achieving super-human performance or +imitating strong but computationally expensive solvers, and (after training) +leveraging the fast inference time of a DNN forward pass to rapidly generate +solutions. Since almost all cluster resource management tasks can be reduced +to canonical CO problems (Bengio et al., 2021), many state-of-the-art resource +management methods utilise recent advances in ML-for-CO. +Job placement. Mirhoseini et al. (2017) were the first to apply ML +to the task of deciding which operations in a computation graph to place +on which devices in a cluster. They used a sequence-to-sequence model +consisting of an LSTM DNN with an attention mechanism trained with the +simple REINFORCE policy gradient RL algorithm (Williams, 1992) such that +the JCT of a deep learning job was minimised, outperforming handcrafted +heuristics when training the Inception-V3 computer vision and LSTM natural +language processing models. Gao et al. (2018) furthered this work by replacing +REINFORCE with the more advanced proximal policy optimisation (PPO) +5 + +RL algorithm (Schulman et al., 2017) with lower variance and reduced training +hardware demands. They demonstrated their method beating Mirhoseini +et al. (2017) on the CIFAR-10 image recognition benchmark in terms of +JCT. Mirhoseini et al. (2018a) proposed a novel hierarchical model which +decomposed the job placement task into a joint group-and-place problem, +reducing the JCT of Inception-V3, ResNet, LSTM, and NMT models by up +to 60% relative to the state-of-the-art. +All works up to this point used DNN architectures restricted to Euclidean- +structured input data. +Consequently, in order to handle non-Euclidean +graph-structured data such as computation graphs and cluster networks, they +had to be re-trained each time a new graph structure was considered. Addanki +et al. (2019a) were the first to instead leverage a GNN, as well as the grouping +scheme of Mirhoseini et al. (2018a), to learn to generalise across different +job types with varying computation graph structures, demonstrating device +placement schemes which were on par with or better than prior approaches +on Inception-V4, NASNet, and NMT after 6.1× fewer training steps. Khadka +et al. (2021) furthered the use of GNNs for job placement by combining +GNNs, RL, and population-based evolutionary search with the hierarchical +group-and-place scheme of Mirhoseini et al. (2018a). Concretely, they replaced +the manually-designed operation grouping heuristic with a learned policy +capable of superior scaling and JCT performance. +Job scheduling. Bao et al. (2018) addressed the job scheduling problem +(the order in which to execute operations placed across a set of devices) +using a primal-dual framework for online job scheduling. They represented +the problem as an integer linear programme (ILP) which their proposed +algorithm could solve in polynomial time in an online fashion such that +the cluster resources were maximally utilised and the JCT minimised. Li +et al. (2021) proposed a placement-aware scheme which leveraged the pre- +determined device placement allocation to decide on a job schedule which +could reduce the average JCT by up to 25% relative to other scheduling +methods. Paliwal et al. (2020) went further by utilising an RL-trained GNN +and a genetic algorithm to jointly optimise both job placement and scheduling, +demonstrating both lower JCT and peak memory usage than other strategies +when distributing TensorFlow computation graphs across a cluster. +Job partitioning. To the best of our knowledge, Khani et al. (2021) +are the only ones to have explicitly considered the question of how much to +distribute a computation graph in the context of an optical network. Like +other works, they assumed that a maximum parallelisation strategy (i.e. +6 + +partition the job across as many workers as possible) is a desirable objective, +and then focused on how best to design the physical layer such that the JCT +could be minimised. +All works discussed in this section have assumed that the JCT is the key +objective to minimise. Consequently, where the question of partitioning is +considered, prior works have assumed that more parallelisation is desirable. +However, we posit that user-critical metrics such as throughput and blocking +rate are compromised by prioritising optimisation of the JCT in a cluster +setting with dynamic job arrivals. To address this shortcoming, we propose a +new ML-based resource management scheme which explicitly addresses the +partitioning question. Concretely, our work leverages the emergent trend +from these other virtual plane fields, namely utilising an RL-trained GNN, +to decide how much to partition different jobs in a dynamic setting with +arbitrary user-defined completion time requirements. +3. Background +3.1. Parallelisation +Types of parallelism. Parallelisation is the process of distributing a +computational job across multiple devices. This is done in order to reduce the +time and/or physical memory needed to complete the job. There are three +main types of deep learning parallelism; data parallelism, model parallelism, +and hybrid parallelism (see Appendix 9.1.1 for extended background infor- +mation on these methods). Although today the most common method for +DNN training parallelisation is data parallelism for its simplicity and limited +network overhead, we focus on the less common but more desirable model +parallelism paradigm for its strong scaling capabilities (Khani et al., 2021). +Our proposed partitioning methods are applicable to hybrid and pipeline par- +allelism, but these require additional simulation complexity and are therefore +beyond the scope of this manuscript. +Computational jobs. A computational job is a directed acyclic graph +(DAG) whose nodes are operations and edges are dependencies. Operations +are computational tasks (e.g. some mathematical reduction, a database +query, etc.). Dependencies are either control dependencies, where the child +operation can only begin once the parent operation has been completed, +or data dependencies, where at least one tensor is output from the parent +operation and required as input to the child operation. In the context of +DNNs, a job DAG is a sequence of forward pass, backward pass, and parameter +7 + +Figure 2: Diagram showing a DNN job DAG being partitioned. Top: A forward pass DAG +where each node has an associated partition degree (how many times it will be divided +when partitioned). Bottom: A partitioned DAG with forward and backward passes handled +consecutively. Green edges in the graph represent data flow (i.e. output to input) between +consecutive operations in the forward pass. Orange edges represent gradient exchanges +processed in the backward pass (backpropagation). Blue edges represent full connectivity +collective operations to synchronise weight updates across partitioned components of an +operation. Note that, for brevity, the top unpartitioned DAG only shows the forward pass +(since, before partitioning, the graph structure is identical to the backward pass), whereas +the bottom partitioned DAG shows both the forward and backwards passes (since, after +partitioning, the graph structures are different). +8 + +Original DAG +(to be partitioned) +Partition Degree: 1 +Partition Degree: 4 +Partition Degree: 4 +Partition Degree: 2 +Partition Degree: 1 +Data Flow +Gradient Flow +RAMP Collective +Partitioned DAG +Forward Pass +3a +Backward Pass +All-to-all +2a +All-to-all +All-to-allupdate operations which need to be performed on data exchanged between +operations. Whether or not this data passes through a communication network +is determined by how the operations are partitioned, placed across a cluster +of workers, and parallelised. +Job partitioning. Job partitioning refers to the process of splitting the +operations of a job DAG into u (the partition degree) smaller sub-operations +which can in turn be placed across u workers, thus reducing their run time +and memory requirements. Partitioning is used in the model, hybrid, and +pipeline parallelisim paradigms. More partitioning can decrease compute time +and memory requirements, but requires more inter-worker communication, +complex intra-worker operation scheduling, and greater resource utilisation, +therefore potentially increasing overall completion time, cluster complexity, +and subsequent job blocking rates. Figure 2 visualises how an initial DAG +for some arbitrary neural network architecture, where each operation has a +partitioning degree, can be re-represented in terms of its partitioned form. +Both forward and backward passes are explicitly represented since inter- +operation information dependencies (i.e. the edges in the graph) are not the +same in each pass. +3.2. Optical Networking +Most current cluster networks use optic fibre communication links, but +the switch devices which interconnect the network are usually electronic. +Limitations of electronic networking. Electronic networks have poor +scalability, bandwidth, latency, and power consumption. Concretely, since the +per-port bandwidth is limited and the power consumption required to cool +active electronic devices is expensive, the bisection bandwidth achievable in +an electronic network is restricted, thus hampering scalability. Consequently, +although the compute power of DCN server nodes, as measured by FLOP/s, +has increased by a factor of 65 over the last 18 years, the bandwidth of the DCN +network facilitating communication between these nodes has only increased +by a factor of 4.8, resulting in an 8-factor decrease in bytes communicated per +FLOP (Bergman, 2018). This has created a performance bottleneck not in +the server nodes themselves, but rather in the network connecting them. This +issue is especially compounded when striving for strong scaling via model +parallelism with distributed computing, and with the trend towards larger +models with ever more parameters as described in Section 1. +Optical circuit switched networks. Cluster networks with optical +switches have the potential to offer significant improvements in performance +9 + +Figure 3: The mean network overhead of the 6 distributed deep learning jobs reported +by (Wang et al., 2022) in Meta’s GPU cluster compared to that of RAMP as reported by +Ottino et al. (2022) on the 5 jobs considered in our work. Note that this is an approximate +comparison, and that the important takeaway is that RAMP retains low network overheads +as jobs become increasingly distributed. +(due to larger bandwidth and lower switching latency) and energy efficiency +(due to the lack of optical-electronic-optical conversion overhead), as well as +the capability to scale to next-generation large-scale distributed compute jobs +with exascale bandwidth and compute (Ottino et al., 2022). OCS networks in +particular offer a promising avenue with which to realise commercial optical +networks due to their non-blocking circuit configurations with high capacity +and scalability and low deterministic switching latency. In contrast to optical +packet switched networks, OCS networks are simpler to implement and they +eliminate the need for in-switch buffering or queuing and addressing. +RAMP. RAMP is a state-of-the-art OCS architecture designed specifically +for cloud data centres and distributed deep learning systems (Ottino et al., +2022). RAMP networks are parameterised by NC communication groups, +NR racks per communication group, and NS servers per rack, resulting in +a NW = NC × NR × NS worker cluster with a colloquially termed ‘RAMP +shape’ defined by tuple ⟨NC, NR, NS⟩. At its core, RAMP proposes a novel +set of message passing interfaces (MPIs) for performing the synchronisation +steps (AllReduce, AllGather, etc.) required by distributed DNN training jobs. +These will be referred to as collective operations. These MPIs are designed +to take full advantage of the high bandwidth provided by optical network +architectures. Consequently, as shown in Figure 3, the network overhead of +RAMP remains remarkably low as the number of workers used to execute +a job increase (see Section 6 for experimental details). The RAMP authors +10 + +(%) +Meta +RAMP +Network Overhead ( +40- +20 - +0.49 +0.99 +2.0 +0.062 +0.13 +0 +101 +102 +# Workersshowed that this low network overhead enables unprecedented scalability with +up to 65 536 worker nodes capable of training O(trillion) parameter DNN +models. +RAMP placement rules. As detailed in Ottino et al. (2022), a group +of workers in a RAMP shape can only undergo collective operations if they +are selected with respect to certain rules, loosely termed here ‘symmetry’ +rules. For shape ⟨NC, NR, NS⟩, these rules are as follows: (1) NS workers per +rack spread over NR racks requires that the set of workers on each rack span +NR distinct communication groups. These NR distinct communication groups +do not have to be the same set across racks. (2) NS workers on NR = 1 rack +must span NS communication groups. (3) NS workers spread over NR racks +(NS = 1 worker per rack) must span NS distinct communication groups. +In our simulations, we use a simple first-fit operation placement heuristic +which conforms to these rules (refer to Appendix 9.4.4 for further details). +3.3. Reinforcement Learning +RL is the study of optimal decision making in natural and artificial systems +(Sutton and Barto, 2018). In the general RL setting shown in Figure 5, an +agent interacts with an environment at each sequential time step t. The +environment can be described by tuple ⟨T, R⟩, where T is a state transition +probability matrix defining the transition probabilities from all states s to all +successor states s′ taking action u where T u +ss′ = P(St+1 = s′|St = s, U t = u), +and R is a scalar reward function giving the expected immediate (next state) +reward given current state s and chosen action u where Ru +s = E(Rt+1|St = +s, U t = u). +Markov decision process. The environment is usually assumed to have +the Markov property whereby P(st+1|st) = P(st+1|ht); that is to say that the +probability of the next state being st+1 given the current state st is the same as +the equivalent probability given all previous states in history ht = {s1, ..., st}. +As such, this RL setting is usually assumed to be a Markov decision process +(MDP) described by tuple ⟨S, U, T, R, γ⟩ where S is a finite set of possible +environment states, U is either a discrete (finite) or continuous (infinite) set +of possible actions, and γ ∈ [0, 1] is a discount factor specifying the factor by +which to multiply future expected rewards to discount their present value. +Since Markov states are stochastic, future rewards are never fully certain and +are therefore expressed as an expectation. +Agent goal. The agent’s goal is to learn to maximise its expected total +discounted future reward, termed the ‘value’ or ‘return’ Gt = �∞ +k=0 γkRt+k+1, +11 + +over the course of an episode (a sequence of decision steps which may or may +not terminate at some point). To do so, the agent can use model-free RL to +avoid explicitly modelling the environment by only using its policy function +and/or its value function to make decisions. The policy function π maps an +observed state st to a corresponding action ut such that some estimated score +objective is maximised. The value function estimates the expected return Gt +from being in state st and following policy π (the state value function v) or +from being in state st, taking action ut, and following policy π (the action +value function q). Crucially, value and policy functions can be approximated +and learned with DNNs, enabling RL to be scaled to large problem instances +(see Appendix 9.1.2 for extended background information on DNNs). +Advantages of RL. Using traditional RL has several advantages over +heuristics and other ML paradigms such as supervised learning. First, no +external data from human-designed or computationally expensive heuristics is +required, enabling an agent to learn super-human policies without potentially +sub-optimal initial biases towards a certain strategy or a costly expert example +collection-and-labelling phase (Silver et al., 2016). Second, a DNN with a finite +number of layers and neurons will have its expressivity constrained (Dong +et al., 2020), restricting the complexity of the set of functions it is capable +of approximating. Because the objective of an RL agent is to maximise its +expected future return which, under the assumption that a suitable reward +function has been crafted, is equivalent to maximising performance on a +given task, RL agents are able to maximise task performance given DNN +expressivity constraints. Third, since RL agents maximise future return, +they are capable of learning sophisticated non-myopic policies which sacrifice +short-term reward in exchange for higher long-term return (Sutton and Barto, +2018). +4. User-Defined Blocking Rate +To motivate our work, we first explore the key metrics to consider when +evaluating a job partitioning strategy with the help of an experiment on +32 GPU workers, and then introduce a new formulation of the user-defined +blocking rate. All experimental details are given in Section 6. +The inadequacy of optimising the job completion time. As dis- +cussed in Section 2, most prior works researching management schemes +for distributed computing aim to minimise JCT; the time taken to com- +plete a given job. If a job j begins running at wall clock time tstart +wc,j and is +12 + +Figure 4: (a-b) Demonstration of how more partitioning can lead to a lower JCT than no +partitioning (i.e. sequentially running the job on a single device), but this may be at the +cost of a higher blocking rate since more cluster resources are occupied when subsequent +jobs arrive. (c-d) Demonstration of how optimising for the cluster throughput leads to +an unfair bias towards more partitioning, because more parallelism creates more work for +the cluster and therefore artificially increases cluster throughput even though, from the +perspective of the user, the original offered throughput may be lower. +completed at time tend +wc,j, researchers usually record the completion time as +JCTj = tend +wc,j − tstart +wc,j . Consequently, most systems maximise the degree to +which they parallelise jobs in order to minimise JCT. While it is true that +end users undoubtedly want this JCT metric to be minimised, it fails to +quantify when a job was blocked, which occurs when no cluster resources were +available to service it. While more parallelism will often lead to a lower JCT +for a given job, it will also use up more of the cluster’s compute and network +resources, potentially blocking future job arrivals (see Figure 4). Therefore in +practice, end-users wish to minimise both the JCT and the overall blocking +rate (the fraction of jobs blocked over a given time period). While maximum +parallelisation will lead to a minimised JCT, we posit that a balance between +these two extreme parallelisation strategies can more aptly optimise for both +the JCT and blocking rate. +Alternative optimisation objectives. One metric which encapsulates +both the JCT and blocking rate is throughput; the information processed per +unit time. There are two issues with using throughput as an optimisation +objective. (1) Operators must be careful how they measure the throughput +to be optimised. If they measure the cluster throughput (the total cluster +information processed per unit time), they will be biased towards more +13 + + Sequential +Paramac +(a) +X104 +(b) +Rate +0.4 - +S +Blocking +0.3 +0.2 +XX~X-X1 +2X +S +(c) +X107 +S +(d) +X107 +B +1.25 +B +Cluster +Offered +1.00 +1.00 +0.75 +00.75 +1.0 +1.5 +1.0 +1.5 +Load +X107 +Load +X107 +Rate (B/s) +Rate (B/s)parallelisation, because when a job is partitioned and parallelised, the edge +dependencies coming in to and out of the partitioned operation node(s) must +be replicated (see Figure 2). This artificially creates more information for +the cluster to process even though, from the end users’ perspective, the total +information processed of their original demand is the same. Therefore, the +offered throughput (the total original demand information (i.e. before parti- +tioning was applied) processed per unit time) is a more suitable throughput +metric to optimise. Figure 4 shows an example of how a ‘maximum parti- +tioning’ strategy, such as that used by SiP-ML (Khani et al., 2021), can have +superior cluster throughput when compared to a ‘no partitioning’ strategy +(sequentially running the job on a single device) despite having lower offered +throughput. However, offered throughput is still an inadequate optimisation +metric, because (2) in practice, different jobs being serviced by the cluster +originating from different client users have different priorities and job comple- +tion time requirements. For example, two identical machine learning training +jobs might be submitted to the cluster, but one from a user who intends to +deploy the model commercially and requires it to be completed overnight, and +the other from a user who is employing the model for research and has less +stringent completion time requirements. Ideally, operators would direct their +clusters to meet flexible user-defined per-job completion time requirements. +The user-defined blocking rate. To enable users to dynamically de- +termine the completion time on a per-job basis whilst also maximising the +number of job demands satisfied, we introduce a new formulation of the +user-defined blocking rate objective for the partitioning algorithm to optimise. +Given a job which, if executed sequentially on one device, would be completed +in JCTseq +j , we define the maximum acceptable JCT as JCTacc +j += β · JCTseq +j , +where {β ∈ R : 0 < β ≤ 1}. Here, β is a parameter chosen by the user which +determines how quickly the job must be completed. If JCTj > β·JCTseq +j , then +the cluster will have failed to complete the job within the required time and +the job will be recorded as having been blocked. The user-defined blocking +rate is therefore the fraction of jobs which failed to meet the JCTj ≤ β·JCTseq +j +requirement over a given period of time. Note that rather than brazenly +optimising for either the JCT or the blocking rate, the user-defined blocking +rate enables the cluster operator to instead dynamically specify their desired +completion time on a per-job basis, and the performance of the cluster is +evaluated according to how well it was able to meet the requirements of the +user. Furthermore, the β parameter corresponds to the speed-up factor being +requested by the user and, since {β ∈ R : 0 < β ≤ 1}, can be given directly +14 + +Figure 5: An overview of our PAC-ML approach transitioning from step t → t + 1. At +each time step t when there is a new job to be placed on the cluster, we: (i) Use a GNN to +generate an embedded representation of the node and edge features in the job’s computation +graph, and a standard feedforward DNN to do the same for the global job and cluster +features; (ii) concatenate the outputs of (i) and use another feedforward DNN to generate +a logit for each action ut ∈ U t; (iii) pass the chosen action ut to the environment and +partition the job accordingly; (iv) apply any internal environment allocation heuristics +(operation and dependency placement and scheduling, etc.) to attempt to host the job +on the cluster; (v) if accepted onto the cluster, perform a lookahead to evaluate the job’s +completion time; (vi) fast-forward the environment’s wall clock time twc to when the next +job arrives, and return the corresponding reward rt+1 and updated state st+1. +as input to a DNN. +5. PAC-ML Partitioning Methodology +RL agents can learn general policies without the need for human guidance. +An RL job partitioner therefore has the potential to take an arbitrary maxi- +mum acceptable JCT provided by the user and automatically decide how much +to distribute the job such that, over a period of time, the number of jobs which +meet the JCT requirements specified by the user is maximised. Such an agent +would therefore be able to minimise the blocking rate whilst also accounting +for the flexible and dynamic JCT specifications of the user. Following this +15 + +Agent +DNN forward pasS +Action selection +Generate job +Generate global +Generate +Action +graph embedding +embedding +logits +scores + = 0 +ui =1 +Concatenate +2 = 2 +ug = 3 +=4 +DNN modules +Node +Edge +Global +Logit +Environment +Input at time t +Environment transition +Job +Cluster +Partitioned job +01 +gc +gu +Allocate +place ops. +: +schedule ops. +4- +place deps. +4- +schedule deps. +Operation +Dependency +Worker +Communication link +NNModule +Actionlogic, we now describe our PAC-ML (partitioning for asynchronous computing +with machine learning) approach for learning to partition computation jobs +with RL and a GNN. +5.1. Markov Decision Process Formulation +Since allocating cluster resources for jobs arriving dynamically in time +is a sequential decision making process, formulating problems such as job +partitioning as an MDP is a natural approach and facilitates the application +of many traditional and state-of-the-art RL algorithms (Mao et al., 2016; +Addanki et al., 2019b; Paliwal et al., 2020). +States. A new job j arriving at time step t is comprised of a DAG +G(O, D, gj) with node operations O, edge dependencies D, and any other +job statistics which might be recorded gj. Similarly, the state of the cluster +at time t is made up of the number of workers available, the jobs currently +running on the cluster, and so on. To compress the state of the cluster and +the job requesting to be placed into a representation suitable as input for a +neural network at time step t, we encode this information into five feature +vectors: +1. Per-operation features oi∀i ∈ {1, ..., |O|} (5 features): (i) The com- +pute cost (run time in seconds on an A100 GPU); (ii) a binary variable +indicating whether the operation has the greatest compute cost in the +job; (iii) the memory cost (byte occupancy); (iv) a binary variable +indicating whether the operation has the greatest memory cost in the +job; and (v) the node depth with respect to the source node. The +compute and memory costs are normalised by the highest compute and +memory cost operations in the job, and the node depth is normalised +by the depth of the deepest node. +2. Per-dependency features di∀i ∈ {1, ..., |D|} (2 features): (i) The size +(in bytes) of the edge dependency normalised by the largest dependency +in the job; and (ii) a binary indicator of whether the dependency is the +largest in the job. +3. Global job features gj (15 features): (i) The number of operations; (ii) +the number of dependencies; (iii) the sequential job completion time; (iv) +the maximum acceptable job completion time; the maximum acceptable +job completion time fraction β both (v) raw and (vi) normalised; (vii) +the total memory cost of all of the operations; (viii) the total size of all +of the dependencies; (ix) the number of training steps which need to +16 + +be performed; the (x) mean and (xi) median of the operation compute +costs; the (xii) mean and (xiii) median of the operation memory costs; +and (xiv) the mean and (xv) median of the dependency sizes. Each +feature is normalised by the highest respective value of the feature across +all job types. +4. Global cluster features gt +C (2 features): (i) The number of occu- +pied workers; and (ii) the number of jobs running. Both features are +normalised by the total number of workers in the cluster NW. +5. Global action features gt +U (NW +2 +features): A binary vector indicating +the validity of each possible partitioning decision given the state of the +cluster and the RAMP rules defined by (Ottino et al., 2022). +Actions. Given the state st encapsulating both the job requesting to +be placed and the current state of the cluster, the partitioning agent uses +a policy π(st) to select a number of times ut up to which to partition each +operation in the job’s computation graph (using a similar minimum operation +run time quantum discretisation scheme to Khani et al. (2021)), where +ut +i∀i ∈ {0, 1, ..., NW +2 } (i.e. there are +� NW +2 + 1 +� +possible discrete actions). Note +that ut = 0 enables the agent to reject a job without placing it, ut = 1 places +the job onto one worker and runs it sequentially, and 1 < ut ≤ NW +2 +attempts +to distribute the job’s operations across up to ut workers. In our setting and +given the RAMP rules of Ottino et al. (2022), an invalid partitioning action +is one which is at least one of: (i) An odd number (except ut = 1), or either +(ii) greater than the number of workers available or (iii) has no valid RAMP +placement shape given the current state of the cluster (see Section 3). +Rewards. As a consequence of the RAMP rules defined by Ottino et al. +(2022), which require that the worker and network resources allocated to a +given job are reserved exclusively for that job for the duration of its run time, +we are able to perform a deterministic lookahead to evaluate what the overall +completion time, JCTj, of the job will be as soon as it is placed. Subsequently, +when a job j arrives at time step t, we can immediately determine whether +or not the cluster met the JCTacc +j +specified by the user. This enables the use +of a simple per-step +1/−1 reward scheme, +rt+1 = +� +1, +if JCTj ≤ β · JCTseq +j +−1, +otherwise +, +(1) +which when aggregated and maximised over the course of an episode +17 + +corresponds to maximally meeting the specified per-job completion time +requirements and therefore minimising the user-defined blocking rate. +Transitions. In our hybrid time- and event-drive simulation, when the +agent makes a partitioning decision at time step t, the environment transitions +to the next step t + 1 by fast-forwarding its internal simulated wall clock time, +twc, to when the next job arrives and requests to be placed, updating the +states of any running and completed jobs and their corresponding compute +and network resources as necessary. The episode terminates when twc = T max +wc . +5.2. PAC-ML Learning Setup +Reinforcement learning algorithm. To find a policy which maximises +the expected return when partitioning jobs, we used the state-of-the-art +Ape-X DQN (Horgan et al., 2018) RL algorithm; a distributed and highly +scalable value-based method (see Appendix 9.7 for algorithm details and +hyperparameters). +Neural network architecture. To make the learning of value and policy +functions tractable in large state-action spaces, we approximated them with +a custom-built message passing GNN implemented using the open-source +PyTorch (Paszke et al., 2019) and DGL (Wang et al., 2019) libraries. Refer +to Appendix 9.6 for further architectural details. +6. Experimental Setup +All code for reproducing the experiments and links to the generated data +sets are provided at https://github.com/cwfparsonson/ddls. +Simulation environment. We built an open-source Gym environment +(Brockman et al., 2016) to simulate the RAMP OCS system of Ottino et al. +(2022) in an RL-compatible manner. We used a hybrid time- and event- +driven simulation approach where we kept track of the internal simulation +wall clock time twc, enabling the measurement of time-based metrics, but +only took a partitioning decision when needed (i.e. when a new job demand +arrived at the cluster), aiding efficiency since no discrete steps were needlessly +simulated. All our experiments used similar cluster parameters to Ottino +et al. (2022). We used NW = 32 (NC = 4, NR = 4, NS = 2) A100 GPUs with +80 GB memory capacity, 2 THz memory frequency, and a peak computational +power of 130 Tflop/s. We assumed an intra-GPU propagation latency of +50 ns, a negligible OCS circuit reconfiguration latency of 1 ns, a worker +input-output latency of 100 ns, and a total worker communication capacity +18 + +Figure 6: The four β distributions used in our experiments in order to measure the capability +of each partitioner to cater to different user-defined maximum acceptable completion time +requirement settings. In each βX experiment setting, each new job generated was assigned +a β value sampled from βX in order to get the maximum acceptable job completion time, +β · JCTseq (see Section 4). +of 1.6 TB/s (resulting in a per-transceiver bandwidth of 1.6×1012 +NC +B/s). All +experiments were run up to a simulated wall clock time of T max +wc += 106 s +(around 12 days) of continuous cluster operation with dynamic job arrivals +and were repeated across 3 random seeds, with the subsequent min-max +confidence intervals for each measurement metric reported. More details of +the simulation environment are provided in Appendix 9.4. +Compute jobs. +We used the computation graph time and memory +profiles of five real deep learning job types open-accessed with Microsoft’s +PipeDream research (Narayanan et al., 2019, 2021) (see Appendix 9.5 for +details). These jobs encompassed image classification (AlexNet (Krizhevsky +et al., 2012), ResNet-18 (He et al., 2016), SqueezeNet-10 (Iandola et al., +2016), and VGG-16 (Simonyan and Zisserman, 2014)) and natural language +processing (GNMT (Wu et al., 2016)) tasks, thereby testing the generality +of the approaches we considered. All jobs arrived to the cluster dynamically +and stochastically throughout the simulation period, with the inter-arrival +time fixed at 1000 s to control the load rate. Each job was ran for Niter = 50 +training iterations, where one training iteration consists of one forward and +backward pass through the neural network. +Partitioning. When partitioning the operations in a job’s computation +graph, we allowed the partitioning agents to split each operation up to NW +2 +19 + +BB +Bc +BD +0.04 +Probability +0.02 +0.00 +1.0 +Probability +0.5 +0.0 +0.2 +3.4 +0.6 +0.8 +1.0 +3times (the environment’s ‘maximum partitioning degree’). We followed Khani +et al. (2021) by (1) assuming a linear dependency between the total number of +operation splits and each split’s compute time; and (2) choosing a minimum +quantum of computation time, τ, and splitting operations up to a number of +times which would result in sub-operations with a compute time no smaller +than τ in order to maximise GPU utilisation. We set τ = 10 ms. As such, a +given partitioning action ut set the maxmimum partitioning degree of the job, +but individual operations within the job could be split fewer times depending +on their initial compute time and τ. Note that although this restricts each +operation to be distributed across a maximum of ut servers, the total number +of workers used by all operations in the job can still be greater than ut +depending on the operation placement heuristic’s choices. +Maximum acceptable job completion times. In our setting, a par- +titioner would ideally be able to take an arbitrary job with an arbitrary +maximum acceptable job completion time, β · JCTseq, and partition the job +such that the completion time requirement is satisfied for as many dynami- +cally arriving jobs as possible (thereby minimising the user-defined blocking +rate; see Section 4). To test each partitioner’s ability to do this, we ran +experiments using four β distributions (βA, βB, βC, and βD; see Figure 6). +For each βX experiment, when one of the five possible jobs was randomly +generated to arrive at the cluster, a β value, discretised to two decimal places, +was randomly sampled from the experiment’s βX distribution and assigned +to the job. By sampling a broad range of β values from a selection of βX +distributions, we ensured that we could analyse the performance of each +partitioning agent under different completion time requirement settings and +subsequently measure the capability of each method to cater for different +user-defined requirements. +Heuristics +RL +Random +Paramax +Paramin +PAC-ML +βA +0.517+0.015 +−0.015 +0.262+0.002 +−0.003 +0.309+0.014 +−0.015 +0.203+0.007 +−0.009 +βB +0.601+0.007 +−0.008 +0.263+0.006 +−0.004 +0.396+0.006 +−0.003 +0.258+0.007 +−0.003 +βC +0.505+0.016 +−0.012 +0.267+0.004 +−0.006 +0.307+0.015 +−0.012 +0.117+0.003 +−0.003 +βD +0.465+0.004 +−0.006 +0.263+0.006 +−0.004 +0.142+0.027 +−0.046 +0.099+0.008 +−0.007 +Table 1: Blocking rate performance of the partitioning agents on the four β distributions +(best in bold). Results are given as the mean across 3 seeds, and error bars denote the +corresponding min-max confidence intervals. +20 + +Partitioner baselines. We considered three heuristic baseline partition- +ing strategies. (1) Most prior works partition a given job across as many +workers as are available up to a pre-defined environment maximum partition +degree (Khani et al., 2021; Wang et al., 2022). We refer to this strategy as +‘Paramax’. (2) Given the low network overhead (see Figure 3) and contention- +less nature of RAMP, and given the operations’ linear split-compute time +dependency of our environment, a reasonable estimate for the completion +time of a job with sequential run time JCTseq distributed across ut workers +would be JCT ≈ JCTseq +ut +. Therefore, in light of our objective to minimise +the user-defined blocking rate, we introduce a new partitioning strategy, +‘Paramin’, which partitions the job up to the estimated minimum amount +of parallelisation needed to satisfy the job’s completion time requirements, +ut = ⌈ 1 +β⌉ (i.e. the estimated speed-up factor needed). (3) For completeness, +we also ran a ‘Random’ partitioning baseline, which selects a partitioning +degree randomly from amongst the number of available workers. +Metrics recorded. To measure the performance of our partitioning +agents, we recorded the following key metrics. (1) User-defined blocking +rate (which we abbreviate to ‘blocking rate’): The fraction of arrived jobs +which had their completion time requirements met by the cluster. (2) Offered +throughput: The total ‘information size’ of the original jobs (i.e. before +partitioning was applied) processed per unit time. Since the open-access +PipeDream job profiles used in our experiments did not contain per-operation +flop/s (computational load) information, we summed the jobs’ operation and +dependency sizes (measured in bytes (B)) to get the total ‘information size’ of +each job. The load rate could then be defined as the rate of job information +arriving at the cluster per unit time, and the corresponding offered throughput +as the rate at which this total job information was processed by the cluster. +For a full list of metric definitions, refer to Appendix 9.2. +7. PAC-ML Partitioning Results & Discussion +7.1. Performance of the PAC-ML Partitioner +Comparison to the baseline partitioners. To test the performance of +each partitioning agent under different completion time requirement settings, +we ran our experiments across four different β distributions (see Section 6). +We visualise the relative blocking rate and throughput performance differences +between the agents in Figure 7, where an agent’s ‘score’ is its normalised +performance relative to the best-performing agent with respect to a given +21 + +Figure 7: Validation performances (higher is better) of each partitioning agent evaluated +across three seeds normalised with respect to the best-performing partitioner in each BX +environment. +metric. We evaluate these scores as scoreblocking = +� best_blocking_rate +blocking_rate +� +, and +scorethroughput = +� +throughput +best_throughput +� +for each agent (refer to Appendix 9.8 for all +raw metric values). As shown in Table 1 and Figure 7, our PAC-ML agent +achieved the best blocking rate across all four β distributions, beating its +nearest rival by 22.5%, 1.90%, 56.2%, and 30.3% for βA,B,C,D respectively. +Comparison amongst the baseline partitioners. Figure 7 visualises +the performance of the best PAC-ML agents on each of the four β distribution +environments compared to the baseline heuristic performances. Interestingly, +the best baseline in terms of blocking rate for βA,B,C is Paramax, but this +switches to Paramin for βD. On βB, PAC-ML achieved roughly equivalent +performance to Paramax by learning that, on this β demand distribution, +maximum parallelisation led to the lowest blocking rates. This shows that +different partitioning strategies have varying relative performances under +different cluster settings. A key advantage of PAC-ML is therefore that the +question of which partitioning strategy is best for a given environment need +not be addressed by sub-optimal hand-crafted heuristics or environment- +specific hyperparameter tuning. Instead, we have demonstrated in Table 1 +and Figure 7 that PAC-ML can automatically learn performant partitioning +strategies in arbitrary environment settings. +7.2. Analysis of the PAC-ML Partitioner +Offered throughput analysis. +One risk of optimising only for the +blocking rate when training the PAC-ML agent is that it maximises the +number of jobs accepted by prioritising small low-information jobs at the +cost of a sub-optimal offered throughput; a key metric when measuring a +cluster’s quality of service to users. Figure 7 shows that the offered throughput +22 + +PAC-ML (Ours) +Paramaa +Paramin +Random +ore +.00 +re +1.0 +Better +L Worse +e +0.75 +Rat +0.50 +0.6 +0.25 +B +βA +βB +Bc +βD +βA +βB +βc +βDFigure 8: Mean per-job blocking rates of the five job types considered for each partitioning +agent under each βX setting plotted against the number of operations (ops.), number of +dependencies (deps.), the total job information size, and the sequential run time of the job +were it ran on a single device (JCTseq). +improves with the blocking rate, with the PAC-ML agent ultimately achieving +the best throughput across all four β distributions. +Bias analysis. An important question is whether there is any bias in the +kinds of jobs the PAC-ML agent learns to prioritise in order to minimise the +blocking rate. To investigate this, Figure 8 shows the blocking rate vs. the +original characteristics for each of the five jobs considered (see Appendix 9.5 +for a summary of these characteristics) for each βX distribution environment. +The PAC-ML agent had little to no bias across the jobs relative to the +other partitioners, with all jobs attaining approximately the same blocking +rate. There was a slight bias towards the larger jobs with greater sequential +completion times and more information to process, which is likely due to the +fact that larger jobs occupy more resources and therefore inherently become +23 + +PAC-ML (Ours) +Paramac +Paramin +Random +βB +βc +βA +βD +0.6 +0.6 +Rate +0.4 +0.4 +b0 0.4 +locking +.0.4 +0.2 +0.2 +0.2 +B +0 +7 +50 +100 +50 +100 +50 +100 +50 +100 +# Ops. +# Ops. +# Ops. +# Ops. +0.6 - +Rate +0.6 - +O +0 +0.4 - +0.4 +b0 0.4 +locking +0.4 +0.2 +0.2 +: +0.2 +O +B +d +50 +100 +150 +50 +100 +150 +50 +100 +150 +50 +100 +150 +# Deps. +# Deps. +# Deps. +# Deps. +0.6 +Rate +0.6 +0.4 +0.4 +b0 0.4 +locking +0.4 +。 +0.2 +0.2 +0.2 +B +cp +2.5 +5.0 +2.5 +5.0 +2.5 +5.0 +2.5 +5.0 +Job Size ×1010 +Job Size ×1010 +Job Size ×1010 +Job Size ×1010 +0.6 +Rate +0.6 - +0.4 +0.4 +bo 0.4 +locking +0.4 +0 +0.2 +0.2 +0.2 +B +1 +1 +1 +1 +2 +3 +1 +2 +3 +1 +2 +3 +1 +2 +3 +JCTseq +×104 +JCTseq +JCTseq +×104 +JCTseq +×104 +×104favoured over smaller jobs. +8. Conclusion & Further Work +In conclusion, we have introduced a new partitioning strategy called PAC- +ML. Leveraging RL and a GNN, PAC-ML learns to partition computation +jobs dynamically arriving at a cluster of machines such that the number of +jobs which meet arbitrary user-defined completion time requirements is max- +imised without the need for hand-crafted heuristics or environment-dependent +hyperparameter tuning. We tested our partitioner on the recently proposed +RAMP optical architecture (Ottino et al., 2022) across four distributions of +user-defined completion time requirements, demonstrating up to 56.2% lower +blocking rates relative to the canonical maximum parallelisation strategies +used by most prior works when partitioning five real deep learning jobs. +We hope that our work will spur a new avenue of research into developing +partitioning strategies for distributed computing. In this section, we outline +potentially interesting areas of further work. +Exceeding completion time expectations. In this work, we rewarded +PAC-ML with +1 for completing a job within the user-defined maximum +acceptable completion time and −1 for failing to do so. Although minimising +the blocking rate is crucial for users, it would also be desirable to minimise +the JCT as much as possible. An interesting area of further study would +therefore be to incorporate this objective into the reward function, perhaps by +combining the JCT speed-up factor or offered throughput with the blocking +rate via multi-objective RL (Hayes et al., 2022). +Real-world experiments. Our work has considered real open-access +deep learning computation graph profiles but on a simulated optical architec- +ture. A natural but significant next step would be to implement PAC-ML in +a real distributed cluster. An important question would be whether an agent +trained in a simulated environment would be capable of inferring in a real +cluster at test time, or if real-world training would be needed. +Generalisation to unseen environments. This study ran PAC-ML in +an environment which had the same load rate, β distribution, cluster network +size, and job computation graphs at train and test time. An interesting +research question would be whether PAC-ML would be able to learn on one +set (or a distribution) of these parameters and then generalise to a new set +at test time, or if it would need to leverage existing or new state-of-the-art +24 + +methods in GNN (Knyazev et al., 2019; Garg et al., 2020; Fan et al., 2021) and +RL (Cobbe et al., 2019; Wang et al., 2020; Kirk et al., 2021) generalisation. +Robustness to stochastic inter-arrival times. In our experiments, +we fixed the inter-arrival rate in order to fix the load rate. However, real +clusters have variable inter-arrival times (Parsonson et al., 2022a). Handling +highly stochastic environments is a known challenge for RL (Mao et al., 2019), +and therefore presents an interesting future research avenue for PAC-ML. +Combining the virtual plane. In our work, we have considered the +job partitioning task in isolation of the job placement and scheduling tasks. +However, prior works have found the merging of the latter sub-tasks into a +single resource management problem beneficial to performance (Paliwal et al., +2020). An interesting area of further work would be to combine PAC-ML +into a a single algorithm which handled job partitioning, placement, and +scheduling via methods such hierarchical RL (Barto and Mahadevan, 2003; +Vezhnevets et al., 2017; Mirhoseini et al., 2018a; Paliwal et al., 2020; Zhang +et al., 2021) or multi-agent RL (Foerster, 2018). +9. Appendix +9.1. Extended Background +9.1.1. Parallelisation +There are three main types of deep learning parallelism; data parallelism, +model parallelism, and hybrid parallelism. +Data parallelism. +Data parallelism (Slotnick et al., 1962) is where +an identical copy of the DNN model is sent to each worker. +The input +training data is parallelised by sampling a training batch, splitting it into +non-overlapping micro-batches, training each worker on its own micro-batch, +and updating the workers’ local model parameters using some method to +synchronise the gradients of the parameters with respect to the training loss +after each training iteration. This synchronisation step is commonly referred to +as AllReduce, and can be performed using various techniques. Data parallelism +can be applied to any DNN model regardless of its architecture, enables the use +of large data sets (which are crucial for scaling model performance (Hoffmann +et al., 2022)), and facilitates the use of large training batch sizes which can +lead to smoother and faster convergence. This is a form of weak scaling, where +the JCT is decreased by reducing the total number of training iterations +needed via increasing the amount of data processed per iteration as the +number of workers is increased (Khani et al., 2021). However, it scales poorly +25 + +for large models with many parameters since all parameters must fit onto a +single worker and then be synchronised at the end of each training step, and +has the constraint that the training data must be i.i.d. in order for parameter +updates to be computed and summed across workers to attain the updated +model parameters. +Model parallelism. Model parallelism (Karakus et al., 2021) is where +the DNN model is partitioned (split) and a part of the model is sent to each +worker. In the DNN forward pass, a training batch is sampled, copied, and +sent to each worker which holds layer-1 of the DNN. The layer-1 worker(s) +then compute the layer-1 output(s) and forward them to the worker(s) which +hold layer-2, and so on. In the backward pass, the gradients of the model +parameters with respect to the training loss are computed by starting at the +worker(s) which hold the final layer and propagating these gradients back +to the layer-1 workers, after which the partitioned model will be globally +synchronised. Layer outputs, gradients, and activations are exchanged during +the training iteration using a synchronisation step commonly referred to as +AllGather. Model parallelism facilitates the use of very large models which +otherwise would not fit onto a single worker and caters for time-efficient +parallelisation of computational operations where possible. This is a form of +strong scaling, where the JCT and per-worker memory utilisation are decreased +via increasingly partitioning different parts the job across more workers as +the number of workers is increased (Khani et al., 2021). However, passing +gradients between workers during training can create a large communication +overhead (Mirhoseini et al., 2017, 2018b), and expert domain knowledge of +the specific model architecture is needed to know how to split the model +across multiple workers. +Hybrid parallelism. Hybrid parallelism (Dean et al., 2012) is where a +combination of data and model parallelism is used to strive for the benefits +of both. This can be extended to include pipeline parallelism (Huang et al., +2019; Narayanan et al., 2019), where intra-batch parallelism (data and model +parallelism) are combined with inter-batch parallelism (pipelining) where +multiple micro-batches are processed simultaneously where possible. Hybrid +parallelism can result in higher worker utilisation and the advantages of both +model and data parallelism, but requires complex bidirectional pipelining +across different inputs, careful model parameter versioning to ensure correct +computations of the gradients during the backward pass, and each stage +allocated across workers must be load-balanced to ensure roughly equivalent +computational times between workers in order to maximise peak pipeline +26 + +throughput. +9.1.2. Neural Networks as Function Approximators +Neural networks. Neural networks are a composition of linear and +non-linear (activation) functions connected in a chain to form a DAG. Each +function in the chain is a layer parameterised by a set of weights and biases +which, given enough parameters, can be trained to approximate any universal +function (Hornik et al., 1989; Montufar et al., 2014). Neural networks with +multiple intermediary (hidden) layers between input and output are referred +to as deep neural networks and have powerful expressivity capabilities when +approximating complex non-linear functions (Hornik et al., 1989; Montufar +et al., 2014). +Graph neural networks. Whereas standard DNNs are restricted to +handling only vector- and grid-structured inputs (e.g. sentences, images, etc.), +GNNs are generalised DNN architectures which can handle graph-structured +data as inputs (e.g. job DAGs). Most current GNNs use the message passing +paradigm by mapping each node and edge onto a vector embedding space +before performing additional graph-level embeddings and readouts if desired. +Specifically, each GNN layer usually performs four stages: (i) On each +edge in the input graph, use a message function to generate a message +(representation) to pass from a source node to a set of destination nodes, +where each node stores the message(s) it receives in its mailbox; (ii) on +each node in the input graph, apply an aggregate function (a vanilla reduce +operation such as mean, sum, max, min, etc., or a trainable function) to the +messages in its mailbox to generate an intermediate aggregate representation +of its neighbourhood; (iii) pass the intermediate aggregate representation +through a trainable function to produce a final vector embedding for each +node; and (optional) (iv) if desired, at the end of the final GNN layer, pass +the node embeddings through a trainable function to produce a graph-level +representation. Crucially, the parameters of all message, aggregation, and +forward pass functions are shared across nodes, enabling GNNs to be inductive +in that they can generalise to unseen nodes and graphs. +9.1.3. Reinforcement Learning Algorithm +Here we break down the key background components of the RL approach +used for PAC-ML. +Q-learning. Q-learning (Watkins, 1989) is the canonical value-based +algorithm which can be applied to a sequential decision making process +27 + +formalised as an MDP. It is an off-policy temporal difference algorithm +whose goal is to learn the action value function mapping state-action pairs to +their expected discounted future return when following a policy π; Qπ(s, u) = +Eπ +� �∞ +t′=t+1 γt′−1r(st′)|st=s, ut=u +� +. By definition, an optimal policy π∗ will se- +lect an action which maximises the true Q-value Q∗(s, u), π∗(s) = arg maxu′ Q∗(s, u′). +Concretely, the classical Q-learning algorithm maintains an action value +look-up table Q(s, u) mapping all possible state-action pairs to their predicted +discounted return. The return is the sum of future rewards over the remainder +of the episode. During training, Q-learning follows an exploration-exploitation +policy. The simplest such policy is ϵ-greedy, where a random action is sampled +with probability ϵ ∈ [0, 1] and the best action, according to the current Q +table, is sampled with probably 1 − ϵ. At each time step t, the agent in state +st uses this policy to select an action ut which it performs in the environment +to transition to the next state st+1 and receive a reward rt+1. Q(s, u) is then +updated according to: +Q(st, ut) ← Q(st, ut) + α · +� +rt + γ · max +u′ +Q(st+1, u′) − Q(st, ut) +� +. +(2) +On the right-hand side of Eq. 2, Q(st, ut) is the agent’s estimate of the +discounted return of taking action ut in state st, α is the learning rate, γ is +the factor by which to discount future rewards to their present value, and +maxu′ Q(st+1, u′) is an estimate of the future value of being in state st+1 and +taking an ‘optimal’ action according to Q. The rt + γ · maxu′ Q(st+1, u′) +term is called the temporal difference target, and and the collective rt + γ · +maxu′ Q(st+1, u′) − Q(st, at) term the temporal difference error. As such, the +maxu′ Q(st+1, u′) term treats Q as an oracle from which optimal actions can +be sampled. Although Q is usually randomly initialised and changes at each +update step, the general idea is that, with stable learning and sufficient +exploration, Q will converge on the true Q∗ function. +As a side note, Q-learning is a temporal difference algorithm because, +rather than using the actual returns to update Q in Eq. 2 as done my +Monte Carlo methods, it uses a bootstrapped estimate of the future returns +maxu′ Q(st+1, u′). Furthermore, it is an off-policy algorithm because the +policy used to select the action ut at the current time step, such as ϵ-greedy +sampling of Q, is different to the policy used to select the next-state action +u′ when evaluating the temporal difference target, such as greedy sampling of +Q. This is as opposed to on-policy temporal difference algorithms, such as +28 + +SARSA, which use the same action selection policy for both the current time +step and for future time steps when bootstrapping. +Deep Q-learning. Many practical problems have an extremely large +number of possible state-action combinations. For example, the game of Go +has over 10700 possible sequences; far more than the number of atoms in the +universe (Silver et al., 2016). As such, modelling the action value function +with a tabular approach is intractable given practical memory constraints. +To enable Q-learning to be scaled to complex problems, deep Q-learning +(DQN) (Mnih et al., 2013) approximates the true Q-function with a DNN +parameterised by θ such that Qθ(s, u) ≈ Q∗(s, u). +Concretely, during training at each time step t, Qθ(s, u) is used with +an exploration strategy such as ϵ-greedy to select an action and add the +observed transition T = (st, ut, rt+1, γt+1, st+1) to a replay memory buffer +(Lin, 1992). The network’s parameters θ are then optimised with stochastic +gradient descent to minimise the mean squared error loss between the online +network’s predictions and a bootstrapped estimate of the Q-value, +JDQN(Q) = +� +rt+1 + γt+1 max +u′ +Q¯θ(st+1, u′) − Qθ(st, ut) +�2, +(3) +where t is a time step uniform randomly sampled from the buffer and +Q¯θ a target network with parameters ¯θ which are periodically copied from +the acting online network. The target network is not directly optimised, but +is used to provide the bootstrapped Q-value estimates for the loss function. +Only periodically updating the target network rather than at each learning +step leads to lower variance in the bootstrapped targets at each step. This +helps helps to stabilise learning and leads to better convergence (Mnih et al., +2013). +Double DQN. In the traditional Q-learning update rule of Eq. 2 and +the DQN loss of Eq. 3, the Q-function used to select and evaluate an action +for the temporal difference target is the same; maxu′ Q(st+1, u′) for Eq. 2, +and maxu′ Q¯θ(st+1, u′) for Eq. 3. However, this can lead to an overestimation +bias where the chosen action u′ is incorrectly over-valued because the same +function which perceives u′ as being best is also being asked to evaluate it. +This can lead to high variance updates, unstable learning, and convergence on +local minima. Double DQN (van Hasselt et al., 2015) reduces overestimation +by decomposing the max operation in the temporal difference target into +action selection and action evaluation and performing these two tasks with +two separate networks. +29 + +Concretely, action u′ is greedily selected according to the online network +Qθ and evaluated with the separate target network Q¯θ. The loss term from +Eq. 3 then becomes: +JDDQN(Q) = +� +rt+1 + γt+1Q¯θ(st+1, max +u′ +Qθ(st+1, u′)) − Qθ(st, ut) +�2. +(4) +Prioritised experience replay. Vanilla DQN replay buffers are sampled +uniformly to obtain transitions for network updates. A preferable approach +is to more frequently sample transitions from which there is much to learn. +Prioritised experience replay (Schaul et al., 2016) deploys this intuition by +sampling transitions with probability pt proportional to the last encountered +absolute temporal difference error, +pt ∝ |rt+1 + γt+1 max +u′ +Q¯θ(st+1, u′) − Qθ(st, ut)|ω, +(5) +where ω is a tuneable hyperparameter for shaping the probability distribu- +tion. New transitions are added to the replay buffer with maximum priority +to ensure all experiences will be sampled at least once to have their errors +evaluated. +n-step Q-learning. Traditional Q-learning uses the target network’s +greedy action at the next step to bootstrap a Q-value estimate for the temporal +difference target. Alternatively, to improve learning speeds and help with +convergence (Sutton and Barto, 2018; Hessel et al., 2017), forward-view +multi-step targets can be used (Sutton and Barto, 2018), where the n-step +discounted return from state s is +r(n) +t += +n−1 +� +k=0 +γ(k) +t rt+k+1, +(6) +resulting in an n-step DQN loss of +JDQNn(Q) = +� +r(n) +t ++ γ(n) +t +max +u′ +Q¯θ(st+n, u′) − Qθ(st, ut) +�2. +(7) +Dueling DQN. Traditional DQN approaches use a DNN architecture +which is not specific to RL. Subsequently, when learning the Q-function, the +entire DNN architecture must learn to estimate the state value and the action +advantage for each action in order to learn the state-action function Qπ(s, u) +of being in state s, taking action u, and following policy π. However, in +30 + +many problems where bootstrapped Q-learning is applied, the most important +objective is to learn to estimate the value of each state rather than the effect +of each action for each state. This is especially true in environments and +individual states where future transitions are mainly influenced by factors +other than the agent’s actions. +Leveraging the insight that in many states it is unnecessary to estimate +the value of each action choice, Wang et al. (2015) developed a new DNN +architecture, termed ‘dueling DQN’, which is better suited to the Q-learning +task. Concretely, the dueling architecture uses the same core DNN as standard +DQN. However, rather than following the initial encoding with a single +sequence of fully connected layers to get a Q-value for each possible action in +the current state, dueling DQN uses two separate streams of fully connected +layers. One stream, parameterised by β, estimates the state value function +Vθ,β(s) (the estimated future discounted return of the current state regardless +of future actions taken), and the other stream, parameterised by α, estimates +the relative action advantage function Aθ,α(s, u) (the relative difference in the +future discounted return of each action). +The outputs of the two streams are then combined via a special aggrega- +tion function to recover the state-action value function Q. Crucially, V (s) and +A(s, u) must be combined into Q(s, u) in such a way that they are indepen- +dently identifiable from the output Q values alone in order for backpropagation +to be able to calculate the appropriate loss and weight updates for the sep- +arate V (s) and A(s, u) streams. As such, a simple Q(s, u) = V (s) + A(s, u) +aggregation function to get the Q-values from the two streams does not suffice. +Instead, the authors tried two different aggregation schemes. +The first aggregation method subtracted the advantage of the maximum +advantage action from all advantages to make the argmax action’s advantage +0 and the rest < 0, +Qθ,α,β = Vθ,β(s) + +� +Aθ,α(s, u) − max +u′ +Aθ,α(s, u′) +� +, +(8) +thus enabling V (s) to be recovered at the argmax action’s Q-value. +The second aggregation method subtracted the mean advantage from all +action advantages to centre the advantage values around 0 (i.e. to have a +mean of 0), +Qθ,α,β = Vθ,β(s) + +� +Aθ,α(s, u) − 1 +|A| +� +u′ +Aθ,α(s, u′) +� +. +(9) +31 + +This makes V (s) recoverable from Q(s, u) by estimating the V (s) value +which, when subtracted from each A(s, u) value, leads to a set of A(s, u) +values which have a mean of 0. In practice, this second approach of using the +mean was found to lead to more stable learning since using a mean operation +resulted in lower variance targets between learning steps compared to when a +max operation was used. +As with standard Q-learning, the output of the dueling network is a set of +Q-values (one for each action), therefore no change to the underlying algorithm +other than a slight adjustment of the network architecture was required. By +decomposing the Q-function approximator in this way, dueling DQN is able to +attain superior policy evaluation in the presence of many similar-value actions, +and the authors demonstrated their architecture achieving state-of-the-art +performance on the Atari 2600 games. +Ape-X DQN. Noting that state-of-the-art ML performance is often +achieved with more computation, more powerful models, and larger training +data sets, Horgan et al. (2018) proposed Ape-X; a parallelisation approach +to off-policy experience replay RL. Concretely, rather than using a single +actor-learner setup, Ape-X decouples acting from learning. It distributes +many actors across a set of CPU cores each with their own instance of the +environment. Each actor retains a copy of a DNN shared across actors which it +uses for action selection to accumulate experiences in parallel with other actors. +These experiences are then communicated to a central shared replay buffer, +where a single learner mounted on a GPU uses prioritised experience replay +to sample the most important experiences for learning. Learner sampling, +gradient computation, and network updates are done asynchronously with one +another on separate threads, as are the periodic updates made to the actors’ +networks with the latest shared learner network. By using multiple actors in +parallel, not only can orders of magnitude more transition data be attained +for learning, but also a broader diversity of experiences can be collected by +allocating a different exploration strategy to each actor and thereby avoid +local optima in difficult exploration and large state-action space settings. +For Nactors distributed actors, Horgan et al. (2018) used a per-actor ϵ-greedy +exploration strategy whereby each actor i had a fixed exploration probability +ϵi = ϵ1+ +i +Nactors−1 ·α where ϵ = 0.4 and α = 0.7. The authors demonstrated their +approach achieving new state-of-the-art results on Atari in a fraction of the +training time of prior works. +32 + +9.2. Metric Definitions +Table 2 summarises the metric jargon used throughout our manuscript. +Metric +Description +Job completion time +Time between job arriving and being +completed. +Sequential job completion time +Time it would take to complete a job +were its operations ran sequentially +on a single device. +Maximum acceptable job completion time +Maximum time allowed to complete +a job. +Speed-up factor +Factor difference between sequential +job completion time and actual job +completion time. +Network overhead +Fraction of the job completion time +spent communicating information be- +tween workers when no computation +was taking place. +Blocking rate +Fraction of the arrived jobs which +were successfully serviced across a +given period of time. +Job information size +Summed sizes (in bytes) of a job’s +operations and dependencies. +Cluster throughput +Total partitioned job information pro- +cessed per unit time by the cluster. +Offered throughput +Total original job information pro- +cessed per unit time by the cluster. +Load rate +Amount of job information arriving +at the cluster per unit time. +Job inter-arrival time +Time between when two jobs arrived +at the cluster. +Table 2: Descriptions of the various metrics referred to throughout this manuscript. +9.3. Experimental Hardware +All environment simulations were ran on Intel Xeon ES-2660 CPUs, and +all learner network training and inference was done on either a V100 or an +A100 GPU. +33 + +9.4. Additional Simulation Details +9.4.1. Code Structure +We built a core RAMP simulation environment which followed a Gym- +like interface (Brockman et al., 2016) but without inheriting from a Gym +environment object to allow additional flexibility. We then built a wrapper +‘job partitioning’ environment which did conform to the Gym interface but +used our core RAMP simulation environment to perform the internal RAMP +simulation logic. Our code base is publicly available at https://github. +com/cwfparsonson/ddls for further practical implementation details. +9.4.2. Job Allocation Procedure +When a job arrives at the cluster, our environment uses the following +ordered sequence of task executions to allocate the job: +1. Op. partitioning: Partition the job DAG’s operations to attain a +‘partitioned’ job DAG. +2. Op. placement: Place the operations in the partitioned job DAG +onto a sub-set of cluster workers. +3. Op. scheduling: For each worker, schedule the priority of its placed +operations to resolve conflicts where ≥ 2 operations are ready to be +executed at the same time. +4. Dep. +placement: Given the placed operations and the data de- +pendencies which must be exchanged between operations, place the +dependencies onto cluster communication links. +5. Dep. scheduling: For each communication link, schedule the priority +of its placed dependencies to resolve conflicts where ≥ 2 dependencies +are ready to be communicated at the same time. +9.4.3. Job Allocation Methods +Each of the above allocation procedure tasks can be performed by any +algorithm, heuristic, or learning agent. In our work, we use the following +methods: +1. Op. partitioning: PAC-ML, Paramax, Paramin, or Random. See the +main manuscript for details. +2. Op. placement: A first-fit heuristic customised for the requirements +of RAMP. See Section 9.4.4 below for details. +34 + +3. Op. scheduling: Shortest remaining processing time (Cai et al., 2016; +Alizadeh et al., 2013; Hong et al., 2012). Given a set of operations +placed on a worker, the operation with the shortest remaining run time +will have the highest priority and therefore be executed first wherever +two operations on the same worker request to be executed at the same +time. +4. Dep. placement: Shortest path & first-fit. Given a set of operation +placements, for any dependencies which need to be transferred through +the network (i.e. for dependencies with size > 0 and whose parent +operation is placed on a separate worker from the child operation), +(1) first-fit select a path from the k−shortest path with available light +channel(s), and (2) first-fit select an available channel. +5. Dep. scheduling: Shortest remaining processing time. Given a set of +dependencies placed on a communication link channel, the dependency +with the shortest remaining processing time (i.e. the lowest amount of +information left to be transferred) will have the highest priority and +therefore be communicated first wherever two dependencies on the same +link channel request to be transported at the same time. +9.4.4. First-Fit Operation Placement in RAMP +The original RAMP paper of Ottino et al. (2022) did not specify an +operation placement heuristic which conformed to the RAMP placement rules +(see Section 3). Here, we propose a simple first-fit heuristic which conforms +to these rules whilst making the placement problem tractable for large cluster +networks. +The basic idea behind partitioning and placement in the scenario described +in this work is to exploit the network efficiencies of RAMP as much as possible. +In particular, this means maximising the use of RAMP’s highly efficient +collective operations. For a generic partitioned DAG, in the backward pass, +collectives happen for each operation when weights/gradients are shared +between sub-operations. If both a parent and child operation are placed on +the same set of (RAMP symmetry adherent) workers, then when the parent +communicates its output to the child’s input in the forward pass this will also +constitute a collective operation. As such the placement heuristic implemented +here seeks to primarily maximise the amount that these two conditions are +encountered. Given some operation, o, that has been partitioned into N equal +sub-operations, oi and needs to be placed, the placement is handled as: +35 + +1. If a parent of o has been partitioned and placed across N servers which +adhere to the RAMP symmetry conditions, and if these servers each +have enough memory to store oi, then place o across this set of N servers. +This ensures collective operations can happen in both the forward and +backward pass. +2. Otherwise, check if a set of N workers can be found in the network +that adheres to the RAMP symmetry requirements. This is achieved by +sliding the various possible symmetric shapes over the topology until a +suitable one (or none) is found. This ensures collective operations in +the backward pass only. +Allocating in this way ensures that every partitioned operation can exploit +RAMP’s efficient collective operation process on the backward pass, and where +possible can also exploit it on the forward pass when receiving information +from (one of) its parents. +9.4.5. Evaluating the job completion time +The time to complete each operation was taken from the real computation +job profiles of the DNN jobs considered (see Section 9.5). To calculate the +communication time of point-to-point information transfers and of the MPI +collectives, we used the equations and code of Ottino et al. (2022). +9.4.6. Possible Causes of a Job Being Blocked +A job is blocked when either JCT > β · JCTseq (i.e. failing to meet user’s +chosen JCT requirement) or when the cluster does not have enough available +resources to service the job. The possible causes of this latter form of blocking +are: +• Prior jobs using up too many cluster resources when later jobs arrive; +• the minimum operation run time quantum not being low enough to +partition the operations enough times to lead to the desired JCT; +• mounted worker operation scheduling conflicts for partitioned operations +mounted on the same worker leading to longer run times, since one +worker can only execute one operation at a time; and +• excessive communication overheads incurring from over-partitioning of +the job. +36 + +Figure 9: Visualisation of the characteristics of the deep learning computation graphs used +for our experiments before partitioning. The bottom left sub-figure contains the model +colour code scheme for all other sub-figures. The statistics shown are for the operations +and dependencies which need to be executed and satisfied to conduct one training iteration. +Therefore, to carry out Niter training steps, the computation graph would need to be +executed Niter times. Computation time units are reported in seconds, and memory units +in bytes. +9.5. Job Computation Graph Data Sets +All computation graphs used in our experiments were taken from the +open-access PipeDream computation graph data set (Narayanan et al., 2019). +Figure 9 shows a visualisation of the key computation graph characteristics +for each neural network model considered, where the numbers reported are +for one training iteration (i.e. one forward and backward pass through the +model). Table 3 reports the same characteristics but in tabular form. Finally, +for completeness, Figure 10 shows the actual job DAGs of the models used. +9.6. Neural Network Architecture +As shown in Fig. 11, we used a message passing GNN similar to Graph- +SAGE with mean pooling (Hamilton et al., 2017) to parameterise the PAC-ML +policy. Table 4 summarises the hyperparameters used for the components of +this DNN. We note that we did not perform extensive hyperparamter tuning +on the GNN architecture. Below is a detailed explanation of this architecture. +GNN. First, the GNN layer takes in the DAG’s node and edge features +and generates an embedding for each node and edge in the graph. Then, +each local node’s nearest neighbour (1-hop away) sends the local node a +37 + +×104 +×109 +600 - +125 - +400 - +1.0 - +iido +75 +2 - +Max. +50 - +200 - +25 - +70 +×1010 +×1010 +×109 +3 +1.5- +150 - + Size ( + azis +# +α +≥ 0.0 +1.0→ +FO'T +Fo +1 +80 - +Graph +0.8 - +60 - +Computation +40 - + 0.4 +0.4 - +20 - +0.4 +E +ELL +ELL +0.2 - +100 +108 +109 +101 +102 +108 +109 +Operation Compute Time +Operation Memory +Dependency MemoryModel +# ops. +JCTseq +Max. op. comp. time +Σ op. mem. +Max. op. mem. +Depth +# deps. +Σ dep. size +Max. dep. size +ResNet-18 +142 +36 668.35 +473.625 +17.258 66 × 109 +0.822 121 2 × 109 +60 +159 +18.733 29 × 109 +0.822 083 6 × 109 +VGG-16 +82 +34 525.35 +113.330 +30.625 30 × 109 +1.644 315 × 109 +80 +83 +29.467 06 × 109 +1.644 167 × 109 +GNMT +96 +4470.80 +15.88 +2.368 447 × 109 +3.269 491 × 108 +30 +117 +1.027 801 × 109 +0.194 437 1 × 109 +SqueezeNet-10 +136 +38 000.15 +474.637 +24.962 62 × 109 +1.168 007 × 109 +102 +153 +27.910 09 × 109 +1.167 950 × 109 +AlexNet +46 +36 061.15 +635.902 +3.046 234 × 109 +0.198 339 6 × 109 +44 +47 +2.422 161 × 109 +0.198 246 4 × 109 +Table 3: Summary of the characteristics of the deep learning computation graphs used for our experiments before partitioning. +The statistics shown are for the operations (‘ops.’) and dependencies (‘deps.’) which need to be executed and satisfied to +conduct one training iteration. Therefore, to carry out Niter training steps, the computation graph would need to be executed +Niter times. Computation (‘comp.’) time units are reported in seconds, and memory (‘mem.’) units in bytes. +38 + +Figure 10: Deep learning computation graphs used for our experiments before partitioning. +Each computation graph represents the operations and dependencies which need to be +executed and satisfied to conduct one forward and one backward pass through the neural +network. Therefore, to carry out Niter training steps, the computation graph would need +to be executed Niter times. +39 + +ResNet-18 +VGG-16 +GNMT +SqueezeNet-10 +AlexNet +国国国Figure 11: Schematic of the DNN architecture with |L| GNN layers used to parameterise +the policy of PAC-ML. The GNN is similar to that of GraphSAGE with mean pooling +(Hamilton et al., 2017). Each GNN layer l ∈ L contains a node, edge, and reduce DNN +module and ultimately learns to create an embedded representation for each node in a +given job DAG. These per-node embeddings are then passed, along with any global job, +cluster, and action features, to a readout module. The readout module ultimately generates +scores for each possible action, which enables an action to be selected following a given +exploration-exploitation policy being followed. For clarity, this figure only shows the GNN +embedding-generation process for node 1. See accompanying text for a detailed explanation +of this architecture and the accompanying figure. +40 + +(1, 3) +Mean +pooling +Mean +pooling +[1,2,3 +4.57 +RLlib FC +Chosen +action +Job +Clustel +Action +Node +Eade +Labelmessage (‘message passing’) which is the neighbouring nodes’ embeddings +concatenated with their connected edges’ embeddings. These messages are +stored in the local node’s ‘mailbox’, which now contains information about the +node’s neighbourhood. To ensure consistent dimensioning with the received +messages, a dummy zero-padded edge embedding is concatenated with the +local node’s embedding. Next, the reduce module takes the local and message +embeddings and generates a reduced representation for each. Finally, to +generate a layer-l output embedding for the local node, the element-wise +mean of the reduced embeddings is taken (‘mean pooling’). Note that this +embedding process is done for each node in the DAG, but for clairty Fig. 11 +only follows node 1. +If l < L (i.e. if this is not the last GNN layer), these final node embeddings +are used as new features for the original DAG’s nodes and are passed to +the next GNN layer. If l ≡ L, then the node embeddings are passed to +the readout module. Note that (1) the node, edge, and reduce modules are +shared across the aforementioned operations within a given GNN layer when +generating node embeddings, but not across different GNN layers, and (2) +the lth-layer’s output node embeddings will contain information about the +node’s neighbourhood from up to l hops away. +Readout. The readout module takes the GNN’s node embeddings and +the job’s and cluster’s global features as input. To convert the node-level +embeddings of the GNN into a representation of the overall job DAG, their +element-wise mean is taken. To generate an embedding capturing the global +job, cluster, and action information, a global DNN module is used. The +DAG and global embeddings are then concatenated and passed to a logit +module, which in turn generates a vector of (optionally masked) scores for +each possible action in the environment. Finally, based on these scores and +the exploration-exploitation policy being followed, an action is selected. +9.7. Reinforcement Learning Algorithm +Approach. Given the stochastic nature of our dynamic cluster environ- +ment setting, we hypothesised that a value-based RL method would be best +suited to our setting (Mao et al., 2019). We did try the PPO (Schulman et al., +2017) actor-critic method but found performance to be worse, although we +leave a full analysis of alternative RL algorithms to future work. +As stated in the main manuscript, we used the state-of-the-art value-based +Ape-X DQN RL algorithm (Horgan et al., 2018) to attain the PAC-ML +policy. Concretely, we used the Ape-X parallelisation approach with double +41 + +Parameter +Value +Message passing # hidden dimensions +64 +Message passing # output dimensions +32 +Reduce module # hidden dimensions +64 +Reduce module # output dimensions +64 if l < L, else 16 +Global module # hidden dimensions +8 +Global module # output dimensions +8 +Logit module RLlib FC net # layers +1 +Logit module RLlib FC net # hidden dimensions +256 +All modules’ activation +ReLU +GNN # layers L +2 +Apply action mask +False +Table 4: Hyperparamters used for the PAC-ML ApeX-DQN DNN policy architecture +shown in Fig. 11. Note that the ‘message passing’ dimensions refer to the dimensions of +the concatenated node and edge modules’ embeddings, so the dimensions of these modules’ +hidden and output embeddings will be half the corresponding ‘message passing’ dimension. +Due to the RLlib implementation of Ape-X DQN, we did not apply an action mask, but +instead included the action mask in the global features given to the model and used the +reward signal to train the agent to avoid selecting invalid actions. +Q-learning action selection-evaluation (van Hasselt et al., 2015) and multi- +step bootstrapped learning targets (Sutton and Barto, 2018; Hessel et al., +2017), prioritised experience replay (Schaul et al., 2016), a dueling DQN +network architecture (Wang et al., 2015), and a per-actor ϵ-greedy exploration +algorithm. For a breakdown of each of these components, refer to Appendix +9.1.3. +Hyperparameters. To select the algorithm hyperparameters, we con- +ducted a Bayesian search across the search space summarised in Table 5, +with simulations conducted in a light 32-worker RAMP environment with a +maximum simulation run time of 2 × 105 seconds to speed up the search. We +adopted similar search ranges to those used by Kurach et al. (2019); Hoffman +et al. (2020); Parsonson et al. (2022b). For each set of hyperparameters, we +ran the algorithm for 100 learner steps (a.k.a. training epochs), and performed +a validation across 3 seeds at each learner step (see Figure 12). We selected +the parameter set with the highest episode return across the 3 seeds (see +Table 5). We also report the importance of each parameter with respect to +the total episode return. The importance is calculated by training a random +forest with all algorithm hyperparameters as inputs and the episode return as +the target output, with the per-feature (hyperparameter) importance values +42 + +Figure 12: Validation performance of the Ape-X DQN hyperparameter sweep. Each agent +was trained for 100 learner steps, and at each learner step a validation was performed +across 3 seeds - the mean metrics with their min-max interval bands are plotted for each +hyperparameter set. +predicted by random forest reported accordingly (fab, 2018; how, 2018). All +our experiments used the same per-actor ϵ-greedy exploration as Horgan et al. +(2018). +We note that our RL algorithms were implemented using the open-source +RLlib library (Liang et al., 2018) and hyperparameter tuning was done using +Weights & Biases (Biewald, 2020). +9.7.1. Final Learning Curves +For completeness, Figure 13 shows the learning curves of the tuned PAC- +ML agents in each βX environment superimposed on the baseline agents’ +performances. At each learner step, the PAC-ML agent was evaluated across +three seeds in the validation environment. +9.8. Additional Experimental Results +Figure 14 shows the performance of the agents in terms of raw blocking +rate, throughput, JCT, and JCT speed-up. +10. Funding and Acknowledgments +Funding +EPSRC Distributed Quantum Computing and Applications EP/W032643/1; +the Innovate UK Project on Quantum Data Centres and the Future 10004793; +OptoCloud EP/T026081/1; TRANSNET EP/R035342/1; the Engineering +and Physical Sciences Research Council EP/R041792/1 and EP/L015455/1; +the Alan Turing Institute; and Horizon Europe Dynamos. +43 + +0.9 +Return +75 +Rate +slocking +0.8 +100 +isode +-125 +B +0.7 +-150 +T +T +20 +40 +60 +80 +100 +20 +40 +60 +80 +100 +Learner Steps +Learner StepsParameter +Search Range +Best Value +Importance +Discount factor γ +{0.99, 0.993, 0.997, 0.999, 0.9999} +0.999 +0.004 +Learning rate +Log-uniform values ( 1 × 10−7, 1 × 10−3 ) +4.121 × 10−7 +0.045 +vmin +{−1, −10, −100, −200, −1000} +−1000 +0.01 +vmax +{1, 10, 100, 200, 1000} +1000 +0.004 +Target network update frequency +{ 1 × 103, 1 × 104, 1 × 105 } +1 × 105 +0.001 +Prioritised replay α +{0.1, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} +0.9 +0.04 +Prioritised replay β +{0.1, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9} +0.1 +0.047 +n-step +{1, 3, 5, 10} +3 +0.227 +# CPU workers +32 +32 +− +# GPU workers +1 +1 +− +Batch mode +Truncated episodes +Truncated episodes +− +Rollout length +50 +50 +− +Train batch size +512 +512 +− +Optimiser +Adam +Adam +− +Dueling +True +True +− +# atoms +1 +1 +− +Noisy +False +False +− +Double Q +True +True +− +Replay buffer capacity +100 000 +100 000 +− +Learning starts +10 000 +10 000 +− +Prioritised replay TD-error ϵ +1 × 10−6 +1 × 10−6 +− +Table 5: Ape-X DQN training parameter sweep search range, best value found, and corresponding parameter importance. +44 + +Figure 13: Validation curves of the PAC-ML agent trained in four different β distribution +environments. At each learner step (update to the GNN), the agent was evaluated across 3 +seeds, with the mean blocking rate, offered throughput, JCT, and JCT speed-up (relative +to the jobs’ sequential run time JCTseq) performance metrics reported as well as their +min-max confidence intervals. For reference, the performances of the baseline heuristic +partitioners are also plotted. +45 + +PAC-ML (Ours) +Paramac +Paramin +Random +βA +βB +βc +βD +Slocking +1.0 +1.0 +1.0 +Rate +0.75 +0.5 +0.5 +0.5 +0.50 +B +0.25 +X107 +X107 +X107 +X107 +2.5 +2.5 +2.5 +S +B +0.0 +0.0 +0.0 +0.0 +X104 +X104 +X104 +X104 +1.5 +1.0 +W +1.5 +1.0 +JCT +1.0 +0.5 +0.5 +0.5 +0.5 - +15 +15 +15 +JCT +10 +10 +10 +10 +5 +5 +5 +5 +S +0 +100 +200 +0 +100 +200 +0 +100 +200 +0 +100 +200 +Learner Steps +Learner Steps +Learner Steps +Learner StepsFigure 14: Validation performances of each partitioning agent evaluated across three seeds, +with the mean blocking rate, offered throughput, JCT, and JCT speed-up (relative to the +jobs’ sequential run time JCTseq) performance metrics reported. +References +, 2018. Intro to machine learning: Lesson 4. URL: https://www.youtube. +com/watch?v=0v93qHDqq_g. +, 2018. Introduction to hyperparameters. 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URL: https://openreview.net/forum?id=r-gPPHEjpmw. +58 + diff --git a/9tFST4oBgHgl3EQfbTje/content/tmp_files/load_file.txt b/9tFST4oBgHgl3EQfbTje/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..920e81cf107024f549ae3c225f810a3940e04ffa --- /dev/null +++ b/9tFST4oBgHgl3EQfbTje/content/tmp_files/load_file.txt @@ -0,0 +1,2568 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf,len=2567 +page_content='Highlights Partitioning Distributed Compute Jobs with Reinforcement Learn- ing and Graph Neural Networks Christopher W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson, Zacharaya Shabka, Alessandro Ottino, Georgios Zervas Demonstrate that deciding how much to partition distributed jobs is a key factor in determining overall system throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Demonstrate that optimising for only the job completion time leads to high blocking rates and poor throughput in dynamic job arrival scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Introduce a new partitioning algorithm which leverages reinforcement learning, a graph neural network, and a novel formulation of the user- defined job completion time specification to automatically learn to partition jobs such that the blocking rate is minimised and user require- ments are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Demonstrate the proposed algorithm out-performing baselines on a state- of-the-art optical network architecture running five real deep learning computation graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='13799v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='LG] 31 Jan 2023 Partitioning Distributed Compute Jobs with Reinforcement Learning and Graph Neural Networks Christopher W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson1,∗, Zacharaya Shabka1, Alessandro Ottino1, Georgios Zervas1 ∗Corresponding author: zciccwf@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='uk 1UCL Partitioning Distributed Compute Jobs with Reinforcement Learning and Graph Neural Networks Christopher W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson1,∗, Zacharaya Shabka1, Alessandro Ottino1, Georgios Zervas1 Abstract From natural language processing to genome sequencing, large-scale ma- chine learning models are bringing advances to a broad range of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Many of these models are too large to be trained on a single machine, and instead must be distributed across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This has motivated the research of new compute and network systems capable of handling such tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In particular, recent work has focused on developing management schemes which decide how to allocate distributed resources such that some overall objective, such as minimising the job completion time (JCT), is optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, such studies omit explicit consideration of how much a job should be distributed, usually assuming that maximum distribution is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In this work, we show that maximum parallelisation is sub-optimal in relation to user-critical metrics such as throughput and blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To address this, we propose PAC-ML (partitioning for asynchronous computing with machine learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' PAC-ML leverages a graph neural network and reinforcement learning to learn how much to partition computation graphs such that the number of jobs which meet arbitrary user-defined JCT requirements is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In experiments with five real deep learning computation graphs on a recently proposed optical architecture across four user-defined JCT requirement distri- butions, we demonstrate PAC-ML achieving up to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2% lower blocking rates in dynamic job arrival settings than the canonical maximum parallelisation strategy used by most prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Keywords: Deep Learning, Reinforcement Learning, Graph Neural Networks, ∗Corresponding author: zciccwf@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='uk 1UCL Preprint submitted to Journal of Parallel and Distributed Computing February 1, 2023 Distributed Asynchronous Computing, Job Partitioning, Optical Networks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Introduction The last decade has seen an exponential increase in the amount of compute demanded by big data jobs such as artificial intelligence (AI) and genome processing, with resource requirements doubling every 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 months since 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 50× faster than Moore’s Law (OpenAI, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This trend is showing no sign of slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The fundamental relationship between neural net- work accuracy and scale (Kaplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020) provides a strong incentive for practitioners seeking performance improvement to further increase their resource requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Moreover, brain-scale AI will require at least as many parameters as the ≈1 000 trillion synapses present in the human brain (Furber, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' several orders of magnitude more than the largest models used today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The compute time and memory requirements of state-of-the-art big data applications already far outstrip the capabilities of any single hardware device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For example, one of the current largest deep neural networks (DNNs), Megatron-Turing natural language generation (MT-NLG) (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022), contains 530 billion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These parameters alone occupy ≈1 000 GB, exceeding the capacity of the largest A100 GPU by over an order of magnitude, and the parameter loss gradients tracked during training occupy several times more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Even if the model could be fitted onto a single device, the training time would be ≈900 years2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To address these compute time and memory demands, rather than using a single device, big data jobs must be distributed and parallelised across a cluster of machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For example, the Selene supercomputing cluster (NVIDIA, 2020) consists of 358 400 A100 GPU tensor cores, bringing the MT-NLG training time from 900 years down to the order of days3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, parallelising jobs across ever-more machines brings its own challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' With any parallelisation strategy, at some point the output of each ‘worker’ (a single device processing at least part of a job) must be collected and synchronised to get the overall result of the parallelised computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This synchronisation requires communication between the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As the number 2Assuming it takes 8 V100 GPUs 36 years to train a 175 billion parameter model (NVIDIA, 2022) and extrapolating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3Assuming a linear parallelisation speedup and 0 communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2 Figure 1: How the network overhead of six distributed deep learning jobs (encompassing object tracking, recommendation, natural language processing, and image recognition) increases with the number of workers used in Meta’s GPU cluster (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' of workers used to execute a job is increased, the per-worker computation demands decrease, but the overall communication overhead between workers grows (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This shifts the performance bottleneck away from the workers themselves and into the network connecting them, and brings additional challenges with managing varying traffic characteristics for different job types and parallelisation strategies (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Benjamin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To address the communication bottleneck in distributed computing, recent works have sought to develop optical clusters (Benjamin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Ballani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' machines interconnected by optical switches (Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Gerard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Compared to their electronic counterparts, optically switched networks offer orders of magnitude improvements in scalability, bandwidth, latency, and power consumption (Ballani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Zervas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Optical clusters are typically operated under the optical circuit switched (OCS) paradigm due to its non-blocking circuit configurations with high capacity and scalability (Raja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' OCS networks are fundamentally different from the electronic packet switched (EPS) architectures used by most current clusters, resulting in entirely new communication patterns and resource demand characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, new compute and network resource management schemes are needed in order to optimally allocate jobs and maximise performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3 Job 1 Job 3 Job 5 Job 2 Job 4 Job 6 60 40: 20 - 101 102 # WorkersOf the many resource management tasks which must be performed in a compute cluster, job partitioning (how to split a job up across how many devices) is key to overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' More partitioning can lead to lower compute times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, it may also increase network overhead and occupancy of cluster resources, possibly leading to future jobs being blocked upon arrival and consequently lower overall cluster throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Prior works such as SiP- ML (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) have introduced simple partitioning heuristics for optical networks which have notably improved cluster performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, they have not been designed under the more realistic setting of dynamic and stochastic job arrivals, have not considered the state of the cluster in a ‘network-aware’ manner when making partitioning decisions, and have been crafted to optimise for the sub-optimal objective of minimising job completion time (JCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In this work, we first argue that simply minimising the JCT is a naive objective because it brazenly encourages more parallelisation of a job re- quest without considering the effect this has on the ability of a cluster to service subsequent jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We then introduce a new more subtle formulation of the optimisation metric, the user-defined blocking rate, which more aptly encompasses the desires of cluster users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Next, we propose a simple modi- fication of the quantised SiP-ML partitioner which, rather than maximally parallelising all jobs, minimally parallelises them such that they meet the user-defined maximum acceptable completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Then, we propose a novel network-aware partitioning strategy (see Figure 5 and Section 5) called PAC-ML (partitioning for asynchronous computing with machine learning) which utilises reinforcement learning (RL) and a graph neural network (GNN) to flexibly meet the demands of the user in an arbitrary manner given the current state of the cluster network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Finally, we demonstrate our method in simulation on the recently propsed RAMP optical architecture (Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022), achieving up to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2% lower blocking rates than the best heuristic baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We show that different user-defined demand environments require different partitioning strategies for optimal results, and that a key advantage of PAC-ML is that it is able to discover performant strategies automatically without the need for handcrafted heuristics or environment-specific tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Related Work Recent years have seen a surge of interest in developing methods to distribute machine learning (ML) tasks across multiple devices (Ben-Nun 4 and Hoefler, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Mayer and Jacobsen, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' One approach has been to optimise the physical plane of the distributed cluster such as its compute and network devices and architectures (Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In this work, we instead focus on optimising the virtual plane, which determines how physical layer resources are allocated to execute a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We divide the virtual plane into three sub-components: Job (1) partitioning (how many devices to use);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2) placement (which devices to use);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (3) scheduling (in which order to use the devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Many prior virtual plane works have considered (2) and (3) (how to distribute), whereas we focus on (1) (how much to distribute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, in this section we comment on recent progress across all these fields, since we leverage this progress throughout our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' ML for discrete optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Many combinatorial optimisation (CO) problems turn out to be NP-hard, rendering exhaustive search techniques intractable for practical application (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, practitioners rely on either approximate algorithms, which give restricted performance guarantees and poor scalability (Williamson and Shmoys, 2011), or heuristics, which have limited solution efficacy (Halim and Ismail, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Since the first application of neural networks to CO by Hopfield and Tank (1985), the last decade has seen a resurgence in ML-for-CO (Bello* et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Gasse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Barrett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The advantages of ML-for-CO over approximation algorithms and heuristics include handling complex problems at scale, learning either without external input and achieving super-human performance or imitating strong but computationally expensive solvers, and (after training) leveraging the fast inference time of a DNN forward pass to rapidly generate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Since almost all cluster resource management tasks can be reduced to canonical CO problems (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021), many state-of-the-art resource management methods utilise recent advances in ML-for-CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2017) were the first to apply ML to the task of deciding which operations in a computation graph to place on which devices in a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' They used a sequence-to-sequence model consisting of an LSTM DNN with an attention mechanism trained with the simple REINFORCE policy gradient RL algorithm (Williams, 1992) such that the JCT of a deep learning job was minimised, outperforming handcrafted heuristics when training the Inception-V3 computer vision and LSTM natural language processing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018) furthered this work by replacing REINFORCE with the more advanced proximal policy optimisation (PPO) 5 RL algorithm (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017) with lower variance and reduced training hardware demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' They demonstrated their method beating Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2017) on the CIFAR-10 image recognition benchmark in terms of JCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018a) proposed a novel hierarchical model which decomposed the job placement task into a joint group-and-place problem, reducing the JCT of Inception-V3, ResNet, LSTM, and NMT models by up to 60% relative to the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All works up to this point used DNN architectures restricted to Euclidean- structured input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, in order to handle non-Euclidean graph-structured data such as computation graphs and cluster networks, they had to be re-trained each time a new graph structure was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Addanki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2019a) were the first to instead leverage a GNN, as well as the grouping scheme of Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018a), to learn to generalise across different job types with varying computation graph structures, demonstrating device placement schemes which were on par with or better than prior approaches on Inception-V4, NASNet, and NMT after 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1× fewer training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Khadka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2021) furthered the use of GNNs for job placement by combining GNNs, RL, and population-based evolutionary search with the hierarchical group-and-place scheme of Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, they replaced the manually-designed operation grouping heuristic with a learned policy capable of superior scaling and JCT performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018) addressed the job scheduling problem (the order in which to execute operations placed across a set of devices) using a primal-dual framework for online job scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' They represented the problem as an integer linear programme (ILP) which their proposed algorithm could solve in polynomial time in an online fashion such that the cluster resources were maximally utilised and the JCT minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2021) proposed a placement-aware scheme which leveraged the pre- determined device placement allocation to decide on a job schedule which could reduce the average JCT by up to 25% relative to other scheduling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Paliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2020) went further by utilising an RL-trained GNN and a genetic algorithm to jointly optimise both job placement and scheduling, demonstrating both lower JCT and peak memory usage than other strategies when distributing TensorFlow computation graphs across a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To the best of our knowledge, Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2021) are the only ones to have explicitly considered the question of how much to distribute a computation graph in the context of an optical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Like other works, they assumed that a maximum parallelisation strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 6 partition the job across as many workers as possible) is a desirable objective, and then focused on how best to design the physical layer such that the JCT could be minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All works discussed in this section have assumed that the JCT is the key objective to minimise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, where the question of partitioning is considered, prior works have assumed that more parallelisation is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, we posit that user-critical metrics such as throughput and blocking rate are compromised by prioritising optimisation of the JCT in a cluster setting with dynamic job arrivals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To address this shortcoming, we propose a new ML-based resource management scheme which explicitly addresses the partitioning question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, our work leverages the emergent trend from these other virtual plane fields, namely utilising an RL-trained GNN, to decide how much to partition different jobs in a dynamic setting with arbitrary user-defined completion time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parallelisation Types of parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parallelisation is the process of distributing a computational job across multiple devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is done in order to reduce the time and/or physical memory needed to complete the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' There are three main types of deep learning parallelism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' data parallelism, model parallelism, and hybrid parallelism (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1 for extended background infor- mation on these methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Although today the most common method for DNN training parallelisation is data parallelism for its simplicity and limited network overhead, we focus on the less common but more desirable model parallelism paradigm for its strong scaling capabilities (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Our proposed partitioning methods are applicable to hybrid and pipeline par- allelism, but these require additional simulation complexity and are therefore beyond the scope of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Computational jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' A computational job is a directed acyclic graph (DAG) whose nodes are operations and edges are dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Operations are computational tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' some mathematical reduction, a database query, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dependencies are either control dependencies, where the child operation can only begin once the parent operation has been completed, or data dependencies, where at least one tensor is output from the parent operation and required as input to the child operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In the context of DNNs, a job DAG is a sequence of forward pass, backward pass, and parameter 7 Figure 2: Diagram showing a DNN job DAG being partitioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Top: A forward pass DAG where each node has an associated partition degree (how many times it will be divided when partitioned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Bottom: A partitioned DAG with forward and backward passes handled consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Green edges in the graph represent data flow (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' output to input) between consecutive operations in the forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Orange edges represent gradient exchanges processed in the backward pass (backpropagation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Blue edges represent full connectivity collective operations to synchronise weight updates across partitioned components of an operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that, for brevity, the top unpartitioned DAG only shows the forward pass (since, before partitioning, the graph structure is identical to the backward pass), whereas the bottom partitioned DAG shows both the forward and backwards passes (since, after partitioning, the graph structures are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 8 Original DAG (to be partitioned) Partition Degree: 1 Partition Degree: 4 Partition Degree: 4 Partition Degree: 2 Partition Degree: 1 Data Flow Gradient Flow RAMP Collective Partitioned DAG Forward Pass 3a Backward Pass All-to-all 2a All-to-all All-to-allupdate operations which need to be performed on data exchanged between operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Whether or not this data passes through a communication network is determined by how the operations are partitioned, placed across a cluster of workers, and parallelised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job partitioning refers to the process of splitting the operations of a job DAG into u (the partition degree) smaller sub-operations which can in turn be placed across u workers, thus reducing their run time and memory requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Partitioning is used in the model, hybrid, and pipeline parallelisim paradigms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' More partitioning can decrease compute time and memory requirements, but requires more inter-worker communication, complex intra-worker operation scheduling, and greater resource utilisation, therefore potentially increasing overall completion time, cluster complexity, and subsequent job blocking rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Figure 2 visualises how an initial DAG for some arbitrary neural network architecture, where each operation has a partitioning degree, can be re-represented in terms of its partitioned form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Both forward and backward passes are explicitly represented since inter- operation information dependencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the edges in the graph) are not the same in each pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Optical Networking Most current cluster networks use optic fibre communication links, but the switch devices which interconnect the network are usually electronic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Limitations of electronic networking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Electronic networks have poor scalability, bandwidth, latency, and power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, since the per-port bandwidth is limited and the power consumption required to cool active electronic devices is expensive, the bisection bandwidth achievable in an electronic network is restricted, thus hampering scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, although the compute power of DCN server nodes, as measured by FLOP/s, has increased by a factor of 65 over the last 18 years, the bandwidth of the DCN network facilitating communication between these nodes has only increased by a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8, resulting in an 8-factor decrease in bytes communicated per FLOP (Bergman, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This has created a performance bottleneck not in the server nodes themselves, but rather in the network connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This issue is especially compounded when striving for strong scaling via model parallelism with distributed computing, and with the trend towards larger models with ever more parameters as described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Optical circuit switched networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Cluster networks with optical switches have the potential to offer significant improvements in performance 9 Figure 3: The mean network overhead of the 6 distributed deep learning jobs reported by (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022) in Meta’s GPU cluster compared to that of RAMP as reported by Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022) on the 5 jobs considered in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that this is an approximate comparison, and that the important takeaway is that RAMP retains low network overheads as jobs become increasingly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (due to larger bandwidth and lower switching latency) and energy efficiency (due to the lack of optical-electronic-optical conversion overhead), as well as the capability to scale to next-generation large-scale distributed compute jobs with exascale bandwidth and compute (Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' OCS networks in particular offer a promising avenue with which to realise commercial optical networks due to their non-blocking circuit configurations with high capacity and scalability and low deterministic switching latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In contrast to optical packet switched networks, OCS networks are simpler to implement and they eliminate the need for in-switch buffering or queuing and addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' RAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' RAMP is a state-of-the-art OCS architecture designed specifically for cloud data centres and distributed deep learning systems (Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' RAMP networks are parameterised by NC communication groups, NR racks per communication group, and NS servers per rack, resulting in a NW = NC × NR × NS worker cluster with a colloquially termed ‘RAMP shape’ defined by tuple ⟨NC, NR, NS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' At its core, RAMP proposes a novel set of message passing interfaces (MPIs) for performing the synchronisation steps (AllReduce, AllGather, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=') required by distributed DNN training jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These will be referred to as collective operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These MPIs are designed to take full advantage of the high bandwidth provided by optical network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, as shown in Figure 3, the network overhead of RAMP remains remarkably low as the number of workers used to execute a job increase (see Section 6 for experimental details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The RAMP authors 10 (%) Meta RAMP Network Overhead ( 40- 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='13 0 101 102 # Workersshowed that this low network overhead enables unprecedented scalability with up to 65 536 worker nodes capable of training O(trillion) parameter DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' RAMP placement rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As detailed in Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022), a group of workers in a RAMP shape can only undergo collective operations if they are selected with respect to certain rules, loosely termed here ‘symmetry’ rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For shape ⟨NC, NR, NS⟩, these rules are as follows: (1) NS workers per rack spread over NR racks requires that the set of workers on each rack span NR distinct communication groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These NR distinct communication groups do not have to be the same set across racks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2) NS workers on NR = 1 rack must span NS communication groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (3) NS workers spread over NR racks (NS = 1 worker per rack) must span NS distinct communication groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our simulations, we use a simple first-fit operation placement heuristic which conforms to these rules (refer to Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Reinforcement Learning RL is the study of optimal decision making in natural and artificial systems (Sutton and Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In the general RL setting shown in Figure 5, an agent interacts with an environment at each sequential time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The environment can be described by tuple ⟨T, R⟩, where T is a state transition probability matrix defining the transition probabilities from all states s to all successor states s′ taking action u where T u ss′ = P(St+1 = s′|St = s, U t = u), and R is a scalar reward function giving the expected immediate (next state) reward given current state s and chosen action u where Ru s = E(Rt+1|St = s, U t = u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The environment is usually assumed to have the Markov property whereby P(st+1|st) = P(st+1|ht);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' that is to say that the probability of the next state being st+1 given the current state st is the same as the equivalent probability given all previous states in history ht = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', st}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such, this RL setting is usually assumed to be a Markov decision process (MDP) described by tuple ⟨S, U, T, R, γ⟩ where S is a finite set of possible environment states, U is either a discrete (finite) or continuous (infinite) set of possible actions, and γ ∈ [0, 1] is a discount factor specifying the factor by which to multiply future expected rewards to discount their present value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Since Markov states are stochastic, future rewards are never fully certain and are therefore expressed as an expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Agent goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The agent’s goal is to learn to maximise its expected total discounted future reward, termed the ‘value’ or ‘return’ Gt = �∞ k=0 γkRt+k+1, 11 over the course of an episode (a sequence of decision steps which may or may not terminate at some point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To do so, the agent can use model-free RL to avoid explicitly modelling the environment by only using its policy function and/or its value function to make decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The policy function π maps an observed state st to a corresponding action ut such that some estimated score objective is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The value function estimates the expected return Gt from being in state st and following policy π (the state value function v) or from being in state st, taking action ut, and following policy π (the action value function q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Crucially, value and policy functions can be approximated and learned with DNNs, enabling RL to be scaled to large problem instances (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 for extended background information on DNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Advantages of RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Using traditional RL has several advantages over heuristics and other ML paradigms such as supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' First, no external data from human-designed or computationally expensive heuristics is required, enabling an agent to learn super-human policies without potentially sub-optimal initial biases towards a certain strategy or a costly expert example collection-and-labelling phase (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Second, a DNN with a finite number of layers and neurons will have its expressivity constrained (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020), restricting the complexity of the set of functions it is capable of approximating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Because the objective of an RL agent is to maximise its expected future return which, under the assumption that a suitable reward function has been crafted, is equivalent to maximising performance on a given task, RL agents are able to maximise task performance given DNN expressivity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Third, since RL agents maximise future return, they are capable of learning sophisticated non-myopic policies which sacrifice short-term reward in exchange for higher long-term return (Sutton and Barto, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' User-Defined Blocking Rate To motivate our work, we first explore the key metrics to consider when evaluating a job partitioning strategy with the help of an experiment on 32 GPU workers, and then introduce a new formulation of the user-defined blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All experimental details are given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The inadequacy of optimising the job completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As dis- cussed in Section 2, most prior works researching management schemes for distributed computing aim to minimise JCT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the time taken to com- plete a given job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If a job j begins running at wall clock time tstart wc,j and is 12 Figure 4: (a-b) Demonstration of how more partitioning can lead to a lower JCT than no partitioning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' sequentially running the job on a single device), but this may be at the cost of a higher blocking rate since more cluster resources are occupied when subsequent jobs arrive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (c-d) Demonstration of how optimising for the cluster throughput leads to an unfair bias towards more partitioning, because more parallelism creates more work for the cluster and therefore artificially increases cluster throughput even though, from the perspective of the user, the original offered throughput may be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' completed at time tend wc,j, researchers usually record the completion time as JCTj = tend wc,j − tstart wc,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Consequently, most systems maximise the degree to which they parallelise jobs in order to minimise JCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' While it is true that end users undoubtedly want this JCT metric to be minimised, it fails to quantify when a job was blocked, which occurs when no cluster resources were available to service it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' While more parallelism will often lead to a lower JCT for a given job, it will also use up more of the cluster’s compute and network resources, potentially blocking future job arrivals (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore in practice, end-users wish to minimise both the JCT and the overall blocking rate (the fraction of jobs blocked over a given time period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' While maximum parallelisation will lead to a minimised JCT, we posit that a balance between these two extreme parallelisation strategies can more aptly optimise for both the JCT and blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Alternative optimisation objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' One metric which encapsulates both the JCT and blocking rate is throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the information processed per unit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' There are two issues with using throughput as an optimisation objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (1) Operators must be careful how they measure the throughput to be optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If they measure the cluster throughput (the total cluster information processed per unit time), they will be biased towards more 13 Sequential Paramac (a) X104 (b) Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 - S Blocking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 XX~X-X1 2X S (c) X107 S (d) X107 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='25 B Cluster Offered 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='75 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 Load X107 Load X107 Rate (B/s) Rate (B/s)parallelisation, because when a job is partitioned and parallelised, the edge dependencies coming in to and out of the partitioned operation node(s) must be replicated (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This artificially creates more information for the cluster to process even though, from the end users’ perspective, the total information processed of their original demand is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore, the offered throughput (the total original demand information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' before parti- tioning was applied) processed per unit time) is a more suitable throughput metric to optimise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Figure 4 shows an example of how a ‘maximum parti- tioning’ strategy, such as that used by SiP-ML (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021), can have superior cluster throughput when compared to a ‘no partitioning’ strategy (sequentially running the job on a single device) despite having lower offered throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, offered throughput is still an inadequate optimisation metric, because (2) in practice, different jobs being serviced by the cluster originating from different client users have different priorities and job comple- tion time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For example, two identical machine learning training jobs might be submitted to the cluster, but one from a user who intends to deploy the model commercially and requires it to be completed overnight, and the other from a user who is employing the model for research and has less stringent completion time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Ideally, operators would direct their clusters to meet flexible user-defined per-job completion time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The user-defined blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To enable users to dynamically de- termine the completion time on a per-job basis whilst also maximising the number of job demands satisfied, we introduce a new formulation of the user-defined blocking rate objective for the partitioning algorithm to optimise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given a job which, if executed sequentially on one device, would be completed in JCTseq j , we define the maximum acceptable JCT as JCTacc j = β · JCTseq j , where {β ∈ R : 0 < β ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Here, β is a parameter chosen by the user which determines how quickly the job must be completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If JCTj > β·JCTseq j , then the cluster will have failed to complete the job within the required time and the job will be recorded as having been blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The user-defined blocking rate is therefore the fraction of jobs which failed to meet the JCTj ≤ β·JCTseq j requirement over a given period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that rather than brazenly optimising for either the JCT or the blocking rate, the user-defined blocking rate enables the cluster operator to instead dynamically specify their desired completion time on a per-job basis, and the performance of the cluster is evaluated according to how well it was able to meet the requirements of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Furthermore, the β parameter corresponds to the speed-up factor being requested by the user and, since {β ∈ R : 0 < β ≤ 1}, can be given directly 14 Figure 5: An overview of our PAC-ML approach transitioning from step t → t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' At each time step t when there is a new job to be placed on the cluster, we: (i) Use a GNN to generate an embedded representation of the node and edge features in the job’s computation graph, and a standard feedforward DNN to do the same for the global job and cluster features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (ii) concatenate the outputs of (i) and use another feedforward DNN to generate a logit for each action ut ∈ U t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iii) pass the chosen action ut to the environment and partition the job accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iv) apply any internal environment allocation heuristics (operation and dependency placement and scheduling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=') to attempt to host the job on the cluster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (v) if accepted onto the cluster, perform a lookahead to evaluate the job’s completion time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (vi) fast-forward the environment’s wall clock time twc to when the next job arrives, and return the corresponding reward rt+1 and updated state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' as input to a DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' PAC-ML Partitioning Methodology RL agents can learn general policies without the need for human guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An RL job partitioner therefore has the potential to take an arbitrary maxi- mum acceptable JCT provided by the user and automatically decide how much to distribute the job such that, over a period of time, the number of jobs which meet the JCT requirements specified by the user is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Such an agent would therefore be able to minimise the blocking rate whilst also accounting for the flexible and dynamic JCT specifications of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Following this 15 Agent DNN forward pasS Action selection Generate job Generate global Generate Action graph embedding embedding logits scores = 0 ui =1 Concatenate 2 = 2 ug = 3 =4 DNN modules Node Edge Global Logit Environment Input at time t Environment transition Job Cluster Partitioned job 01 gc gu Allocate place ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' : schedule ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4- place deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4- schedule deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Operation Dependency Worker Communication link NNModule Actionlogic, we now describe our PAC-ML (partitioning for asynchronous computing with machine learning) approach for learning to partition computation jobs with RL and a GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Markov Decision Process Formulation Since allocating cluster resources for jobs arriving dynamically in time is a sequential decision making process, formulating problems such as job partitioning as an MDP is a natural approach and facilitates the application of many traditional and state-of-the-art RL algorithms (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Addanki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Paliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' A new job j arriving at time step t is comprised of a DAG G(O, D, gj) with node operations O, edge dependencies D, and any other job statistics which might be recorded gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Similarly, the state of the cluster at time t is made up of the number of workers available, the jobs currently running on the cluster, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To compress the state of the cluster and the job requesting to be placed into a representation suitable as input for a neural network at time step t, we encode this information into five feature vectors: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Per-operation features oi∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', |O|} (5 features): (i) The com- pute cost (run time in seconds on an A100 GPU);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (ii) a binary variable indicating whether the operation has the greatest compute cost in the job;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iii) the memory cost (byte occupancy);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iv) a binary variable indicating whether the operation has the greatest memory cost in the job;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (v) the node depth with respect to the source node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The compute and memory costs are normalised by the highest compute and memory cost operations in the job, and the node depth is normalised by the depth of the deepest node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Per-dependency features di∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', |D|} (2 features): (i) The size (in bytes) of the edge dependency normalised by the largest dependency in the job;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (ii) a binary indicator of whether the dependency is the largest in the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Global job features gj (15 features): (i) The number of operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (ii) the number of dependencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iii) the sequential job completion time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iv) the maximum acceptable job completion time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the maximum acceptable job completion time fraction β both (v) raw and (vi) normalised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (vii) the total memory cost of all of the operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (viii) the total size of all of the dependencies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (ix) the number of training steps which need to 16 be performed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the (x) mean and (xi) median of the operation compute costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the (xii) mean and (xiii) median of the operation memory costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (xiv) the mean and (xv) median of the dependency sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each feature is normalised by the highest respective value of the feature across all job types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Global cluster features gt C (2 features): (i) The number of occu- pied workers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (ii) the number of jobs running.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Both features are normalised by the total number of workers in the cluster NW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Global action features gt U (NW 2 features): A binary vector indicating the validity of each possible partitioning decision given the state of the cluster and the RAMP rules defined by (Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given the state st encapsulating both the job requesting to be placed and the current state of the cluster, the partitioning agent uses a policy π(st) to select a number of times ut up to which to partition each operation in the job’s computation graph (using a similar minimum operation run time quantum discretisation scheme to Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2021)), where ut i∀i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', NW 2 } (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' there are � NW 2 + 1 � possible discrete actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that ut = 0 enables the agent to reject a job without placing it, ut = 1 places the job onto one worker and runs it sequentially, and 1 < ut ≤ NW 2 attempts to distribute the job’s operations across up to ut workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our setting and given the RAMP rules of Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022), an invalid partitioning action is one which is at least one of: (i) An odd number (except ut = 1), or either (ii) greater than the number of workers available or (iii) has no valid RAMP placement shape given the current state of the cluster (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As a consequence of the RAMP rules defined by Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022), which require that the worker and network resources allocated to a given job are reserved exclusively for that job for the duration of its run time, we are able to perform a deterministic lookahead to evaluate what the overall completion time, JCTj, of the job will be as soon as it is placed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Subsequently, when a job j arrives at time step t, we can immediately determine whether or not the cluster met the JCTacc j specified by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This enables the use of a simple per-step +1/−1 reward scheme, rt+1 = � 1, if JCTj ≤ β · JCTseq j −1, otherwise , (1) which when aggregated and maximised over the course of an episode 17 corresponds to maximally meeting the specified per-job completion time requirements and therefore minimising the user-defined blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our hybrid time- and event-drive simulation, when the agent makes a partitioning decision at time step t, the environment transitions to the next step t + 1 by fast-forwarding its internal simulated wall clock time, twc, to when the next job arrives and requests to be placed, updating the states of any running and completed jobs and their corresponding compute and network resources as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The episode terminates when twc = T max wc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' PAC-ML Learning Setup Reinforcement learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To find a policy which maximises the expected return when partitioning jobs, we used the state-of-the-art Ape-X DQN (Horgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2018) RL algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' a distributed and highly scalable value-based method (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7 for algorithm details and hyperparameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To make the learning of value and policy functions tractable in large state-action spaces, we approximated them with a custom-built message passing GNN implemented using the open-source PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019) and DGL (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019) libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Refer to Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 for further architectural details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Experimental Setup All code for reproducing the experiments and links to the generated data sets are provided at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='com/cwfparsonson/ddls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Simulation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We built an open-source Gym environment (Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016) to simulate the RAMP OCS system of Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022) in an RL-compatible manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We used a hybrid time- and event- driven simulation approach where we kept track of the internal simulation wall clock time twc, enabling the measurement of time-based metrics, but only took a partitioning decision when needed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' when a new job demand arrived at the cluster), aiding efficiency since no discrete steps were needlessly simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All our experiments used similar cluster parameters to Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We used NW = 32 (NC = 4, NR = 4, NS = 2) A100 GPUs with 80 GB memory capacity, 2 THz memory frequency, and a peak computational power of 130 Tflop/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We assumed an intra-GPU propagation latency of 50 ns, a negligible OCS circuit reconfiguration latency of 1 ns, a worker input-output latency of 100 ns, and a total worker communication capacity 18 Figure 6: The four β distributions used in our experiments in order to measure the capability of each partitioner to cater to different user-defined maximum acceptable completion time requirement settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In each βX experiment setting, each new job generated was assigned a β value sampled from βX in order to get the maximum acceptable job completion time, β · JCTseq (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 TB/s (resulting in a per-transceiver bandwidth of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6×1012 NC B/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All experiments were run up to a simulated wall clock time of T max wc = 106 s (around 12 days) of continuous cluster operation with dynamic job arrivals and were repeated across 3 random seeds, with the subsequent min-max confidence intervals for each measurement metric reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' More details of the simulation environment are provided in Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Compute jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We used the computation graph time and memory profiles of five real deep learning job types open-accessed with Microsoft’s PipeDream research (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019, 2021) (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These jobs encompassed image classification (AlexNet (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2012), ResNet-18 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016), SqueezeNet-10 (Iandola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016), and VGG-16 (Simonyan and Zisserman, 2014)) and natural language processing (GNMT (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016)) tasks, thereby testing the generality of the approaches we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All jobs arrived to the cluster dynamically and stochastically throughout the simulation period, with the inter-arrival time fixed at 1000 s to control the load rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each job was ran for Niter = 50 training iterations, where one training iteration consists of one forward and backward pass through the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' When partitioning the operations in a job’s computation graph, we allowed the partitioning agents to split each operation up to NW 2 19 BB Bc BD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='04 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 3times (the environment’s ‘maximum partitioning degree’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We followed Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2021) by (1) assuming a linear dependency between the total number of operation splits and each split’s compute time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (2) choosing a minimum quantum of computation time, τ, and splitting operations up to a number of times which would result in sub-operations with a compute time no smaller than τ in order to maximise GPU utilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We set τ = 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such, a given partitioning action ut set the maxmimum partitioning degree of the job, but individual operations within the job could be split fewer times depending on their initial compute time and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that although this restricts each operation to be distributed across a maximum of ut servers, the total number of workers used by all operations in the job can still be greater than ut depending on the operation placement heuristic’s choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Maximum acceptable job completion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our setting, a par- titioner would ideally be able to take an arbitrary job with an arbitrary maximum acceptable job completion time, β · JCTseq, and partition the job such that the completion time requirement is satisfied for as many dynami- cally arriving jobs as possible (thereby minimising the user-defined blocking rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To test each partitioner’s ability to do this, we ran experiments using four β distributions (βA, βB, βC, and βD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For each βX experiment, when one of the five possible jobs was randomly generated to arrive at the cluster, a β value, discretised to two decimal places, was randomly sampled from the experiment’s βX distribution and assigned to the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' By sampling a broad range of β values from a selection of βX distributions, we ensured that we could analyse the performance of each partitioning agent under different completion time requirement settings and subsequently measure the capability of each method to cater for different user-defined requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Heuristics RL Random Paramax Paramin PAC-ML βA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='517+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='262+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='002 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='309+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='014 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='203+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='009 βB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='601+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='263+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='396+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='258+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='003 βC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='505+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='016 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='267+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='307+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='015 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='117+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='003 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='003 βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='465+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='263+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='006 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='142+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='027 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='099+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='007 Table 1: Blocking rate performance of the partitioning agents on the four β distributions (best in bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Results are given as the mean across 3 seeds, and error bars denote the corresponding min-max confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 20 Partitioner baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We considered three heuristic baseline partition- ing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (1) Most prior works partition a given job across as many workers as are available up to a pre-defined environment maximum partition degree (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We refer to this strategy as ‘Paramax’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2) Given the low network overhead (see Figure 3) and contention- less nature of RAMP, and given the operations’ linear split-compute time dependency of our environment, a reasonable estimate for the completion time of a job with sequential run time JCTseq distributed across ut workers would be JCT ≈ JCTseq ut .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore, in light of our objective to minimise the user-defined blocking rate, we introduce a new partitioning strategy, ‘Paramin’, which partitions the job up to the estimated minimum amount of parallelisation needed to satisfy the job’s completion time requirements, ut = ⌈ 1 β⌉ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the estimated speed-up factor needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (3) For completeness, we also ran a ‘Random’ partitioning baseline, which selects a partitioning degree randomly from amongst the number of available workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Metrics recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To measure the performance of our partitioning agents, we recorded the following key metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (1) User-defined blocking rate (which we abbreviate to ‘blocking rate’): The fraction of arrived jobs which had their completion time requirements met by the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2) Offered throughput: The total ‘information size’ of the original jobs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' before partitioning was applied) processed per unit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Since the open-access PipeDream job profiles used in our experiments did not contain per-operation flop/s (computational load) information, we summed the jobs’ operation and dependency sizes (measured in bytes (B)) to get the total ‘information size’ of each job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The load rate could then be defined as the rate of job information arriving at the cluster per unit time, and the corresponding offered throughput as the rate at which this total job information was processed by the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For a full list of metric definitions, refer to Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' PAC-ML Partitioning Results & Discussion 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Performance of the PAC-ML Partitioner Comparison to the baseline partitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To test the performance of each partitioning agent under different completion time requirement settings, we ran our experiments across four different β distributions (see Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We visualise the relative blocking rate and throughput performance differences between the agents in Figure 7, where an agent’s ‘score’ is its normalised performance relative to the best-performing agent with respect to a given 21 Figure 7: Validation performances (higher is better) of each partitioning agent evaluated across three seeds normalised with respect to the best-performing partitioner in each BX environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We evaluate these scores as scoreblocking = � best_blocking_rate blocking_rate � , and scorethroughput = � throughput best_throughput � for each agent (refer to Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8 for all raw metric values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As shown in Table 1 and Figure 7, our PAC-ML agent achieved the best blocking rate across all four β distributions, beating its nearest rival by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='90%, 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2%, and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3% for βA,B,C,D respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Comparison amongst the baseline partitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Figure 7 visualises the performance of the best PAC-ML agents on each of the four β distribution environments compared to the baseline heuristic performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Interestingly, the best baseline in terms of blocking rate for βA,B,C is Paramax, but this switches to Paramin for βD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' On βB, PAC-ML achieved roughly equivalent performance to Paramax by learning that, on this β demand distribution, maximum parallelisation led to the lowest blocking rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This shows that different partitioning strategies have varying relative performances under different cluster settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' A key advantage of PAC-ML is therefore that the question of which partitioning strategy is best for a given environment need not be addressed by sub-optimal hand-crafted heuristics or environment- specific hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Instead, we have demonstrated in Table 1 and Figure 7 that PAC-ML can automatically learn performant partitioning strategies in arbitrary environment settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Analysis of the PAC-ML Partitioner Offered throughput analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' One risk of optimising only for the blocking rate when training the PAC-ML agent is that it maximises the number of jobs accepted by prioritising small low-information jobs at the cost of a sub-optimal offered throughput;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' a key metric when measuring a cluster’s quality of service to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Figure 7 shows that the offered throughput 22 PAC-ML (Ours) Paramaa Paramin Random ore .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='00 re 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 Better L Worse e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='75 Rat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='25 B βA βB Bc βD βA βB βc βDFigure 8: Mean per-job blocking rates of the five job types considered for each partitioning agent under each βX setting plotted against the number of operations (ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' ), number of dependencies (deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' ), the total job information size, and the sequential run time of the job were it ran on a single device (JCTseq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' improves with the blocking rate, with the PAC-ML agent ultimately achieving the best throughput across all four β distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Bias analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An important question is whether there is any bias in the kinds of jobs the PAC-ML agent learns to prioritise in order to minimise the blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To investigate this, Figure 8 shows the blocking rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the original characteristics for each of the five jobs considered (see Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 for a summary of these characteristics) for each βX distribution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The PAC-ML agent had little to no bias across the jobs relative to the other partitioners, with all jobs attaining approximately the same blocking rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' There was a slight bias towards the larger jobs with greater sequential completion times and more information to process, which is likely due to the fact that larger jobs occupy more resources and therefore inherently become 23 PAC-ML (Ours) Paramac Paramin Random βB βc βA βD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 b0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 locking .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 B 0 7 50 100 50 100 50 100 50 100 # Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 - Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 - O 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 b0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 locking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 O B d 50 100 150 50 100 150 50 100 150 50 100 150 # Deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' # Deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 b0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 locking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 B cp 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 Job Size ×1010 Job Size ×1010 Job Size ×1010 Job Size ×1010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 bo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 locking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 B 1 1 1 1 2 3 1 2 3 1 2 3 1 2 3 JCTseq ×104 JCTseq JCTseq ×104 JCTseq ×104 ×104favoured over smaller jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Conclusion & Further Work In conclusion, we have introduced a new partitioning strategy called PAC- ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Leveraging RL and a GNN, PAC-ML learns to partition computation jobs dynamically arriving at a cluster of machines such that the number of jobs which meet arbitrary user-defined completion time requirements is max- imised without the need for hand-crafted heuristics or environment-dependent hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We tested our partitioner on the recently proposed RAMP optical architecture (Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022) across four distributions of user-defined completion time requirements, demonstrating up to 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2% lower blocking rates relative to the canonical maximum parallelisation strategies used by most prior works when partitioning five real deep learning jobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We hope that our work will spur a new avenue of research into developing partitioning strategies for distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In this section, we outline potentially interesting areas of further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Exceeding completion time expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In this work, we rewarded PAC-ML with +1 for completing a job within the user-defined maximum acceptable completion time and −1 for failing to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Although minimising the blocking rate is crucial for users, it would also be desirable to minimise the JCT as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An interesting area of further study would therefore be to incorporate this objective into the reward function, perhaps by combining the JCT speed-up factor or offered throughput with the blocking rate via multi-objective RL (Hayes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Real-world experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Our work has considered real open-access deep learning computation graph profiles but on a simulated optical architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' A natural but significant next step would be to implement PAC-ML in a real distributed cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An important question would be whether an agent trained in a simulated environment would be capable of inferring in a real cluster at test time, or if real-world training would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Generalisation to unseen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This study ran PAC-ML in an environment which had the same load rate, β distribution, cluster network size, and job computation graphs at train and test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An interesting research question would be whether PAC-ML would be able to learn on one set (or a distribution) of these parameters and then generalise to a new set at test time, or if it would need to leverage existing or new state-of-the-art 24 methods in GNN (Knyazev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) and RL (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Robustness to stochastic inter-arrival times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our experiments, we fixed the inter-arrival rate in order to fix the load rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, real clusters have variable inter-arrival times (Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Handling highly stochastic environments is a known challenge for RL (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019), and therefore presents an interesting future research avenue for PAC-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Combining the virtual plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our work, we have considered the job partitioning task in isolation of the job placement and scheduling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, prior works have found the merging of the latter sub-tasks into a single resource management problem beneficial to performance (Paliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' An interesting area of further work would be to combine PAC-ML into a a single algorithm which handled job partitioning, placement, and scheduling via methods such hierarchical RL (Barto and Mahadevan, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Vezhnevets et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Paliwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) or multi-agent RL (Foerster, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Extended Background 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parallelisation There are three main types of deep learning parallelism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' data parallelism, model parallelism, and hybrid parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Data parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Data parallelism (Slotnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 1962) is where an identical copy of the DNN model is sent to each worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The input training data is parallelised by sampling a training batch, splitting it into non-overlapping micro-batches, training each worker on its own micro-batch, and updating the workers’ local model parameters using some method to synchronise the gradients of the parameters with respect to the training loss after each training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This synchronisation step is commonly referred to as AllReduce, and can be performed using various techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Data parallelism can be applied to any DNN model regardless of its architecture, enables the use of large data sets (which are crucial for scaling model performance (Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2022)), and facilitates the use of large training batch sizes which can lead to smoother and faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is a form of weak scaling, where the JCT is decreased by reducing the total number of training iterations needed via increasing the amount of data processed per iteration as the number of workers is increased (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, it scales poorly 25 for large models with many parameters since all parameters must fit onto a single worker and then be synchronised at the end of each training step, and has the constraint that the training data must be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' in order for parameter updates to be computed and summed across workers to attain the updated model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Model parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Model parallelism (Karakus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021) is where the DNN model is partitioned (split) and a part of the model is sent to each worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In the DNN forward pass, a training batch is sampled, copied, and sent to each worker which holds layer-1 of the DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The layer-1 worker(s) then compute the layer-1 output(s) and forward them to the worker(s) which hold layer-2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In the backward pass, the gradients of the model parameters with respect to the training loss are computed by starting at the worker(s) which hold the final layer and propagating these gradients back to the layer-1 workers, after which the partitioned model will be globally synchronised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Layer outputs, gradients, and activations are exchanged during the training iteration using a synchronisation step commonly referred to as AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Model parallelism facilitates the use of very large models which otherwise would not fit onto a single worker and caters for time-efficient parallelisation of computational operations where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is a form of strong scaling, where the JCT and per-worker memory utilisation are decreased via increasingly partitioning different parts the job across more workers as the number of workers is increased (Khani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, passing gradients between workers during training can create a large communication overhead (Mirhoseini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017, 2018b), and expert domain knowledge of the specific model architecture is needed to know how to split the model across multiple workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hybrid parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hybrid parallelism (Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2012) is where a combination of data and model parallelism is used to strive for the benefits of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This can be extended to include pipeline parallelism (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019), where intra-batch parallelism (data and model parallelism) are combined with inter-batch parallelism (pipelining) where multiple micro-batches are processed simultaneously where possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hybrid parallelism can result in higher worker utilisation and the advantages of both model and data parallelism, but requires complex bidirectional pipelining across different inputs, careful model parameter versioning to ensure correct computations of the gradients during the backward pass, and each stage allocated across workers must be load-balanced to ensure roughly equivalent computational times between workers in order to maximise peak pipeline 26 throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Neural Networks as Function Approximators Neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Neural networks are a composition of linear and non-linear (activation) functions connected in a chain to form a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each function in the chain is a layer parameterised by a set of weights and biases which, given enough parameters, can be trained to approximate any universal function (Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Montufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Neural networks with multiple intermediary (hidden) layers between input and output are referred to as deep neural networks and have powerful expressivity capabilities when approximating complex non-linear functions (Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Montufar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Graph neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Whereas standard DNNs are restricted to handling only vector- and grid-structured inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' sentences, images, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' ), GNNs are generalised DNN architectures which can handle graph-structured data as inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' job DAGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Most current GNNs use the message passing paradigm by mapping each node and edge onto a vector embedding space before performing additional graph-level embeddings and readouts if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Specifically, each GNN layer usually performs four stages: (i) On each edge in the input graph, use a message function to generate a message (representation) to pass from a source node to a set of destination nodes, where each node stores the message(s) it receives in its mailbox;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (ii) on each node in the input graph, apply an aggregate function (a vanilla reduce operation such as mean, sum, max, min, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', or a trainable function) to the messages in its mailbox to generate an intermediate aggregate representation of its neighbourhood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (iii) pass the intermediate aggregate representation through a trainable function to produce a final vector embedding for each node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and (optional) (iv) if desired, at the end of the final GNN layer, pass the node embeddings through a trainable function to produce a graph-level representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Crucially, the parameters of all message, aggregation, and forward pass functions are shared across nodes, enabling GNNs to be inductive in that they can generalise to unseen nodes and graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Reinforcement Learning Algorithm Here we break down the key background components of the RL approach used for PAC-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Q-learning (Watkins, 1989) is the canonical value-based algorithm which can be applied to a sequential decision making process 27 formalised as an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' It is an off-policy temporal difference algorithm whose goal is to learn the action value function mapping state-action pairs to their expected discounted future return when following a policy π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Qπ(s, u) = Eπ � �∞ t′=t+1 γt′−1r(st′)|st=s, ut=u � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' By definition, an optimal policy π∗ will se- lect an action which maximises the true Q-value Q∗(s, u), π∗(s) = arg maxu′ Q∗(s, u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, the classical Q-learning algorithm maintains an action value look-up table Q(s, u) mapping all possible state-action pairs to their predicted discounted return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The return is the sum of future rewards over the remainder of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' During training, Q-learning follows an exploration-exploitation policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The simplest such policy is ϵ-greedy, where a random action is sampled with probability ϵ ∈ [0, 1] and the best action, according to the current Q table, is sampled with probably 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' At each time step t, the agent in state st uses this policy to select an action ut which it performs in the environment to transition to the next state st+1 and receive a reward rt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Q(s, u) is then updated according to: Q(st, ut) ← Q(st, ut) + α · � rt + γ · max u′ Q(st+1, u′) − Q(st, ut) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2) On the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2, Q(st, ut) is the agent’s estimate of the discounted return of taking action ut in state st, α is the learning rate, γ is the factor by which to discount future rewards to their present value, and maxu′ Q(st+1, u′) is an estimate of the future value of being in state st+1 and taking an ‘optimal’ action according to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The rt + γ · maxu′ Q(st+1, u′) term is called the temporal difference target, and and the collective rt + γ · maxu′ Q(st+1, u′) − Q(st, at) term the temporal difference error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such, the maxu′ Q(st+1, u′) term treats Q as an oracle from which optimal actions can be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Although Q is usually randomly initialised and changes at each update step, the general idea is that, with stable learning and sufficient exploration, Q will converge on the true Q∗ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As a side note, Q-learning is a temporal difference algorithm because, rather than using the actual returns to update Q in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2 as done my Monte Carlo methods, it uses a bootstrapped estimate of the future returns maxu′ Q(st+1, u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Furthermore, it is an off-policy algorithm because the policy used to select the action ut at the current time step, such as ϵ-greedy sampling of Q, is different to the policy used to select the next-state action u′ when evaluating the temporal difference target, such as greedy sampling of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is as opposed to on-policy temporal difference algorithms, such as 28 SARSA, which use the same action selection policy for both the current time step and for future time steps when bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Deep Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Many practical problems have an extremely large number of possible state-action combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For example, the game of Go has over 10700 possible sequences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' far more than the number of atoms in the universe (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such, modelling the action value function with a tabular approach is intractable given practical memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To enable Q-learning to be scaled to complex problems, deep Q-learning (DQN) (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2013) approximates the true Q-function with a DNN parameterised by θ such that Qθ(s, u) ≈ Q∗(s, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, during training at each time step t, Qθ(s, u) is used with an exploration strategy such as ϵ-greedy to select an action and add the observed transition T = (st, ut, rt+1, γt+1, st+1) to a replay memory buffer (Lin, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The network’s parameters θ are then optimised with stochastic gradient descent to minimise the mean squared error loss between the online network’s predictions and a bootstrapped estimate of the Q-value, JDQN(Q) = � rt+1 + γt+1 max u′ Q¯θ(st+1, u′) − Qθ(st, ut) �2, (3) where t is a time step uniform randomly sampled from the buffer and Q¯θ a target network with parameters ¯θ which are periodically copied from the acting online network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The target network is not directly optimised, but is used to provide the bootstrapped Q-value estimates for the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Only periodically updating the target network rather than at each learning step leads to lower variance in the bootstrapped targets at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This helps helps to stabilise learning and leads to better convergence (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Double DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In the traditional Q-learning update rule of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2 and the DQN loss of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3, the Q-function used to select and evaluate an action for the temporal difference target is the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' maxu′ Q(st+1, u′) for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2, and maxu′ Q¯θ(st+1, u′) for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, this can lead to an overestimation bias where the chosen action u′ is incorrectly over-valued because the same function which perceives u′ as being best is also being asked to evaluate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This can lead to high variance updates, unstable learning, and convergence on local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Double DQN (van Hasselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2015) reduces overestimation by decomposing the max operation in the temporal difference target into action selection and action evaluation and performing these two tasks with two separate networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 29 Concretely, action u′ is greedily selected according to the online network Qθ and evaluated with the separate target network Q¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The loss term from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3 then becomes: JDDQN(Q) = � rt+1 + γt+1Q¯θ(st+1, max u′ Qθ(st+1, u′)) − Qθ(st, ut) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (4) Prioritised experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Vanilla DQN replay buffers are sampled uniformly to obtain transitions for network updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' A preferable approach is to more frequently sample transitions from which there is much to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Prioritised experience replay (Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016) deploys this intuition by sampling transitions with probability pt proportional to the last encountered absolute temporal difference error, pt ∝ |rt+1 + γt+1 max u′ Q¯θ(st+1, u′) − Qθ(st, ut)|ω, (5) where ω is a tuneable hyperparameter for shaping the probability distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' New transitions are added to the replay buffer with maximum priority to ensure all experiences will be sampled at least once to have their errors evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' n-step Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Traditional Q-learning uses the target network’s greedy action at the next step to bootstrap a Q-value estimate for the temporal difference target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Alternatively, to improve learning speeds and help with convergence (Sutton and Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hessel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017), forward-view multi-step targets can be used (Sutton and Barto, 2018), where the n-step discounted return from state s is r(n) t = n−1 � k=0 γ(k) t rt+k+1, (6) resulting in an n-step DQN loss of JDQNn(Q) = � r(n) t + γ(n) t max u′ Q¯θ(st+n, u′) − Qθ(st, ut) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (7) Dueling DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Traditional DQN approaches use a DNN architecture which is not specific to RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Subsequently, when learning the Q-function, the entire DNN architecture must learn to estimate the state value and the action advantage for each action in order to learn the state-action function Qπ(s, u) of being in state s, taking action u, and following policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, in 30 many problems where bootstrapped Q-learning is applied, the most important objective is to learn to estimate the value of each state rather than the effect of each action for each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is especially true in environments and individual states where future transitions are mainly influenced by factors other than the agent’s actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Leveraging the insight that in many states it is unnecessary to estimate the value of each action choice, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2015) developed a new DNN architecture, termed ‘dueling DQN’, which is better suited to the Q-learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, the dueling architecture uses the same core DNN as standard DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' However, rather than following the initial encoding with a single sequence of fully connected layers to get a Q-value for each possible action in the current state, dueling DQN uses two separate streams of fully connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' One stream, parameterised by β, estimates the state value function Vθ,β(s) (the estimated future discounted return of the current state regardless of future actions taken), and the other stream, parameterised by α, estimates the relative action advantage function Aθ,α(s, u) (the relative difference in the future discounted return of each action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The outputs of the two streams are then combined via a special aggrega- tion function to recover the state-action value function Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Crucially, V (s) and A(s, u) must be combined into Q(s, u) in such a way that they are indepen- dently identifiable from the output Q values alone in order for backpropagation to be able to calculate the appropriate loss and weight updates for the sep- arate V (s) and A(s, u) streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such, a simple Q(s, u) = V (s) + A(s, u) aggregation function to get the Q-values from the two streams does not suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Instead, the authors tried two different aggregation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The first aggregation method subtracted the advantage of the maximum advantage action from all advantages to make the argmax action’s advantage 0 and the rest < 0, Qθ,α,β = Vθ,β(s) + � Aθ,α(s, u) − max u′ Aθ,α(s, u′) � , (8) thus enabling V (s) to be recovered at the argmax action’s Q-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The second aggregation method subtracted the mean advantage from all action advantages to centre the advantage values around 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' to have a mean of 0), Qθ,α,β = Vθ,β(s) + � Aθ,α(s, u) − 1 |A| � u′ Aθ,α(s, u′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (9) 31 This makes V (s) recoverable from Q(s, u) by estimating the V (s) value which, when subtracted from each A(s, u) value, leads to a set of A(s, u) values which have a mean of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In practice, this second approach of using the mean was found to lead to more stable learning since using a mean operation resulted in lower variance targets between learning steps compared to when a max operation was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As with standard Q-learning, the output of the dueling network is a set of Q-values (one for each action), therefore no change to the underlying algorithm other than a slight adjustment of the network architecture was required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' By decomposing the Q-function approximator in this way, dueling DQN is able to attain superior policy evaluation in the presence of many similar-value actions, and the authors demonstrated their architecture achieving state-of-the-art performance on the Atari 2600 games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Ape-X DQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Noting that state-of-the-art ML performance is often achieved with more computation, more powerful models, and larger training data sets, Horgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018) proposed Ape-X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' a parallelisation approach to off-policy experience replay RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely, rather than using a single actor-learner setup, Ape-X decouples acting from learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' It distributes many actors across a set of CPU cores each with their own instance of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each actor retains a copy of a DNN shared across actors which it uses for action selection to accumulate experiences in parallel with other actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These experiences are then communicated to a central shared replay buffer, where a single learner mounted on a GPU uses prioritised experience replay to sample the most important experiences for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Learner sampling, gradient computation, and network updates are done asynchronously with one another on separate threads, as are the periodic updates made to the actors’ networks with the latest shared learner network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' By using multiple actors in parallel, not only can orders of magnitude more transition data be attained for learning, but also a broader diversity of experiences can be collected by allocating a different exploration strategy to each actor and thereby avoid local optima in difficult exploration and large state-action space settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For Nactors distributed actors, Horgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018) used a per-actor ϵ-greedy exploration strategy whereby each actor i had a fixed exploration probability ϵi = ϵ1+ i Nactors−1 ·α where ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The authors demonstrated their approach achieving new state-of-the-art results on Atari in a fraction of the training time of prior works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 32 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Metric Definitions Table 2 summarises the metric jargon used throughout our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Metric Description Job completion time Time between job arriving and being completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Sequential job completion time Time it would take to complete a job were its operations ran sequentially on a single device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Maximum acceptable job completion time Maximum time allowed to complete a job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Speed-up factor Factor difference between sequential job completion time and actual job completion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Network overhead Fraction of the job completion time spent communicating information be- tween workers when no computation was taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Blocking rate Fraction of the arrived jobs which were successfully serviced across a given period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job information size Summed sizes (in bytes) of a job’s operations and dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Cluster throughput Total partitioned job information pro- cessed per unit time by the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Offered throughput Total original job information pro- cessed per unit time by the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Load rate Amount of job information arriving at the cluster per unit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job inter-arrival time Time between when two jobs arrived at the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Table 2: Descriptions of the various metrics referred to throughout this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Experimental Hardware All environment simulations were ran on Intel Xeon ES-2660 CPUs, and all learner network training and inference was done on either a V100 or an A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 33 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Additional Simulation Details 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Code Structure We built a core RAMP simulation environment which followed a Gym- like interface (Brockman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016) but without inheriting from a Gym environment object to allow additional flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We then built a wrapper ‘job partitioning’ environment which did conform to the Gym interface but used our core RAMP simulation environment to perform the internal RAMP simulation logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Our code base is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' com/cwfparsonson/ddls for further practical implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job Allocation Procedure When a job arrives at the cluster, our environment uses the following ordered sequence of task executions to allocate the job: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' partitioning: Partition the job DAG’s operations to attain a ‘partitioned’ job DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' placement: Place the operations in the partitioned job DAG onto a sub-set of cluster workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' scheduling: For each worker, schedule the priority of its placed operations to resolve conflicts where ≥ 2 operations are ready to be executed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' placement: Given the placed operations and the data de- pendencies which must be exchanged between operations, place the dependencies onto cluster communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' scheduling: For each communication link, schedule the priority of its placed dependencies to resolve conflicts where ≥ 2 dependencies are ready to be communicated at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job Allocation Methods Each of the above allocation procedure tasks can be performed by any algorithm, heuristic, or learning agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In our work, we use the following methods: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' partitioning: PAC-ML, Paramax, Paramin, or Random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' See the main manuscript for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' placement: A first-fit heuristic customised for the requirements of RAMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' See Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 below for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' scheduling: Shortest remaining processing time (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Alizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given a set of operations placed on a worker, the operation with the shortest remaining run time will have the highest priority and therefore be executed first wherever two operations on the same worker request to be executed at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' placement: Shortest path & first-fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given a set of operation placements, for any dependencies which need to be transferred through the network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' for dependencies with size > 0 and whose parent operation is placed on a separate worker from the child operation), (1) first-fit select a path from the k−shortest path with available light channel(s), and (2) first-fit select an available channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' scheduling: Shortest remaining processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given a set of dependencies placed on a communication link channel, the dependency with the shortest remaining processing time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the lowest amount of information left to be transferred) will have the highest priority and therefore be communicated first wherever two dependencies on the same link channel request to be transported at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' First-Fit Operation Placement in RAMP The original RAMP paper of Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022) did not specify an operation placement heuristic which conformed to the RAMP placement rules (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Here, we propose a simple first-fit heuristic which conforms to these rules whilst making the placement problem tractable for large cluster networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The basic idea behind partitioning and placement in the scenario described in this work is to exploit the network efficiencies of RAMP as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' In particular, this means maximising the use of RAMP’s highly efficient collective operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For a generic partitioned DAG, in the backward pass, collectives happen for each operation when weights/gradients are shared between sub-operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If both a parent and child operation are placed on the same set of (RAMP symmetry adherent) workers, then when the parent communicates its output to the child’s input in the forward pass this will also constitute a collective operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As such the placement heuristic implemented here seeks to primarily maximise the amount that these two conditions are encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given some operation, o, that has been partitioned into N equal sub-operations, oi and needs to be placed, the placement is handled as: 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If a parent of o has been partitioned and placed across N servers which adhere to the RAMP symmetry conditions, and if these servers each have enough memory to store oi, then place o across this set of N servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This ensures collective operations can happen in both the forward and backward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Otherwise, check if a set of N workers can be found in the network that adheres to the RAMP symmetry requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This is achieved by sliding the various possible symmetric shapes over the topology until a suitable one (or none) is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' This ensures collective operations in the backward pass only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Allocating in this way ensures that every partitioned operation can exploit RAMP’s efficient collective operation process on the backward pass, and where possible can also exploit it on the forward pass when receiving information from (one of) its parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Evaluating the job completion time The time to complete each operation was taken from the real computation job profiles of the DNN jobs considered (see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To calculate the communication time of point-to-point information transfers and of the MPI collectives, we used the equations and code of Ottino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Possible Causes of a Job Being Blocked A job is blocked when either JCT > β · JCTseq (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' failing to meet user’s chosen JCT requirement) or when the cluster does not have enough available resources to service the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The possible causes of this latter form of blocking are: Prior jobs using up too many cluster resources when later jobs arrive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the minimum operation run time quantum not being low enough to partition the operations enough times to lead to the desired JCT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' mounted worker operation scheduling conflicts for partitioned operations mounted on the same worker leading to longer run times, since one worker can only execute one operation at a time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and excessive communication overheads incurring from over-partitioning of the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 36 Figure 9: Visualisation of the characteristics of the deep learning computation graphs used for our experiments before partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The bottom left sub-figure contains the model colour code scheme for all other sub-figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The statistics shown are for the operations and dependencies which need to be executed and satisfied to conduct one training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore, to carry out Niter training steps, the computation graph would need to be executed Niter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Computation time units are reported in seconds, and memory units in bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Job Computation Graph Data Sets All computation graphs used in our experiments were taken from the open-access PipeDream computation graph data set (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Figure 9 shows a visualisation of the key computation graph characteristics for each neural network model considered, where the numbers reported are for one training iteration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' one forward and backward pass through the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Table 3 reports the same characteristics but in tabular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Finally, for completeness, Figure 10 shows the actual job DAGs of the models used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Neural Network Architecture As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 11, we used a message passing GNN similar to Graph- SAGE with mean pooling (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017) to parameterise the PAC-ML policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Table 4 summarises the hyperparameters used for the components of this DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We note that we did not perform extensive hyperparamter tuning on the GNN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Below is a detailed explanation of this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' First, the GNN layer takes in the DAG’s node and edge features and generates an embedding for each node and edge in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Then, each local node’s nearest neighbour (1-hop away) sends the local node a 37 ×104 ×109 600 - 125 - 400 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 - iido 75 2 - Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 50 - 200 - 25 - 70 ×1010 ×1010 ×109 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5- 150 - Size ( azis # α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content="0→ FO'T Fo 1 80 - Graph 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8 - 60 - Computation 40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 - 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4 E ELL ELL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='2 - 100 108 109 101 102 108 109 Operation Compute Time Operation Memory Dependency MemoryModel # ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' JCTseq Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' time Σ op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Depth # deps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Σ dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' size Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' size ResNet-18 142 36 668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='35 473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='625 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='258 66 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='822 121 2 × 109 60 159 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='733 29 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='822 083 6 × 109 VGG-16 82 34 525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='35 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='330 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='625 30 × 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='644 315 × 109 80 83 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='467 06 × 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='644 167 × 109 GNMT 96 4470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='80 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='368 447 × 109 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='269 491 × 108 30 117 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='027 801 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='194 437 1 × 109 SqueezeNet-10 136 38 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='15 474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='637 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='962 62 × 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='168 007 × 109 102 153 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='910 09 × 109 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='167 950 × 109 AlexNet 46 36 061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='15 635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='902 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='046 234 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='198 339 6 × 109 44 47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='422 161 × 109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='198 246 4 × 109 Table 3: Summary of the characteristics of the deep learning computation graphs used for our experiments before partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The statistics shown are for the operations (‘ops.’) and dependencies (‘deps.’) which need to be executed and satisfied to conduct one training iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore, to carry out Niter training steps, the computation graph would need to be executed Niter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Computation (‘comp.’) time units are reported in seconds, and memory (‘mem.’) units in bytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 38 Figure 10: Deep learning computation graphs used for our experiments before partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each computation graph represents the operations and dependencies which need to be executed and satisfied to conduct one forward and one backward pass through the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Therefore, to carry out Niter training steps, the computation graph would need to be executed Niter times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 39 ResNet-18 VGG-16 GNMT SqueezeNet-10 AlexNet 国国国Figure 11: Schematic of the DNN architecture with |L| GNN layers used to parameterise the policy of PAC-ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The GNN is similar to that of GraphSAGE with mean pooling (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each GNN layer l ∈ L contains a node, edge, and reduce DNN module and ultimately learns to create an embedded representation for each node in a given job DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These per-node embeddings are then passed, along with any global job, cluster, and action features, to a readout module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The readout module ultimately generates scores for each possible action, which enables an action to be selected following a given exploration-exploitation policy being followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For clarity, this figure only shows the GNN embedding-generation process for node 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' See accompanying text for a detailed explanation of this architecture and the accompanying figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 40 (1, 3) Mean pooling Mean pooling [1,2,3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='57 RLlib FC Chosen action Job Clustel Action Node Eade Labelmessage (‘message passing’) which is the neighbouring nodes’ embeddings concatenated with their connected edges’ embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' These messages are stored in the local node’s ‘mailbox’, which now contains information about the node’s neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To ensure consistent dimensioning with the received messages, a dummy zero-padded edge embedding is concatenated with the local node’s embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Next, the reduce module takes the local and message embeddings and generates a reduced representation for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Finally, to generate a layer-l output embedding for the local node, the element-wise mean of the reduced embeddings is taken (‘mean pooling’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that this embedding process is done for each node in the DAG, but for clairty Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 11 only follows node 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If l < L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' if this is not the last GNN layer), these final node embeddings are used as new features for the original DAG’s nodes and are passed to the next GNN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' If l ≡ L, then the node embeddings are passed to the readout module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that (1) the node, edge, and reduce modules are shared across the aforementioned operations within a given GNN layer when generating node embeddings, but not across different GNN layers, and (2) the lth-layer’s output node embeddings will contain information about the node’s neighbourhood from up to l hops away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The readout module takes the GNN’s node embeddings and the job’s and cluster’s global features as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To convert the node-level embeddings of the GNN into a representation of the overall job DAG, their element-wise mean is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To generate an embedding capturing the global job, cluster, and action information, a global DNN module is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The DAG and global embeddings are then concatenated and passed to a logit module, which in turn generates a vector of (optionally masked) scores for each possible action in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Finally, based on these scores and the exploration-exploitation policy being followed, an action is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Reinforcement Learning Algorithm Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Given the stochastic nature of our dynamic cluster environ- ment setting, we hypothesised that a value-based RL method would be best suited to our setting (Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We did try the PPO (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017) actor-critic method but found performance to be worse, although we leave a full analysis of alternative RL algorithms to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' As stated in the main manuscript, we used the state-of-the-art value-based Ape-X DQN RL algorithm (Horgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2018) to attain the PAC-ML policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Concretely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' we used the Ape-X parallelisation approach with double 41 Parameter Value Message passing # hidden dimensions 64 Message passing # output dimensions 32 Reduce module # hidden dimensions 64 Reduce module # output dimensions 64 if l < L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' else 16 Global module # hidden dimensions 8 Global module # output dimensions 8 Logit module RLlib FC net # layers 1 Logit module RLlib FC net # hidden dimensions 256 All modules’ activation ReLU GNN # layers L 2 Apply action mask False Table 4: Hyperparamters used for the PAC-ML ApeX-DQN DNN policy architecture shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Note that the ‘message passing’ dimensions refer to the dimensions of the concatenated node and edge modules’ embeddings, so the dimensions of these modules’ hidden and output embeddings will be half the corresponding ‘message passing’ dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Due to the RLlib implementation of Ape-X DQN, we did not apply an action mask, but instead included the action mask in the global features given to the model and used the reward signal to train the agent to avoid selecting invalid actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Q-learning action selection-evaluation (van Hasselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2015) and multi- step bootstrapped learning targets (Sutton and Barto, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hessel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2017), prioritised experience replay (Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2016), a dueling DQN network architecture (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2015), and a per-actor ϵ-greedy exploration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For a breakdown of each of these components, refer to Appendix 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' To select the algorithm hyperparameters, we con- ducted a Bayesian search across the search space summarised in Table 5, with simulations conducted in a light 32-worker RAMP environment with a maximum simulation run time of 2 × 105 seconds to speed up the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We adopted similar search ranges to those used by Kurach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Parsonson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For each set of hyperparameters, we ran the algorithm for 100 learner steps (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' training epochs), and performed a validation across 3 seeds at each learner step (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We selected the parameter set with the highest episode return across the 3 seeds (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We also report the importance of each parameter with respect to the total episode return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' The importance is calculated by training a random forest with all algorithm hyperparameters as inputs and the episode return as the target output, with the per-feature (hyperparameter) importance values 42 Figure 12: Validation performance of the Ape-X DQN hyperparameter sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Each agent was trained for 100 learner steps, and at each learner step a validation was performed across 3 seeds - the mean metrics with their min-max interval bands are plotted for each hyperparameter set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' predicted by random forest reported accordingly (fab, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' how, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' All our experiments used the same per-actor ϵ-greedy exploration as Horgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' We note that our RL algorithms were implemented using the open-source RLlib library (Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=', 2018) and hyperparameter tuning was done using Weights & Biases (Biewald, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Final Learning Curves For completeness, Figure 13 shows the learning curves of the tuned PAC- ML agents in each βX environment superimposed on the baseline agents’ performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' At each learner step, the PAC-ML agent was evaluated across three seeds in the validation environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Additional Experimental Results Figure 14 shows the performance of the agents in terms of raw blocking rate, throughput, JCT, and JCT speed-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Funding and Acknowledgments Funding EPSRC Distributed Quantum Computing and Applications EP/W032643/1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the Innovate UK Project on Quantum Data Centres and the Future 10004793;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' OptoCloud EP/T026081/1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' TRANSNET EP/R035342/1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the Engineering and Physical Sciences Research Council EP/R041792/1 and EP/L015455/1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' the Alan Turing Institute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and Horizon Europe Dynamos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='9 Return 75 Rate slocking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8 100 isode 125 B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7 150 T T 20 40 60 80 100 20 40 60 80 100 Learner Steps Learner StepsParameter Search Range Best Value Importance Discount factor γ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='99, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='993, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='997, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='9999} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 Learning rate Log-uniform values ( 1 × 10−7, 1 × 10−3 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='121 × 10−7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='045 vmin {−1, −10, −100, −200, −1000} −1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='01 vmax {1, 10, 100, 200, 1000} 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='004 Target network update frequency { 1 × 103, 1 × 104, 1 × 105 } 1 × 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='001 Prioritised replay α {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='9} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='04 Prioritised replay β {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='9} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='047 n-step {1, 3, 5, 10} 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='227 # CPU workers 32 32 − # GPU workers 1 1 − Batch mode Truncated episodes Truncated episodes − Rollout length 50 50 − Train batch size 512 512 − Optimiser Adam Adam − Dueling True True − # atoms 1 1 − Noisy False False − Double Q True True − Replay buffer capacity 100 000 100 000 − Learning starts 10 000 10 000 − Prioritised replay TD-error ϵ 1 × 10−6 1 × 10−6 − Table 5: Ape-X DQN training parameter sweep search range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' best value found,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' and corresponding parameter importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 44 Figure 13: Validation curves of the PAC-ML agent trained in four different β distribution environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' At each learner step (update to the GNN), the agent was evaluated across 3 seeds, with the mean blocking rate, offered throughput, JCT, and JCT speed-up (relative to the jobs’ sequential run time JCTseq) performance metrics reported as well as their min-max confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' For reference, the performances of the baseline heuristic partitioners are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 45 PAC-ML (Ours) Paramac Paramin Random βA βB βc βD Slocking 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='0 Rate 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='5 - 15 15 15 JCT 10 10 10 10 5 5 5 5 S 0 100 200 0 100 200 0 100 200 0 100 200 Learner Steps Learner Steps Learner Steps Learner StepsFigure 14: Validation performances of each partitioning agent evaluated across three seeds, with the mean blocking rate, offered throughput, JCT, and JCT speed-up (relative to the jobs’ sequential run time JCTseq) performance metrics reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' References , 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' Intro to machine learning: Lesson 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' URL: https://www.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' URL: https://openreview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='net/forum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content='id=r-gPPHEjpmw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} +page_content=' 58' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFST4oBgHgl3EQfbTje/content/2301.13799v1.pdf'} diff --git a/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/2301.04667v1.pdf.txt b/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/2301.04667v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c28c971c7b94cd3f695fd38b5bf58e5c5797f663 --- /dev/null +++ b/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/2301.04667v1.pdf.txt @@ -0,0 +1,1522 @@ +MNRAS 000, 1–13 (2023) +Preprint 13 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Unravelling the mass spectrum of destroyed dwarf galaxies with the +metallicity distribution function +Alis J. Deason1,2★, Sergey E. Koposov3,4,5†, Azadeh Fattahi1, Robert J. J. Grand6,7 +1 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK +2 Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK +3Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK +4 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +5 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +6Instituto de Astrofisica de Canarias, Calle Via Lactea s/n, E-38205 La Laguna, Tenerife, Spain +7Departamento de Astrofisica, Universidad de La Laguna, Av. del Astrofisico Francisco Sanchez s/n, E-38206, La Laguna, Tenerife, Spain +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Accreted stellar populations are comprised of the remnants of destroyed galaxies, and often dominate the ‘stellar haloes’ of +galaxies such as the Milky Way (MW). This ensemble of external contributors is a key indicator of the past assembly history +of a galaxy. We introduce a novel statistical method that uses the unbinned metallicity distribution function (MDF) of a stellar +population to estimate the mass spectrum of its progenitors. Our model makes use of the well-known mass-metallicity relation of +galaxies and assumes Gaussian MDF distributions for individual progenitors: the overall MDF is thus a mixture of MDFs from +smaller galaxies. We apply the method to the stellar halo of the MW, as well as the classical MW satellite galaxies. The stellar +components of the satellite galaxies have relatively small sample sizes, but we do not find any evidence for accreted populations +with 𝐿 > 𝐿host/100. We find that the MW stellar halo has 𝑁 ∼ 1 − 3 massive progenitors (𝐿 ≳ 108𝐿 ⊙) within 10 kpc, and likely +several hundred progenitors in total. We also test our method on simulations of MW-mass haloes, and find that our method is +able to recover the true accreted population within a factor of two. Future datasets will provide MDFs with orders of magnitude +more stars, and this method could be a powerful technique to quantify the accreted populations down to the ultra-faint dwarf +mass-scale for both the MW and its satellites. +Key words: Galaxies: dwarf – Galaxy: halo – Local Group – galaxies: luminosity function +1 INTRODUCTION +Dark matter haloes of all shapes and sizes grow by accumulating +lower mass constituents (or subhaloes). The galaxies at the centres +of these haloes grow via ongoing star formation, but can also form +diffuse ‘stellar haloes’ from the stellar material deposited by the +accretion of subhaloes (if they contain stars). Depending on the mass- +scale, this accreted stellar material can amount to significant (e.g. +clusters, ∼ 20 − 30%) or minuscule (e.g. dwarfs, ∼ 0 − 5%) fractions +of the overall stellar mass of the central galaxy (Purcell et al. 2007). +Despite having a relatively low stellar mass and surface brightness, +stellar haloes retain a record of the lower mass systems that have been +digested by haloes over time, and quantifying and understanding this +accreted relic has been a major research focus in astronomy for several +decades (see e.g. Helmi 2008; Belokurov 2013). +The most-studied stellar halo is, unsurprisingly, that of our own +Milky Way (MW) galaxy. However, despite significant progress in +recent years, we still only have a qualitative view of the mass spectrum +of dwarf galaxies that have been consumed by the MW. Most notably, +it has become clear since the game-changing Gaia mission (Gaia +★ E-mail: alis.j.deason@durham.ac.uk (AD) +† E-mail: sergey.koposov@ed.ac.uk (SK) +Collaboration et al. 2016), that the inner stellar halo (within ∼ 20 +kpc) is dominated by one ancient, massive accretion event, dubbed +the Gaia-Enceladus-Sausage (GES, Belokurov et al. 2018; Helmi +et al. 2018). There is also some evidence that an additional massive +structure resides in the very central regions of the galaxy (within ∼ 4 +kpc), and was accreted even earlier than the GES (Kruijssen et al. +2019; Horta et al. 2021a). However, it is debated whether or not this +is really an accreted structure, or rather in-situ MW material (see +e.g. Myeong et al. 2022; Rix et al. 2022). These massive progenitors, +join the already discovered streams and substructures, such as the +Sagittarius and Orphan streams (e.g. Newberg et al. 2003; Majewski +et al. 2004; Belokurov et al. 2007b), and the Virgo (Jurić et al. 2008) +and Hercules-Aquila (Belokurov et al. 2007a) clouds (although the +latter structures may be related to the GES, see e.g. Simion et al. +2019; Chandra et al. 2022), and more stellar structures in the halo are +continuously being discovered (e.g. Naidu et al. 2020). The overall +inventory of the Galactic stellar halo is evolving, but the picture +is far from complete, and we have no quantitative ‘mass-spectrum’ +of destroyed dwarfs akin to the surviving satellite dwarf luminosity +function (Koposov et al. 2008; Tollerud et al. 2008; Drlica-Wagner +et al. 2020; Nadler et al. 2020), which is a pillar of the field. +Many of the halo structures that have been discovered in the +MW are identified in phase-space and/or action-angle space. This, +© 2023 The Authors +arXiv:2301.04667v1 [astro-ph.GA] 11 Jan 2023 + +2 +Deason, Koposov et al. +of course, is where an astrometric mission such as Gaia has en- +abled a deeper understanding of the phase-space structure of the halo +by providing 6D measurements (at least for the inner halo). How- +ever, even with perfect 6D data, robustly identifying distinct halo +substructures is challenging. Indeed, massive progenitors can have +several ‘clumps’ in dynamical spaces which cannot be unambigu- +ously disentangled (e.g. Callingham et al. 2022) and when the stellar +material is fully phase-mixed it becomes more difficult to identify +from the background (e.g. Johnston et al. 2008). Furthermore, even +in the space of conserved quantities the clumps may not stay com- +pact due to perturbations from massive systems such as the LMC +(Koposov et al. 2022b). This is where chemical information can be +crucial, as galaxies of different mass (and star formation history) can +have distinct chemical signatures (e.g. Venn et al. 2004; Tolstoy et al. +2009). Most notable, is the well-known mass-metallicity relation for +galaxies, which extends down to the dwarf mass-scales (e.g. Skillman +et al. 1989; Kirby et al. 2011). +More massive galaxies are, on average, more metal-rich, and the +relation between mass and metallicity exists over several orders of +magnitude in mass (e.g. Tremonti et al. 2004; Kirby et al. 2013). +This relation can, to first order, be explained by the larger gravi- +tational wells of more massive galaxies, which are able to retain +metals (Dekel & Silk 1986). Lower mass galaxies lack the gravity to +resist the expulsion of metals due to feedback mechanisms. On the +dwarf mass-scale, not only does the average metallicity vary with +mass, but the width of the metallicity distribution function (MDF) +also varies, with the lowest mass dwarfs having a wider spread of +metallicities (e.g. Kirby et al. 2011). The combined MDF of a popu- +lation of accreted dwarf galaxies, such as a stellar halo, is therefore +the superposition of several individual MDFs. Thus, in principle, +metallicity measurements alone contain a unique record of the mass +spectrum of accreted dwarfs. Indeed, the disentangling of a MDF +into its individual components is the main focus of this work. Fi- +nally, it is worth noting that previous work on the MDFs of dwarf +galaxies has focused on surviving dwarfs, which, depending on the +largely unknown redshift evolution of the mass-metallicity relation, +may or may not be relevant for the destroyed dwarfs that make-up +stellar haloes (see e.g. Fattahi et al. 2020; Naidu et al. 2022). +In this work, we consider Galactic-sized stellar haloes as well as +the (potential) stellar haloes of dwarf galaxies. In principle, dwarf +galaxies themselves can cannibalise lower-mass dwarfs, and form +what we classically think of as a ‘stellar halo’. However, unlike larger +mass-scales where the merging dark matter clumps all contain stars, +at lower mass scales (below ∼ 109𝑀⊙ in halo mass) dark matter sub- +haloes may not have any stars at all (e.g. Benitez-Llambay & Frenk +2020). A recent study by Deason et al. (2022) showed that the very +existence of a stellar halo around a dwarf galaxy can have important +implications for both small-scale galaxy formation and the nature of +dark matter. For example, the mass-threshold for galaxy formation, +which is largely determined by the epoch of reionization, can have +a major effect on the stellar haloes of dwarf galaxies: for models +with a high mass threshold for galaxy formation (≳ 109𝑀⊙) dwarf +galaxies should not have stellar haloes at all! Thus, the detection +or non-detection of lower mass accretion events surrounding dwarf +galaxies, particularly at the ultra-faint mass scale (𝑀star ≲ 105𝑀⊙), +is of utmost importance. +In order to study the MDFs of accreted populations, we need +large, ideally unbiased, spectroscopic samples with metallicity mea- +surements. For both the Galactic halo, and dwarf satellite galaxies in +the MW, extensive samples are hard to come by, but there has been +significant progress in recent years (e.g. Kirby et al. 2011; Zhao et al. +2012; Kunder et al. 2017; Majewski et al. 2017; Walker et al. 2007; +Conroy et al. 2019; Taibi et al. 2022). Moreover, and importantly, +we are entering a new era of spectroscopic surveys in the MW, with +several projects such as DESI, WEAVE, and 4MOST on the horizon +(Cooper et al. 2022; Dalton et al. 2012; de Jong et al. 2019). Thus, +with these new surveys in mind, we develop a new modeling method +to extract the mass spectrum of accreted components from a sample +of [Fe/H] measurements and apply this to current datasets. +The paper is organised as follows. In Section 2 we outline our +methodology and introduce the statistical model. This is a fairly tech- +nical section that some readers may want to skip over! The method is +applied to spectroscopic samples of classical dwarf satellite galaxies, +and Galactic halo data in Section 3. We test the method on state-of- +the-art cosmological simulations of MW-mass galaxies in Section +4, and discuss caveats and future prospects in Section 5. Finally, we +summarise our main findings in Section 6. +2 MDF MODELING +In this Section, we present the methodology that allows us to take +samples with measured [Fe/H], and some estimate of the total lu- +minosity of the system, and use them to provide constraints on the +number of discrete stellar systems of different luminosities that can +contribute to a given galaxy. +This next Section is fairly technical, so a less statistically-minded +reader may want to skip it and continue with Section 3. The Python +code implementing the inference method presented in this section is +released on GitHub1. +2.1 General statistical model +We construct a generative model that allows us to represent the +metallicity distribution function (MDF) as a mixture of MDFs from +smaller galaxies. Throughout this work, we will assume that the MDF +of each smaller galaxy can be represented by a Gaussian. +The generic model, where the sample of stars for the MDF is +coming from several galaxies, can be described with these model +parameters: +• Number of galaxies N +• 𝐿𝑖 individual galaxy luminosities (where 1 < 𝑖 < 𝑁) +• 𝜇𝑖 mean galaxy metallicities +• 𝜎𝑖 widths of MDF of individual galaxies. +We can then assume that the number of stars in the sample scales +linearly with galaxy luminosity. This assumption is accurate for stel- +lar populations of similar ages. For that assumption to hold, our +sample must not be biased towards one galaxy or another (e.g. if +our sample comes from a small volume that has an unrepresentative +subsample of certain galaxies). If the proportionality holds, one can +write the MDF as +𝑃(𝑧|𝑁, {𝐿𝑖}, {𝜇𝑖}, {𝜎𝑖}) = +1 +� 𝐿𝑖 +𝑖=𝑁 +∑︁ +𝑖=1 +𝐿𝑖N (𝑧|𝜇𝑖, 𝜎𝑖) +(1) +Here, for clarity, we use 𝑧 as a short-hand notation of [Fe/H]. Given +our expectation that galaxy luminosities and metallicities are corre- +lated (Tremonti et al. 2004; Kirby et al. 2011), we can assume that +galaxies follow a mass metallicity relation (or luminosity metallicity +relation) +𝜇𝑖 ∼ N (𝐴 + 𝐵 log 𝐿𝑖|S) +(2) +1 https://github.com/segasai/mdf_modeling_paper +MNRAS 000, 1–13 (2023) + +Destroyed dwarfs with the MDF +3 +where 𝐴 and 𝐵 are constants i.e. taken from the mass metallicity +relation presented in Kirby et al. (2011) and Simon (2019). S is a +constant representing a scatter in the relation (found to be 0.15 dex +by Simon 2019, for MW satellites). +The individual widths 𝜎𝑖 of MDFs differ from galaxy to galaxy +but have been approximated to be slowly dependent on the galaxy +luminosity 𝜎 = 𝐶 + 𝐷 log 𝐿 (see Simon 2019). If we specify the +constants 𝐴, 𝐵, 𝐶, 𝐷, and S we have a model for the distribution +of metallicities, and this model has an integer parameter 𝑁 and +2𝑁 floating point parameters for luminosities and metallicities of 𝑁 +individual galaxies. +While this model for the MDF is valid and can be applied to real +data, it has the problem of having a variable number of parameters +and therefore is difficult to sample in practice (i.e. Green 1995). +Therefore, it would be beneficial to reformulate the model in a way +that makes the number of parameters fixed. +The first modification we can do is to group galaxies in 𝑀 lumi- +nosity bins, so that rather than represent their luminosities by discrete +parameters we represent the number of galaxies in certain luminosity +bins. Now we define: +• ˆ𝐿 𝑗 are the grid of galaxy luminosities 1 ≤ 𝑗 ≤ 𝑀 +• 𝑁 𝑗 are the numbers of galaxies with luminosities ˆ𝐿 𝑗. +• 𝜇 𝑗,𝑘 are mean metallicities of k-th galaxy with luminosity ˆ𝐿 𝑗. +1 < 𝑘 < 𝑁 𝑗 +Where due to mass metallicity relation +𝜇 𝑗,𝑘 ∼ N (𝐴 + 𝐵 log ˆ𝐿 𝑗 |S) +or +𝜇 𝑗,𝑘 = 𝐴 + 𝐵 log ˆ𝐿 𝑗 + S𝜖 𝑗,𝑘 +where 𝜖 𝑗,𝑘 ∼ N (0, 1). Here S could either be a constant or a deter- +ministic function of ˆ𝐿 𝑗 +The MDF model is now +𝑃 �𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}� = +1 +� 𝑁 𝑗 𝐿 𝑗 +𝑗=𝑀 +∑︁ +𝑗=1 +ˆ𝐿 𝑗 +������ +𝑘=𝑁𝑗 +∑︁ +𝑘=1 +N (𝑧|𝜇 𝑗,𝑘, 𝜎𝑗,𝑘) +������ +. +The likelihood of the data consisting of (for simplicity) a single +star with metallicity z would be exactly 𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}). The only +problem with this formulation is that this likelihood still depends +on a variable number of parameters 𝜖 𝑗,𝑘 so one would prefer to +marginalise over these. +𝑃(𝑧|{𝑁 𝑗}) = +∫ +𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘})N ({𝜖 𝑗,𝑘}|0, 1)𝑑𝜖 𝑗,𝑘 +While this marginalisation is difficult, and may be impossible to +do analytically, one can simply perform a Monte-Carlo integration +over 𝑄 samples from a normal distribution, where 𝜖 𝑗,𝑘,𝑞 are the q-th +sample 1 ≤ 𝑞 ≤ 𝑄 from N (0, 1) +𝑃(𝑧|{𝑁 𝑗}) ≈ 1 +𝑄 +𝑞=𝑄 +∑︁ +𝑞=1 +𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘,𝑞}) +Finally, instead of directly doing the summation we can simply +treat this as likelihood with integer parameter 𝑞 +𝑃(𝑧|{𝑁 𝑗}, 𝑞) = 𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘,𝑞}) +(3) +where 𝑞 is a nuisance seed parameter that we marginalise over +−4.0 +−3.5 +−3.0 +−2.5 +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +[Fe/H] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +dP +d[Fe/H] +MV =-4.0 +MV =-4.0 +MV =-10.0 +MV =-10.0 +MV =-10.0, +20 x {-4.0} +Figure 1. The simulated MDFs for a few systems of different luminosities. +The black lines show the expected MDFs in our model for a system with +𝑀𝑉 = −4, with solid and dashed curves showing the MDFs when using a +different random seed that controls the offset of the galaxy with respect to +the mass metallicity relation. Red curves similarly show the MDF of a single +𝑀𝑉 = −10 galaxy with different random seeds. The green curve shows the +MDF for a synthetic galaxy that consists of stars coming from one galaxy +with 𝑀𝑉 = −10 and 20 galaxies with 𝑀𝑉 = −4. +under the uniform prior 𝑈(1, 𝑄). Here, we assume that 𝜖 𝑗,𝑘,𝑞 are +coming from a pseudo-random number generator that is seeded by +𝑞 and provides normally distributed samples. We then will need to +sample the posterior over {𝑁 𝑗} and 𝑞, which gives the model with +𝑀 + 1 parameters. Armed with Eqn 3 that specifies the likelihood +function for the metallicity distribution, the only missing ingredient +for the model are the priors. +We assume that occupation numbers {𝑁𝑖} (i.e. numbers of galaxies +in luminosity bins) have a prior distribution of ⌊10𝑥⌋ where 𝑥 ∼ +𝑈(−1, 4). This is essentially the log uniform of integers distribution +with 20% prior volume at 𝑁𝑖 = 0 and 20% for 1 ≤ 𝑁𝑖 ≤ 10, and +20 ≤ 𝑁𝑖 ≤ 100 etc. +Finally, we complement the model with the constraint on the total +luminosity of the system. Specifically, we require that the combined +luminosity of multiple galaxies must match certain known total lu- +minosity log 𝐿tot with some uncertainty 𝜎𝐿. This provides a term for +the log of the posterior. +log( +∑︁ +𝑁𝑖 ˆ𝐿𝑖) ∼ 𝑁(log 𝐿tot|𝜎𝐿) +A final remark that despite the introduction of the formalism based +on binned number of galaxies, we have found the model is more stable +when at least one contributor to the MDF (likely the one being the +most massive main progenitor) is represented directly (rather than +in a bin) by the satellite luminosity 𝐿main, metallicity 𝑧main and that +also obeys the mass-metallicity relation. +To illustrate our modeling approach, in Figure 1 we show the +expected [Fe/H] distributions given our model. Specifically, solid +black and red curves show possible MDFs for a single galaxy of +𝑀𝑉 = −4 and 𝑀𝑉 = −10, respectively. Dashed lines of same colours +show the MDFs when different random seeds are used. The green +curve shows a distribution that we might expect if we observe stars +coming from a single 𝑀𝑉 = −10 galaxy and 20 𝑀𝑉 = −4 systems. +This shows a prominent tail towards low metallicities, and this is +exactly what allows us to probe the number of possible mergers with +low luminosity systems. +MNRAS 000, 1–13 (2023) + +4 +Deason, Koposov et al. +2.2 Sampling +In the previous section, we have introduced the likelihood function +for the metallicity distribution that is conditional on the number of +different dwarf galaxies 𝑁 𝑗 on a grid of luminosities. The model +also has an integer seed parameter 𝑞. It is not trivial to sample inte- +ger parameters, especially if we expect multiple modes. To perform +the sampling we decide to use the dynamic nested sampling as im- +plemented in the dynesty package (Speagle 2020; Koposov et al. +2022a). As nested sampling is technically invalid if the likelihood +surface has plateaus (Fowlie et al. 2020), we add a a small level of +deterministic noise with standard deviation of 0.01 to the likelihoods, +which should not affect the inference. +3 APPLICATIONS +We now apply the method described above to observational data. +Here, we focus on the classical MW satellites (§3.1) and the MW +stellar halo (§3.2). +3.1 Classical dwarf satellite galaxies +We start from the homogeneous sample of dwarf galaxy members +presented in Kirby et al. (2011) as provided in the Strasbourg astro- +nomical Data Center (CDS). As mentioned in the previous section, +the key assumption that we rely on for our method is that the abun- +dances that we model are random samples from the system. This is +likely not technically correct for the data at hand since the stellar +samples in dwarfs tend to be biased towards the centres of systems +(see e.g. Walker & Peñarrubia 2011), and may have slight metallic- +ity biases caused by the colour-magnitude selection of spectroscopic +targets. We will, however, proceed ignoring these issues. +We take the sample of stars from Kirby et al. (2011) and only +consider stars with small metallicity uncertainty 𝜎[Fe/H] < 0.2. This +catalogue has measurements of 10 MW satellites with more than +10 stars: Canes Venatici I, Draco, Fornax, Hercules, Leo I, Leo II, +Sculptor, Sextans, Ursa Minor and Ursa Major I. We then proceed to +model each of the dwarfs with the machinery presented in Section 2. +We take the luminosities of each system from McConnachie (2012) +(using an updated catalogue from January 2021) and adopt an 𝑀𝑉 +uncertainty for each system of 0.1 mag. For each system, we use +the luminosity bins that are 1 magnitude wide from 𝑀𝑉 = 0 to the +luminosity of the dwarf itself minus 2.5 magnitudes. +The posterior samples on the number of possible dwarf galaxies +that contributed to the systems’ MDF are shown in Figures 2 and +3. We show measurements for 8 out of 10 systems spanning the +luminosity range from 𝑀𝑉 ∼ −5 for Ursa Major I to 𝑀𝑉 ∼ −13 +for Fornax. The panels are ordered by system luminosity. The total +number of stars varies from 𝑁 = 15 for Ursa Major I to 𝑁 = 789 +for Leo I. Figure 2 shows the constraints on the differential number +of systems that have contributed to the dwarfs’ MDF, while Figure 3 +shows constraints on the cumulative counts of the number of systems +brighter than a certain value. The blue/orange bands show the 16/84 +and 1/99 percentiles, and the black line shows the median of the +posterior. The green bands show the constraints if we do not use +metallicities at all. This is essentially a prior and corresponds to the +case where the only constraint comes from ensuring the combination +of galaxies matches the total luminosity of the system. Note that, +because we include all the stars in the galaxy, we expect to measure +𝑁merged = 1 at around the total luminosity of the dwarf galaxy (shown +with the solid red line). Although technically this is an ‘in-situ’ rather +than an accreted component, what we are actually constraining are +the contributors to the MDF, regardless of their origin. +We now look at the posteriors in more detail. First, we focus on +clear cases where the data is particularly constraining. These are the +cases of Fornax, Leo I, Leo II, and Draco, where the spectroscopic +samples have hundreds of members. We see that their differential +posterior distributions (Figure 2) have a peak with a value of one +next to the system luminosity (highlighted in red) and show the value +consistent with zero for 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host+5. Thus, the data +suggests that these systems did not experience a merger with a dwarf +that is larger than 1% of the system luminosity. This is also seen in the +cumulative plots, where we see the implied number 𝑁merged(< 𝑀𝑉 ) +is flat and equal to one in the range 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host + 5. +Looking at the implications for the number of faint contributors to +the MDF for Fornax, Leo I, Leo II, and Draco systems, we can see +that our constraints on 𝑁merged shoot up and become significantly +broader. The differential counts are essentially unconstrained. For +example, for the Fornax MDF contributors at 𝑀𝑉 = 0 (top left panel +of Figure 2) the 1-𝜎 confidence interval is 0 < 𝑁merged < 100 as the +data allows many faint dwarfs before the observed MDF is affected +significantly. The behavior of the cumulative counts for the faint +MDF contributors is somewhat misleading as it rises at faint 𝑀𝑉 +purely because we are summing over bins with non-negative values. +Fainter dwarf galaxies like CVnI or UMa have a smaller number of +spectroscopic observations. In Figure 2 we see that the posteriors on +the number of MDF contributors start to rise next to 𝑀𝑉 = 𝑀𝑉 ,host, +which indicates that we cannot even rule out that the galaxy is a +product of a merger of two systems with similar luminosities. The +constraints on the cumulative number of mergers for fainter dwarfs +do not show a flat 𝑁merged = 1 part next to the luminosity of the +system and instead rises to faint luminosities. We also see that for +faint systems, the posteriors basically look very close to priors. +3.2 Galactic stellar halo +We next apply our method to the Galactic stellar halo. It has been +realised for some time that the stellar halo of the MW comprises +an assortment of destroyed dwarf debris, and thus the metallicity +distribution of these halo stars retains a memory of their dwarf galaxy +progenitors. +Large, homogeneous samples of halo stars with metallicity mea- +surements are hard to come by, and this is a significant limitation of +our current study. At present, we build a sample of halo stars based +on several spectroscopic surveys and use the latest Gaia data (Gaia +Collaboration et al. 2021), to help select a clean halo sample. We +begin by cross-matching stars with spectroscopic data from SDSS +(Abolfathi et al. 2018), RAVE (Kunder et al. 2017), LAMOST (Zhao +et al. 2012), APOGEE (Majewski et al. 2017), and GALAH (Buder +et al. 2021) with Gaia DR3. This results in 𝑁 = 656, 0819 stars. To +estimate distances to the stars we use the Bailer-Jones et al. (2021) +photogeometric distances computed from Gaia EDR3. We only con- +sider stars with reasonable parallax 𝜎𝜛/𝜛 < 0.5, and restrict our +sample to 𝑟 < 10 kpc and |𝑧| > 1 kpc. Finally, to avoid disk con- +tamination, we apply a cut on the rotational velocity of the stars. We +impose a fairly strict cut to remove the majority of thick disk and/or +splash stars (Belokurov et al. 2020), and only include those with ret- +rograde orbits 𝑣𝜙 < −50 km/s. The resulting spatial (top-panel) and +metallicity distribution (bottom-panel) of the stars are shown in Fig. +4. In the bottom panel, we also show the MDF for the stars without the +𝑣𝜙 cut in grey. Our restriction to retrograde orbits is fairly stringent +but, as can be seen in the figure, it is effective at removing disk stars, +which have prograde orbits and are generally more metal-rich. We +MNRAS 000, 1–13 (2023) + +Destroyed dwarfs with the MDF +5 +−10 +−5 +100 +101 +102 +Nmerged +For +−10 +−5 +MV +LeoI +−10 +−5 +MV +LeoII +−10 +−5 +MV +Sex +−10 +−5 +MV +100 +101 +102 +Nmerged +UMi +−10 +−5 +MV +Dra +−10 +−5 +MV +CVnI +−10 +−5 +MV +UMaI +Figure 2. The inferred contributions from systems to the MDF of different dwarf galaxies from our analysis. In each panel, the black curve shows the median +number of galaxies of a given luminosity that could have contributed to the MDF. The blue and orange bands show the 16/84 and 1/99 percentiles, respectively. +The green band shows the sampling of the prior with only the constraint on total luminosity of the system. The vertical red line on each panel shows the +luminosity of each system. Note that the logarithmic y-axis is cut-off at 𝑁merged = 10−1, so median values at this level are consistent with zero. +−10 +−5 +100 +101 +102 +Nmerged(< MV ) +For +−10 +−5 +MV +LeoI +−10 +−5 +MV +LeoII +−10 +−5 +MV +Sex +−10 +−5 +MV +100 +101 +102 +Nmerged(< MV ) +UMi +−10 +−5 +MV +Dra +−10 +−5 +MV +CVnI +−10 +−5 +MV +UMaI +Figure 3. The inferred contributions to the MDF from our analysis. This is similar to Figure 2 but shows the cumulative numbers. Each panel shows a different +dwarf galaxy. In each panel, the black curve shows the median number of galaxies of a given luminosity or brighter that could have contributed to the MDF. +The blue and orange bands show the 16/84 and 1/99 percentiles, respectively. The green band shows the sampling of the prior with only the constraint on total +luminosity of the system. The vertical red line on each panel shows the luminosity of each system. +apply our modelling procedure to stars with −4 < [Fe/H] < −0.82, +and 𝜎([Fe/H]) < 0.2, which results in a sample of 𝑁 = 21, 813 stars. +2 Note that this metallicity cut is made in both the data and model, so there +is no metallicity bias introduced with our selection. +Our sample is comprised of 5 different spectroscopic surveys, with +varying selection functions. Here, we aim to maximise the number of +halo stars with metallicity measurements by combining these surveys +but note that, ideally, a more homogeneous sample would be used. +For now, we continue on, under the assumption that there are no +MNRAS 000, 1–13 (2023) + +6 +Deason, Koposov et al. +0 +2 +4 +6 +8 +10 +R [kpc] +-10 +-5 +0 +5 +10 +|z| [kpc] +0 +2 +4 +6 +8 +10 +-10 +-5 +0 +5 +10 +-4 +-3 +-2 +-1 +0 +1 +[Fe/H] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +dN +All +vφ < -50 km/s +log(g) < 3.5 +log(g) > 3.5 +Figure 4. Top panel: The spatial distribution in the 𝑧 vs. 𝑅 plane of our MW +halo sample. Bottom panel: The metallicity distribution of the sample. The +grey line indicates the MDF without a cut in 𝑣𝜙, which leads to the inclusion of +(metal-rich) disk stars. The dashed red line indicates the median metallicity +of our halo sample ([Fe/H] = −1.5). We also show the MDFs of our halo +sample split by log(𝑔) with the blue and purple dotted lines, respectively. +The gray line-filled region indicates the metal-rich regime ([Fe/H] > −0.8) +that is excluded in our modelling. +significant metallicity biases in this combined sample. However, we +stress that future work with upcoming spectroscopic surveys such as +DESI (Cooper et al. 2022) and WEAVE (Dalton et al. 2012) will be +much better suited for this type of analysis. +In our analysis, we adopt a total halo luminosity of 𝑀𝑉 += +−17.7 ± 0.5. This is consistent with recent measurements which +suggest 𝐿 ∼ 1 × 109𝐿⊙, and also allows a range around this value +encompassing the majority of observational constraints and their un- +certainties (Deason et al. 2019; Mackereth & Bovy 2020; Horta et al. +2021b). The top panels of Fig. 5 show the resulting number of de- +stroyed dwarfs in the MW halo as a function of 𝑀𝑉 . We consider +dwarfs with 0 > 𝑀𝑉 > −22, using 22 bins with 1 mag bin size. Note +that the size of our sample means that we are unlikely constraining +dwarfs with 𝑀𝑉 ≳ −10, which will not be represented by a large +enough number of stars (see e.g. Section 5.2). In blue, we show the +results when the Kirby et al. (2011) mass-metallicity relation is used, +which is appropriate for surviving dwarf galaxies in the MW. In re- +cent work, Naidu et al. (2022) (see also Fattahi et al. 2020) argue +that destroyed dwarfs may not lie on this relation, and a relation with +∼ 0.3 dex offset to lower metallicities is more appropriate. We show +the results with this offset applied in orange. +Our model predicts several hundred (𝑁 ∼ 400) destroyed dwarfs +with 𝑀𝑉 ≲ −10. However, the different mass-metallicity relations +(relevant for either ‘surviving’ or ‘destroyed’ dwarfs) predict different +distributions of progenitor masses, particularly at larger masses. For +example, when using the Kirby et al. (2011) mass-metallicity relation +applicable for surviving dwarf galaxies, we estimate 𝑁 = 1 massive +dwarf progenitor with 𝐿 ∼ 108.5𝐿⊙, but this rises to 𝑁 = 3 when the +relation more relevant to destroyed dwarf galaxies is used instead. +This seems to be at odds with our adopted total halo luminosity of 𝐿 ∼ +1 × 109𝐿⊙. Indeed, by summing the predicted numbers of destroyed +dwarfs we find that the total luminosity when the Kirby et al. (2011) +relation is used is 1.1+0.2 +−0.2 × 109𝐿⊙, but this rises to 3.4+7.2 +−2.3 × 109𝐿⊙ +when an 0.3 dex offset is applied to the mass-metallicity relation. +Clearly, in this latter case, the bias in metallicity has pushed the +progenitor masses higher, and, because we have allowed a fairly +flexible total luminosity, resulted in a high halo luminosity. However, +it is still consistent with the input luminosity within 1 − 𝜎. +In the bottom panel of Fig. 5 we show the results when the total +halo luminosity is fixed to 𝑀𝑉 = −17.7 (technically, an uncertainty +of 0.01 dex is adopted). Here, the ‘fiducial’ result using the Kirby +et al. (2011) mass-metallicity relation is only slightly changed. For +example, the most massive progenitor is shifted to a slightly lower +luminosity (by ∼ 1 dex in 𝑀𝑉 ), and the total number of dwarfs with +𝑀𝑉 < −10 is reduced (𝑁 ∼ 300). In general, the changes are within +the predicted uncertainties. When an 0.3 dex metallicity offset is +applied to the mass-metallicity relation, fixing the halo luminosity +has a greater effect. This is unsurprising given that allowing for a +more flexible halo luminosity favours a higher value than the fiducial +1×109𝐿⊙. In this case, the most massive progenitor has 𝐿 ∼ 108.1𝐿⊙ +(compared to 𝑁 ∼ 3 with 𝐿 ∼ 108.5𝐿⊙ when the total luminosity +is more flexible). The number of low mass dwarfs is also reduced, +with 𝑁 ∼ 110 with 𝑀𝑉 < −10. This exercise emphasizes how +important the assumed total halo luminosity, as well as the adopted +mass-metallicity relation are for this type of analysis. +We also show the surviving dwarf satellite luminosity function for +comparison in Fig. 5. Here, we show the observed (solid purple) and +completeness-corrected (dashed purple) cumulative number counts +given by Drlica-Wagner et al. (2020). The numbers of low luminosity +satellite systems are much lower than the predicted number of de- +stroyed dwarfs. This is perhaps unsurprising given that our estimates +are likely overestimated at low luminosities, owing both to sample +size and our assumption of Gaussian MDFs (see Section 5.1). Inter- +estingly, the (cumulative) number counts are similar at intermediate +luminosities (−16 ≲ 𝑀𝑉 ≲ −12) but destroyed dwarfs as massive +as the LMC are not favoured unless the adopted mass-metallicity +relation is adjusted from the fiducial 𝑧 = 0 form. +Fattahi et al. (2020) show using the Auriga simulation suite that +the number of destroyed dwarfs in MW-mass haloes is larger than +MNRAS 000, 1–13 (2023) + +Destroyed dwarfs with the MDF +7 +20 +15 +10 +5 +0 +MV +10 +1 +100 +101 +102 +103 +N(merged) +MW halo: MV = +17.7 ± 0.5 +20 +15 +10 +5 +0 +MV +10 +1 +100 +101 +102 +103 +N(merged < MV) +Satellite dwarf LF +Mass-Metallicity relation: +Kirby+2011 +Kirby+2011 +0.3 dex +20 +15 +10 +5 +0 +MV +10 +1 +100 +101 +102 +103 +N(merged) +MW halo: MV = +17.7 (fixed) +20 +15 +10 +5 +0 +MV +10 +1 +100 +101 +102 +103 +N(merged < MV) +Satellite dwarf LF +Mass-Metallicity relation: +Kirby+2011 +Kirby+2011 +0.3 dex +Figure 5. The estimated differential (left) and cumulative (right) number of destroyed dwarfs in the MW halo. The dark(light) shaded regions show the +16-84(1-99) percentiles, and the solid lines are the medians. The dashed black line indicates the assumed total stellar halo luminosity (𝑀𝑉 = −17.7). In the top +panels, the total luminosity has a flexible uncertainty of ±0.5 dex, whereas in the bottom panel the total luminosity is kept fixed. The results in blue are for when +the 𝑧 = 0 mass-metallicity relation for dwarfs is assumed (Kirby et al. 2011). In orange, we show the results when an −0.3 dex offset is applied to the relation, +which has been postulated to be more applicable to destroyed dwarfs (Naidu et al. 2022). For comparison, we show the surviving dwarf satellite luminosity +function in purple. The dashed line indicates the completeness-corrected LF derived by Drlica-Wagner et al. (2020). +the number of surviving satellites, at least down to 𝑀𝑉 ∼ −8. This is +in agreement with our results, however, our estimated total number +of destroyed dwarfs is far higher than these models (by a factor of +∼ 3 − 10, see also Fig. 7). This could be a genuine tension with +the models, but it is worth stressing that our number estimates at +low luminosities are likely biased high, and the numbers could be +reduced if we had larger sample sizes and/or the metal-poor tails of +higher mass systems are taken into account (see Section 5.1). +Finally, given the heterogeneous nature of our sample of halo stars, +we consider how different cuts in surface gravity affect the results. +Namely, dwarf stars and giants can have different metallicity biases, +and probe different volumes in magnitude-limited surveys. The MDF +of our halo sample split by log(𝑔) was shown in Fig. 4. Here, we +MNRAS 000, 1–13 (2023) + +8 +Deason, Koposov et al. +22 +20 +18 +16 +14 +12 +10 +8 +6 +4 +2 +0 +MV +10 +1 +100 +101 +102 +103 +N(merged < MV) +MV = +17.7 (fixed) +All +log(g) > 3.5 +log(g) < 3.5 +Figure 6. The estimated cumulative number of destroyed dwarfs in the MW +halo. Same as Fig. 5, but split into two bins with low (log(g) < 3.5) and high +(log(g) > 3.5) surface gravity stars. The thick gray line indicates the overall +sample. The stellar halo luminosity is fixed (𝑀𝑉 = −17.7). +can see there are slight differences for low and high log(𝑔), and now +we consider how our inferred number counts of destroyed dwarfs +are affected. The cumulative number of destroyed dwarfs is shown +in Fig. 6 with two different bins of log(𝑔), appropriate for dwarf +stars (log(𝑔) > 3.5) and giants (log(𝑔) < 3.5). It is worth bearing in +mind that our overall sample is dominated by the high surface gravity +dwarf stars (approximately ∼ 2/3 have log(𝑔) > 3.5). Note that here +we only use the Kirby et al. (2011) mass-metallicity relation, and +the total halo luminosity is fixed. Encouragingly, the total number of +progenitors (for 𝑀𝑉 ≲ −10) is very similar for the two bins of log(𝑔). +However, massive progenitors (𝐿 ≳ 108𝐿⊙) are only favoured in the +high log(𝑔) sample. This is likely because the MDF is biased towards +lower metallicities for the giant star sample (see Fig. 4). Moreover, +the giant and dwarfs are probing slightly different volumes, with +the high surface gravity dwarfs more concentrated around the solar +neighbourhood. This exercise highlights the difficulty of using a +‘hodge-podge’ of halo stars for our analysis, and it will clearly be +preferable for future work to have a more homogeneous sample, +where the selection function is clearly defined. +4 AURIGA SIMULATIONS +Our modeling procedure makes various assumptions and simplifica- +tions. For example, it assumes each progenitor galaxy is sampled in a +representative way, and that their MDFs are adequately described by a +Gaussian distribution. In reality, this may not be the case, particularly +for volume-limited Galactic-sized stellar haloes. To this end, we test +our model on simulated MW stellar haloes, which are representative +of ‘realistic’ accreted populations. We apply our modeling procedure +to halo stars in the Auriga simulations (Grand et al. 2017); these cos- +mological hydrodynamical simulations are a suite of 𝑁 ∼ 30 high +resolution (𝑚 𝑝 ∼ 5×104𝑀⊙) MW-mass (1−2×1012𝑀⊙) haloes. In +this work, we make use of the 𝑁 = 28 haloes studied in Fattahi et al. +(2019), which omits two haloes currently undergoing major mergers. +We only consider accreted halo stars, which are identified in Fattahi +et al. (2019) as those that formed in subhaloes other than the main +progenitor galaxy. +For each halo, we construct a sample of halo star particles within +𝑟 < 20 kpc. This is chosen to roughly mimic the volume limit of +current observations, and ensure large enough sample sizes. The in- +put into the model is the [Fe/H] values of the stellar particles. Of +course, in the simulations, we also know the progenitor galaxy of +each star particle, and can thus test the estimated mass spectrum of +accreted dwarfs from our modeling procedure. The final ingredient +we need to define is the mass-metallicity relation for the Auriga sim- +ulations. Grand et al. (2021) show that the mass-metallicity relation +for dwarf galaxies in Auriga is in good agreement with low mass +dwarfs (𝑀star ∼ 106𝑀⊙), but is too metal-rich by ∼ 0.5 dex for more +massive dwarfs (see Figure 13 in Grand et al. 2021). We use all the +destroyed dwarf progenitors across the 𝑁 = 28 Auriga haloes to cali- +brate this relation3. However, we do exclude dwarfs that are accreted +recently (less than 5 Gyr ago) as these can have significantly different +metallicities due to ongoing star formation. The debris from these +events is still included in the analysis, but our calibration is only based +on the relatively old dwarf galaxies. Note that we only consider dwarf +progenitors with 𝑀𝑉 > −7, which corresponds to a stellar mass of +𝑀star > 105𝑀⊙ or 𝑁 > 2 star particles. We use the ‘peak’ stellar +mass of each dwarf, which corresponds to the maximum stellar mass +the progenitor has reached. Note that we get similar results if the +stellar mass at infall is used instead. The resulting mass-metallicity4 +relation for Auriga is: [Fe/H] = −1.69 + 0.39 × (log10𝐿 − 6). To es- +timate the scatter around this mean relation, we calculate the scatter +for each individual halo, and use the median value across all haloes. +This results in a scatter around the mean [Fe/H] relation of 0.3 dex. +Finally, we consider the spread in [Fe/H] for individual dwarfs. Un- +like the observations, we find no strong evidence for a variation with +dwarf mass, so instead adopt a constant dispersion of 0.4 dex of +the MDF for all dwarfs. Armed with the mass-metallicity relation +appropriate for Auriga, we can now test our modeling procedure on +these cosmological haloes. +When applying our method to the Auriga haloes, we assume the +total luminosity of the halo is known. This of course results in addi- +tional uncertainty in the real observations, but we particularly want to +investigate the systematic influences present in the cosmological sim- +ulations. We consider accreted dwarfs in the range −7 > 𝑀𝑉 > −22, +and estimate the number of dwarfs in 15 bins with bin size of 1 mag. +Fig. 7 shows the resulting cumulative number of destroyed dwarfs +in the Auriga haloes. Each panel shows a different halo, and our +estimated numbers are shown with the solid black lines (median), +and blue/orange shaded regions (16-84/1-99 percentiles). The points +with error bars are the true values, with Poisson noise adopted for +the uncertainties in each 𝑀𝑉 bin. Note that the ‘true’ values include +all dwarfs that have deposited any material within 20 kpc of the +host halo. Thus, there can be cases where only a small fraction of +a destroyed dwarf is included in the sample (see below). The green +values in Fig. 7 are for all progenitors, while the purple are only +those accreted earlier than 5 Gyr ago. In many cases, there is little +difference between the green and purple values, because most dwarfs +are accreted at earlier times. However, we highlight the most recently +accreted dwarfs because these are likely not fully phase-mixed, and +can significantly deviate from the mass-metallicity relation for (old) +dwarf galaxies in Auriga (see above). In reality, we find that these +recently accreted dwarfs only cause a significant effect if the progen- +itors are relatively massive (e.g. Halo 25). +3 To clarify, all destroyed dwarfs are used, not just those that have debris +within 20 kpc of the host halo +4 Note that we assume a stellar mass-to-light ratio of (𝑀/𝐿) = 2 to convert +stellar mass to luminosity. +MNRAS 000, 1–13 (2023) + +Destroyed dwarfs with the MDF +9 +1 +10 +100 +N(merged < MV) +halo_2 +halo_3 +halo_21 +halo_23 +halo_1 +halo_22 +halo_26 +1 +10 +100 +N(merged < MV) +halo_4 +halo_5 +halo_27 +halo_19 +halo_25 +halo_7 +halo_6 +1 +10 +100 +N(merged < MV) +halo_24 +halo_30 +halo_18 +halo_15 +halo_29 +halo_28 +halo_14 +10 +15 +20 +MV +1 +10 +100 +N(merged < MV) +halo_16 +10 +15 +20 +MV +halo_8 +10 +15 +20 +MV +halo_9 +10 +15 +20 +MV +halo_17 +10 +15 +20 +MV +halo_13 +10 +15 +20 +MV +halo_12 +10 +15 +20 +MV +halo_10 +Figure 7. The estimated cumulative number of destroyed dwarfs for the 𝑁 = 28 Auriga haloes. The solid black line shows the median, and the shaded +blue(orange) regions the 16-84(1-99) percentiles. The red dashed line indicates the assumed total luminosity of the halo. For each halo, accreted star particles are +selected within 𝑟 < 20 kpc. The points with (Poisson) error bars indicate the ‘truth’, with all progenitors shown in green, and only those accreted > 5 Gyr ago +in purple. The latter are shown because recently accreted dwarfs are likely (i) not fully phase-mixed, and (ii) can significantly deviate from the mass-metallicity +relation for (old) dwarf galaxies in Auriga. +We discuss these results more quantitatively below, but first cast +a qualitative eye on Fig. 7. In general, our estimates agree well with +the true mass spectrum of accreted dwarfs. However, in some cases, +there can be notable differences. We find that the most significant +deviations are due to the following: (1) relatively massive progenitors +that lie off the mass-metallicity relation (e.g. Halo 2, 6) and/or (2) +progenitors with a low fraction of their material within the given +radial range (e.g. Halo 15, 27). These systematics, and sometimes the +combination of both, are most likely to cause our method to fail. On +the other hand, there are a significant number of haloes for which we +recover the mass spectrum very well, which is encouraging given the +complexity of these hydrodynamic simulations, and the cosmological +nature of their assembly histories. +In Fig. 8 we give a more quantitative summary of our tests of +the Auriga haloes. Here, for each halo (identified in the x-axis) we +show the fraction of 𝑀𝑉 bins that have number estimates that agree +within the 16-84, 5-95, and 1-99 percentile confidence limits. The +median recovery fractions across all haloes are 0.61, 0.77, and 0.83, +respectively. These fractions are below the expected fractions for a +‘perfect’ procedure, but this is unsurprising given the various sys- +tematic influences present in the simulations, such as deviations from +the adopted mass-metallicity relation and the presence of stellar de- +bris that does not fully occupy the available phase-space. These, of +course, are realistic effects that could be present in the observational +data. +In Fig. 9 we explore the halo-to-halo scatter more closely. In the +left-hand panel, we show the difference between the estimated and +1 +2 +3 +4 +5 +6 +7 +8 +9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 +Halo ID +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction +16-84 +5-95 +1-99 +Figure 8. Quantifying the test with Auriga haloes. For each halo, we show +the fraction of 𝑀𝑉 bins (1 mag wide) that have estimated numbers that +agree within the 16 − 84 (red-filled circles), 5 − 95 (blue-filled squares), and +1−99 (green-filled diamonds) percentage confidence limits, respectively. The +median values are shown with the horizontal coloured lines. +true cumulative numbers of destroyed dwarfs as a function of 𝑀𝑉 . +The black line shows the median of the 𝑁 = 28 haloes, and the +blue and orange shaded regions show the 16-84 and 1-99 percentiles, +respectively. The deviation from the true numbers is fairly symmet- +MNRAS 000, 1–13 (2023) + +10 +Deason, Koposov et al. +22 +20 +18 +16 +14 +12 +10 +8 +MV +40 +20 +0 +20 +40 +N( −10). We find that MW stellar halo +samples with 𝑁 ∼ 106 tracers will allow us to probe down to 𝑀𝑉 > +−10; encouragingly, this should be feasible with upcoming surveys +such as DESI and WEAVE. Moreover, with sample sizes exceeding +𝑁 ∼ 5000 we should be able to probe the lower mass accretion events +associated with classical dwarf satellites in the MW. Our ability to +probe down to these puny stellar systems will enable us to address +fundamental questions about galaxy formation at the lowest mass +scales and, potentially, the nature of dark matter. +We have shown that using only the MDF of an (accreted) stellar +population, the mass-spectrum of its progenitors can be uncovered. +This is encouraging for the upcoming generation of spectroscopic sur- +veys of the MW. However, a possible extension of this work would +be to combine the MDF modeling with phase-space data and/or ad- +ditional chemical dimensions (see e.g. Cunningham et al. 2022). +The addition of dynamical information could provide tighter con- +straints on the luminosity function of destroyed dwarfs. In particular, +where the MDF modeling is weakest, i.e. when the stellar material +is un-mixed in phase-space, is likely where the dynamical data is the +most informative. Moving forward, modeling in the chemodynami- +cal space is the next logical step, and, importantly, we will have the +data to do this. Thus, it is clear that future datasets combined with +modeling methods such as that presented here will provide all the +tools needed to finally quantify the accretion history of the Galaxy +and its satellite population. +ACKNOWLEDGEMENTS +AD is supported by a Royal Society University Research Fellow- +ship. AD acknowledges support from the Leverhulme Trust and the +Science and Technology Facilities Council (STFC) [grant numbers +ST/P000541/1, ST/T000244/1]. AF is supported by a UKRI Future +Leaders Fellowship (grant no MR/T042362/1). RG acknowledges +financial support from the Spanish Ministry of Science and Innova- +tion (MICINN) through the Spanish State Research Agency, under +the Severo Ochoa Program 2020-2023 (CEX2019-000920-S). +This work used the DiRAC@Durham facility managed by the In- +stitute for Computational Cosmology on behalf of the STFC DiRAC +HPC Facility (www.dirac.ac.uk). The equipment was funded +by BEIS capital funding via STFC capital grants ST/K00042X/1, +ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham Univer- +sity and STFC operations grant ST/R000832/1. DiRAC is part of the +National e-Infrastructure. +For the purpose of open access, the author has applied a Cre- +ative Commons Attribution (CC BY) licence to any Author Accepted +Manuscript version arising from this submission. +AD thanks Ethan Nadler for providing the completeness-corrected +estimates of the MW dwarf satellite luminosity function. +MNRAS 000, 1–13 (2023) + +Destroyed dwarfs with the MDF +13 +DATA AVAILABILITY +The data analysed in this article can be made available upon reason- +able request to the corresponding authors. +The code used to perform the MDF modeling is available on +Github5 +REFERENCES +Abolfathi B., et al., 2018, ApJS, 235, 42 +Bailer-Jones C. A. L., Rybizki J., Fouesneau M., Demleitner M., Andrae R., +2021, AJ, 161, 147 +Belokurov V., 2013, New Astron. Rev., 57, 100 +Belokurov V., et al., 2007a, ApJ, 657, L89 +Belokurov V., et al., 2007b, ApJ, 658, 337 +Belokurov V., Erkal D., Evans N. W., Koposov S. E., Deason A. 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S., et al., 2019, The Messenger, 175, 3 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–13 (2023) + diff --git a/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/load_file.txt b/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..815d3155ddd683e4cbbda4add80112895f021850 --- /dev/null +++ b/ANE3T4oBgHgl3EQfsQvt/content/tmp_files/load_file.txt @@ -0,0 +1,1213 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf,len=1212 +page_content='MNRAS 000, 1–13 (2023) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 Unravelling the mass spectrum of destroyed dwarf galaxies with the metallicity distribution function Alis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Deason1,2★, Sergey E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Koposov3,4,5†, Azadeh Fattahi1, Robert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Grand6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 1 Institute for Computational Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' South Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Durham DH1 3LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' UK 2 Centre for Extragalactic Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' South Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Durham DH1 3LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' UK 3Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Royal Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Blackford Hill,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Edinburgh EH9 3HJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' UK 4 Institute of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' UK 5 Kavli Institute for Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Madingley Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Cambridge CB3 0HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' UK 6Instituto de Astrofisica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Calle Via Lactea s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Spain 7Departamento de Astrofisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' del Astrofisico Francisco Sanchez s/n, E-38206, La Laguna, Tenerife, Spain Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' in original form ZZZ ABSTRACT Accreted stellar populations are comprised of the remnants of destroyed galaxies, and often dominate the ‘stellar haloes’ of galaxies such as the Milky Way (MW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This ensemble of external contributors is a key indicator of the past assembly history of a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We introduce a novel statistical method that uses the unbinned metallicity distribution function (MDF) of a stellar population to estimate the mass spectrum of its progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Our model makes use of the well-known mass-metallicity relation of galaxies and assumes Gaussian MDF distributions for individual progenitors: the overall MDF is thus a mixture of MDFs from smaller galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We apply the method to the stellar halo of the MW, as well as the classical MW satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The stellar components of the satellite galaxies have relatively small sample sizes, but we do not find any evidence for accreted populations with 𝐿 > 𝐿host/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We find that the MW stellar halo has 𝑁 ∼ 1 − 3 massive progenitors (𝐿 ≳ 108𝐿 ⊙) within 10 kpc, and likely several hundred progenitors in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We also test our method on simulations of MW-mass haloes, and find that our method is able to recover the true accreted population within a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Future datasets will provide MDFs with orders of magnitude more stars, and this method could be a powerful technique to quantify the accreted populations down to the ultra-faint dwarf mass-scale for both the MW and its satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Key words: Galaxies: dwarf – Galaxy: halo – Local Group – galaxies: luminosity function 1 INTRODUCTION Dark matter haloes of all shapes and sizes grow by accumulating lower mass constituents (or subhaloes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The galaxies at the centres of these haloes grow via ongoing star formation, but can also form diffuse ‘stellar haloes’ from the stellar material deposited by the accretion of subhaloes (if they contain stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Depending on the mass- scale, this accreted stellar material can amount to significant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' clusters, ∼ 20 − 30%) or minuscule (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' dwarfs, ∼ 0 − 5%) fractions of the overall stellar mass of the central galaxy (Purcell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Despite having a relatively low stellar mass and surface brightness, stellar haloes retain a record of the lower mass systems that have been digested by haloes over time, and quantifying and understanding this accreted relic has been a major research focus in astronomy for several decades (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Helmi 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Belokurov 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The most-studied stellar halo is, unsurprisingly, that of our own Milky Way (MW) galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, despite significant progress in recent years, we still only have a qualitative view of the mass spectrum of dwarf galaxies that have been consumed by the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Most notably, it has become clear since the game-changing Gaia mission (Gaia ★ E-mail: alis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='deason@durham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='uk (AD) † E-mail: sergey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='koposov@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='uk (SK) Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2016), that the inner stellar halo (within ∼ 20 kpc) is dominated by one ancient, massive accretion event, dubbed the Gaia-Enceladus-Sausage (GES, Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' There is also some evidence that an additional massive structure resides in the very central regions of the galaxy (within ∼ 4 kpc), and was accreted even earlier than the GES (Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, it is debated whether or not this is really an accreted structure, or rather in-situ MW material (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Myeong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Rix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' These massive progenitors, join the already discovered streams and substructures, such as the Sagittarius and Orphan streams (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Newberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2007b), and the Virgo (Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2008) and Hercules-Aquila (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2007a) clouds (although the latter structures may be related to the GES, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Simion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Chandra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022), and more stellar structures in the halo are continuously being discovered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The overall inventory of the Galactic stellar halo is evolving, but the picture is far from complete, and we have no quantitative ‘mass-spectrum’ of destroyed dwarfs akin to the surviving satellite dwarf luminosity function (Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Tollerud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Nadler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020), which is a pillar of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Many of the halo structures that have been discovered in the MW are identified in phase-space and/or action-angle space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This, © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='04667v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='GA] 11 Jan 2023 2 Deason, Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' of course, is where an astrometric mission such as Gaia has en- abled a deeper understanding of the phase-space structure of the halo by providing 6D measurements (at least for the inner halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' How- ever, even with perfect 6D data, robustly identifying distinct halo substructures is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Indeed, massive progenitors can have several ‘clumps’ in dynamical spaces which cannot be unambigu- ously disentangled (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022) and when the stellar material is fully phase-mixed it becomes more difficult to identify from the background (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Johnston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Furthermore, even in the space of conserved quantities the clumps may not stay com- pact due to perturbations from massive systems such as the LMC (Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is where chemical information can be crucial, as galaxies of different mass (and star formation history) can have distinct chemical signatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Venn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Most notable, is the well-known mass-metallicity relation for galaxies, which extends down to the dwarf mass-scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' More massive galaxies are, on average, more metal-rich, and the relation between mass and metallicity exists over several orders of magnitude in mass (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This relation can, to first order, be explained by the larger gravi- tational wells of more massive galaxies, which are able to retain metals (Dekel & Silk 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Lower mass galaxies lack the gravity to resist the expulsion of metals due to feedback mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' On the dwarf mass-scale, not only does the average metallicity vary with mass, but the width of the metallicity distribution function (MDF) also varies, with the lowest mass dwarfs having a wider spread of metallicities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The combined MDF of a popu- lation of accreted dwarf galaxies, such as a stellar halo, is therefore the superposition of several individual MDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Thus, in principle, metallicity measurements alone contain a unique record of the mass spectrum of accreted dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Indeed, the disentangling of a MDF into its individual components is the main focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Fi- nally, it is worth noting that previous work on the MDFs of dwarf galaxies has focused on surviving dwarfs, which, depending on the largely unknown redshift evolution of the mass-metallicity relation, may or may not be relevant for the destroyed dwarfs that make-up stellar haloes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In this work, we consider Galactic-sized stellar haloes as well as the (potential) stellar haloes of dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In principle, dwarf galaxies themselves can cannibalise lower-mass dwarfs, and form what we classically think of as a ‘stellar halo’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, unlike larger mass-scales where the merging dark matter clumps all contain stars, at lower mass scales (below ∼ 109𝑀⊙ in halo mass) dark matter sub- haloes may not have any stars at all (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Benitez-Llambay & Frenk 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' A recent study by Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2022) showed that the very existence of a stellar halo around a dwarf galaxy can have important implications for both small-scale galaxy formation and the nature of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For example, the mass-threshold for galaxy formation, which is largely determined by the epoch of reionization, can have a major effect on the stellar haloes of dwarf galaxies: for models with a high mass threshold for galaxy formation (≳ 109𝑀⊙) dwarf galaxies should not have stellar haloes at all!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Thus, the detection or non-detection of lower mass accretion events surrounding dwarf galaxies, particularly at the ultra-faint mass scale (𝑀star ≲ 105𝑀⊙), is of utmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In order to study the MDFs of accreted populations, we need large, ideally unbiased, spectroscopic samples with metallicity mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For both the Galactic halo, and dwarf satellite galaxies in the MW, extensive samples are hard to come by, but there has been significant progress in recent years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Walker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Taibi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Moreover, and importantly, we are entering a new era of spectroscopic surveys in the MW, with several projects such as DESI, WEAVE, and 4MOST on the horizon (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' de Jong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Thus, with these new surveys in mind, we develop a new modeling method to extract the mass spectrum of accreted components from a sample of [Fe/H] measurements and apply this to current datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In Section 2 we outline our methodology and introduce the statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is a fairly tech- nical section that some readers may want to skip over!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The method is applied to spectroscopic samples of classical dwarf satellite galaxies, and Galactic halo data in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We test the method on state-of- the-art cosmological simulations of MW-mass galaxies in Section 4, and discuss caveats and future prospects in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Finally, we summarise our main findings in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2 MDF MODELING In this Section, we present the methodology that allows us to take samples with measured [Fe/H], and some estimate of the total lu- minosity of the system, and use them to provide constraints on the number of discrete stellar systems of different luminosities that can contribute to a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This next Section is fairly technical, so a less statistically-minded reader may want to skip it and continue with Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The Python code implementing the inference method presented in this section is released on GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 General statistical model We construct a generative model that allows us to represent the metallicity distribution function (MDF) as a mixture of MDFs from smaller galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Throughout this work, we will assume that the MDF of each smaller galaxy can be represented by a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The generic model, where the sample of stars for the MDF is coming from several galaxies, can be described with these model parameters: Number of galaxies N 𝐿𝑖 individual galaxy luminosities (where 1 < 𝑖 < 𝑁) 𝜇𝑖 mean galaxy metallicities 𝜎𝑖 widths of MDF of individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We can then assume that the number of stars in the sample scales linearly with galaxy luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This assumption is accurate for stel- lar populations of similar ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For that assumption to hold, our sample must not be biased towards one galaxy or another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' if our sample comes from a small volume that has an unrepresentative subsample of certain galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' If the proportionality holds, one can write the MDF as 𝑃(𝑧|𝑁, {𝐿𝑖}, {𝜇𝑖}, {𝜎𝑖}) = 1 � 𝐿𝑖 𝑖=𝑁 ∑︁ 𝑖=1 𝐿𝑖N (𝑧|𝜇𝑖, 𝜎𝑖) (1) Here, for clarity, we use 𝑧 as a short-hand notation of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Given our expectation that galaxy luminosities and metallicities are corre- lated (Tremonti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2011), we can assume that galaxies follow a mass metallicity relation (or luminosity metallicity relation) 𝜇𝑖 ∼ N (𝐴 + 𝐵 log 𝐿𝑖|S) (2) 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='com/segasai/mdf_modeling_paper MNRAS 000, 1–13 (2023) Destroyed dwarfs with the MDF 3 where 𝐴 and 𝐵 are constants i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' taken from the mass metallicity relation presented in Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) and Simon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' S is a constant representing a scatter in the relation (found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 dex by Simon 2019, for MW satellites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The individual widths 𝜎𝑖 of MDFs differ from galaxy to galaxy but have been approximated to be slowly dependent on the galaxy luminosity 𝜎 = 𝐶 + 𝐷 log 𝐿 (see Simon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' If we specify the constants 𝐴, 𝐵, 𝐶, 𝐷, and S we have a model for the distribution of metallicities, and this model has an integer parameter 𝑁 and 2𝑁 floating point parameters for luminosities and metallicities of 𝑁 individual galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' While this model for the MDF is valid and can be applied to real data, it has the problem of having a variable number of parameters and therefore is difficult to sample in practice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Green 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Therefore, it would be beneficial to reformulate the model in a way that makes the number of parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The first modification we can do is to group galaxies in 𝑀 lumi- nosity bins, so that rather than represent their luminosities by discrete parameters we represent the number of galaxies in certain luminosity bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Now we define: ˆ𝐿 𝑗 are the grid of galaxy luminosities 1 ≤ 𝑗 ≤ 𝑀 𝑁 𝑗 are the numbers of galaxies with luminosities ˆ𝐿 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 𝜇 𝑗,𝑘 are mean metallicities of k-th galaxy with luminosity ˆ𝐿 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 1 < 𝑘 < 𝑁 𝑗 Where due to mass metallicity relation 𝜇 𝑗,𝑘 ∼ N (𝐴 + 𝐵 log ˆ𝐿 𝑗 |S) or 𝜇 𝑗,𝑘 = 𝐴 + 𝐵 log ˆ𝐿 𝑗 + S𝜖 𝑗,𝑘 where 𝜖 𝑗,𝑘 ∼ N (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here S could either be a constant or a deter- ministic function of ˆ𝐿 𝑗 The MDF model is now 𝑃 �𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}� = 1 � 𝑁 𝑗 𝐿 𝑗 𝑗=𝑀 ∑︁ 𝑗=1 ˆ𝐿 𝑗 ������ 𝑘=𝑁𝑗 ∑︁ 𝑘=1 N (𝑧|𝜇 𝑗,𝑘, 𝜎𝑗,𝑘) ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The likelihood of the data consisting of (for simplicity) a single star with metallicity z would be exactly 𝑃(𝑧|{𝑁 𝑗}, {𝜖 𝑗,𝑘}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The only problem with this formulation is that this likelihood still depends on a variable number of parameters 𝜖 𝑗,𝑘 so one would prefer to marginalise over these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 𝑃(𝑧|{𝑁 𝑗}) = ∫ 𝑃(𝑧|{𝑁 𝑗},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' {𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘})N ({𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘}|0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 1)𝑑𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘 While this marginalisation is difficult,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' and may be impossible to do analytically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' one can simply perform a Monte-Carlo integration over 𝑄 samples from a normal distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' where 𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑞 are the q-th sample 1 ≤ 𝑞 ≤ 𝑄 from N (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 1) 𝑃(𝑧|{𝑁 𝑗}) ≈ 1 𝑄 𝑞=𝑄 ∑︁ 𝑞=1 𝑃(𝑧|{𝑁 𝑗},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' {𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑞}) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' instead of directly doing the summation we can simply treat this as likelihood with integer parameter 𝑞 𝑃(𝑧|{𝑁 𝑗},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 𝑞) = 𝑃(𝑧|{𝑁 𝑗},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' {𝜖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='𝑞}) (3) where 𝑞 is a nuisance seed parameter that we marginalise over −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 dP d[Fe/H] MV =-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 MV =-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 MV =-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 MV =-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 MV =-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0, 20 x {-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0} Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The simulated MDFs for a few systems of different luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The black lines show the expected MDFs in our model for a system with 𝑀𝑉 = −4, with solid and dashed curves showing the MDFs when using a different random seed that controls the offset of the galaxy with respect to the mass metallicity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Red curves similarly show the MDF of a single 𝑀𝑉 = −10 galaxy with different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green curve shows the MDF for a synthetic galaxy that consists of stars coming from one galaxy with 𝑀𝑉 = −10 and 20 galaxies with 𝑀𝑉 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' under the uniform prior 𝑈(1, 𝑄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, we assume that 𝜖 𝑗,𝑘,𝑞 are coming from a pseudo-random number generator that is seeded by 𝑞 and provides normally distributed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We then will need to sample the posterior over {𝑁 𝑗} and 𝑞, which gives the model with 𝑀 + 1 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Armed with Eqn 3 that specifies the likelihood function for the metallicity distribution, the only missing ingredient for the model are the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We assume that occupation numbers {𝑁𝑖} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' numbers of galaxies in luminosity bins) have a prior distribution of ⌊10𝑥⌋ where 𝑥 ∼ 𝑈(−1, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is essentially the log uniform of integers distribution with 20% prior volume at 𝑁𝑖 = 0 and 20% for 1 ≤ 𝑁𝑖 ≤ 10, and 20 ≤ 𝑁𝑖 ≤ 100 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Finally, we complement the model with the constraint on the total luminosity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Specifically, we require that the combined luminosity of multiple galaxies must match certain known total lu- minosity log 𝐿tot with some uncertainty 𝜎𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This provides a term for the log of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' log( ∑︁ 𝑁𝑖 ˆ𝐿𝑖) ∼ 𝑁(log 𝐿tot|𝜎𝐿) A final remark that despite the introduction of the formalism based on binned number of galaxies, we have found the model is more stable when at least one contributor to the MDF (likely the one being the most massive main progenitor) is represented directly (rather than in a bin) by the satellite luminosity 𝐿main, metallicity 𝑧main and that also obeys the mass-metallicity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' To illustrate our modeling approach, in Figure 1 we show the expected [Fe/H] distributions given our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Specifically, solid black and red curves show possible MDFs for a single galaxy of 𝑀𝑉 = −4 and 𝑀𝑉 = −10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Dashed lines of same colours show the MDFs when different random seeds are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green curve shows a distribution that we might expect if we observe stars coming from a single 𝑀𝑉 = −10 galaxy and 20 𝑀𝑉 = −4 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This shows a prominent tail towards low metallicities, and this is exactly what allows us to probe the number of possible mergers with low luminosity systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' MNRAS 000, 1–13 (2023) 4 Deason, Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 Sampling In the previous section, we have introduced the likelihood function for the metallicity distribution that is conditional on the number of different dwarf galaxies 𝑁 𝑗 on a grid of luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The model also has an integer seed parameter 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' It is not trivial to sample inte- ger parameters, especially if we expect multiple modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' To perform the sampling we decide to use the dynamic nested sampling as im- plemented in the dynesty package (Speagle 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' As nested sampling is technically invalid if the likelihood surface has plateaus (Fowlie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020), we add a a small level of deterministic noise with standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='01 to the likelihoods, which should not affect the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 3 APPLICATIONS We now apply the method described above to observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, we focus on the classical MW satellites (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1) and the MW stellar halo (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 Classical dwarf satellite galaxies We start from the homogeneous sample of dwarf galaxy members presented in Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) as provided in the Strasbourg astro- nomical Data Center (CDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' As mentioned in the previous section, the key assumption that we rely on for our method is that the abun- dances that we model are random samples from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is likely not technically correct for the data at hand since the stellar samples in dwarfs tend to be biased towards the centres of systems (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Walker & Peñarrubia 2011), and may have slight metallic- ity biases caused by the colour-magnitude selection of spectroscopic targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We will, however, proceed ignoring these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We take the sample of stars from Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) and only consider stars with small metallicity uncertainty 𝜎[Fe/H] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This catalogue has measurements of 10 MW satellites with more than 10 stars: Canes Venatici I, Draco, Fornax, Hercules, Leo I, Leo II, Sculptor, Sextans, Ursa Minor and Ursa Major I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We then proceed to model each of the dwarfs with the machinery presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We take the luminosities of each system from McConnachie (2012) (using an updated catalogue from January 2021) and adopt an 𝑀𝑉 uncertainty for each system of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For each system, we use the luminosity bins that are 1 magnitude wide from 𝑀𝑉 = 0 to the luminosity of the dwarf itself minus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The posterior samples on the number of possible dwarf galaxies that contributed to the systems’ MDF are shown in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We show measurements for 8 out of 10 systems spanning the luminosity range from 𝑀𝑉 ∼ −5 for Ursa Major I to 𝑀𝑉 ∼ −13 for Fornax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The panels are ordered by system luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The total number of stars varies from 𝑁 = 15 for Ursa Major I to 𝑁 = 789 for Leo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Figure 2 shows the constraints on the differential number of systems that have contributed to the dwarfs’ MDF, while Figure 3 shows constraints on the cumulative counts of the number of systems brighter than a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The blue/orange bands show the 16/84 and 1/99 percentiles, and the black line shows the median of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green bands show the constraints if we do not use metallicities at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is essentially a prior and corresponds to the case where the only constraint comes from ensuring the combination of galaxies matches the total luminosity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that, because we include all the stars in the galaxy, we expect to measure 𝑁merged = 1 at around the total luminosity of the dwarf galaxy (shown with the solid red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Although technically this is an ‘in-situ’ rather than an accreted component, what we are actually constraining are the contributors to the MDF, regardless of their origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We now look at the posteriors in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' First, we focus on clear cases where the data is particularly constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' These are the cases of Fornax, Leo I, Leo II, and Draco, where the spectroscopic samples have hundreds of members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We see that their differential posterior distributions (Figure 2) have a peak with a value of one next to the system luminosity (highlighted in red) and show the value consistent with zero for 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Thus, the data suggests that these systems did not experience a merger with a dwarf that is larger than 1% of the system luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is also seen in the cumulative plots, where we see the implied number 𝑁merged(< 𝑀𝑉 ) is flat and equal to one in the range 𝑀𝑉 ,host ≲ 𝑀𝑉 ≲ 𝑀𝑉 ,host + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Looking at the implications for the number of faint contributors to the MDF for Fornax, Leo I, Leo II, and Draco systems, we can see that our constraints on 𝑁merged shoot up and become significantly broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The differential counts are essentially unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For example, for the Fornax MDF contributors at 𝑀𝑉 = 0 (top left panel of Figure 2) the 1-𝜎 confidence interval is 0 < 𝑁merged < 100 as the data allows many faint dwarfs before the observed MDF is affected significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The behavior of the cumulative counts for the faint MDF contributors is somewhat misleading as it rises at faint 𝑀𝑉 purely because we are summing over bins with non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Fainter dwarf galaxies like CVnI or UMa have a smaller number of spectroscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In Figure 2 we see that the posteriors on the number of MDF contributors start to rise next to 𝑀𝑉 = 𝑀𝑉 ,host, which indicates that we cannot even rule out that the galaxy is a product of a merger of two systems with similar luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The constraints on the cumulative number of mergers for fainter dwarfs do not show a flat 𝑁merged = 1 part next to the luminosity of the system and instead rises to faint luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We also see that for faint systems, the posteriors basically look very close to priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 Galactic stellar halo We next apply our method to the Galactic stellar halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' It has been realised for some time that the stellar halo of the MW comprises an assortment of destroyed dwarf debris, and thus the metallicity distribution of these halo stars retains a memory of their dwarf galaxy progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Large, homogeneous samples of halo stars with metallicity mea- surements are hard to come by, and this is a significant limitation of our current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' At present, we build a sample of halo stars based on several spectroscopic surveys and use the latest Gaia data (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2021), to help select a clean halo sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We begin by cross-matching stars with spectroscopic data from SDSS (Abolfathi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2018), RAVE (Kunder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2017), LAMOST (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2012), APOGEE (Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2017), and GALAH (Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2021) with Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This results in 𝑁 = 656, 0819 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' To estimate distances to the stars we use the Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2021) photogeometric distances computed from Gaia EDR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We only con- sider stars with reasonable parallax 𝜎𝜛/𝜛 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5, and restrict our sample to 𝑟 < 10 kpc and |𝑧| > 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Finally, to avoid disk con- tamination, we apply a cut on the rotational velocity of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We impose a fairly strict cut to remove the majority of thick disk and/or splash stars (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020), and only include those with ret- rograde orbits 𝑣𝜙 < −50 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The resulting spatial (top-panel) and metallicity distribution (bottom-panel) of the stars are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In the bottom panel, we also show the MDF for the stars without the 𝑣𝜙 cut in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Our restriction to retrograde orbits is fairly stringent but, as can be seen in the figure, it is effective at removing disk stars, which have prograde orbits and are generally more metal-rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We MNRAS 000, 1–13 (2023) Destroyed dwarfs with the MDF 5 −10 −5 100 101 102 Nmerged For −10 −5 MV LeoI −10 −5 MV LeoII −10 −5 MV Sex −10 −5 MV 100 101 102 Nmerged UMi −10 −5 MV Dra −10 −5 MV CVnI −10 −5 MV UMaI Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The inferred contributions from systems to the MDF of different dwarf galaxies from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In each panel, the black curve shows the median number of galaxies of a given luminosity that could have contributed to the MDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The blue and orange bands show the 16/84 and 1/99 percentiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green band shows the sampling of the prior with only the constraint on total luminosity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The vertical red line on each panel shows the luminosity of each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that the logarithmic y-axis is cut-off at 𝑁merged = 10−1, so median values at this level are consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' −10 −5 100 101 102 Nmerged(< MV ) For −10 −5 MV LeoI −10 −5 MV LeoII −10 −5 MV Sex −10 −5 MV 100 101 102 Nmerged(< MV ) UMi −10 −5 MV Dra −10 −5 MV CVnI −10 −5 MV UMaI Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The inferred contributions to the MDF from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is similar to Figure 2 but shows the cumulative numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Each panel shows a different dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In each panel, the black curve shows the median number of galaxies of a given luminosity or brighter that could have contributed to the MDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The blue and orange bands show the 16/84 and 1/99 percentiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green band shows the sampling of the prior with only the constraint on total luminosity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The vertical red line on each panel shows the luminosity of each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' apply our modelling procedure to stars with −4 < [Fe/H] < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='82, and 𝜎([Fe/H]) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2, which results in a sample of 𝑁 = 21, 813 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2 Note that this metallicity cut is made in both the data and model, so there is no metallicity bias introduced with our selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Our sample is comprised of 5 different spectroscopic surveys, with varying selection functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, we aim to maximise the number of halo stars with metallicity measurements by combining these surveys but note that, ideally, a more homogeneous sample would be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For now, we continue on, under the assumption that there are no MNRAS 000, 1–13 (2023) 6 Deason, Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 0 2 4 6 8 10 R [kpc] 10 5 0 5 10 |z| [kpc] 0 2 4 6 8 10 10 5 0 5 10 4 3 2 1 0 1 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4 dN All vφ < -50 km/s log(g) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 log(g) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Top panel: The spatial distribution in the 𝑧 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 𝑅 plane of our MW halo sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Bottom panel: The metallicity distribution of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The grey line indicates the MDF without a cut in 𝑣𝜙, which leads to the inclusion of (metal-rich) disk stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The dashed red line indicates the median metallicity of our halo sample ([Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We also show the MDFs of our halo sample split by log(𝑔) with the blue and purple dotted lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The gray line-filled region indicates the metal-rich regime ([Fe/H] > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='8) that is excluded in our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' significant metallicity biases in this combined sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, we stress that future work with upcoming spectroscopic surveys such as DESI (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022) and WEAVE (Dalton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2012) will be much better suited for this type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In our analysis, we adopt a total halo luminosity of 𝑀𝑉 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is consistent with recent measurements which suggest 𝐿 ∼ 1 × 109𝐿⊙, and also allows a range around this value encompassing the majority of observational constraints and their un- certainties (Deason et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Mackereth & Bovy 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The top panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 5 show the resulting number of de- stroyed dwarfs in the MW halo as a function of 𝑀𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We consider dwarfs with 0 > 𝑀𝑉 > −22, using 22 bins with 1 mag bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that the size of our sample means that we are unlikely constraining dwarfs with 𝑀𝑉 ≳ −10, which will not be represented by a large enough number of stars (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In blue, we show the results when the Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) mass-metallicity relation is used, which is appropriate for surviving dwarf galaxies in the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In re- cent work, Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2022) (see also Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2020) argue that destroyed dwarfs may not lie on this relation, and a relation with ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex offset to lower metallicities is more appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We show the results with this offset applied in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Our model predicts several hundred (𝑁 ∼ 400) destroyed dwarfs with 𝑀𝑉 ≲ −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, the different mass-metallicity relations (relevant for either ‘surviving’ or ‘destroyed’ dwarfs) predict different distributions of progenitor masses, particularly at larger masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For example, when using the Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) mass-metallicity relation applicable for surviving dwarf galaxies, we estimate 𝑁 = 1 massive dwarf progenitor with 𝐿 ∼ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5𝐿⊙, but this rises to 𝑁 = 3 when the relation more relevant to destroyed dwarf galaxies is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This seems to be at odds with our adopted total halo luminosity of 𝐿 ∼ 1 × 109𝐿⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Indeed, by summing the predicted numbers of destroyed dwarfs we find that the total luminosity when the Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) relation is used is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 × 109𝐿⊙, but this rises to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 × 109𝐿⊙ when an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex offset is applied to the mass-metallicity relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Clearly, in this latter case, the bias in metallicity has pushed the progenitor masses higher, and, because we have allowed a fairly flexible total luminosity, resulted in a high halo luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, it is still consistent with the input luminosity within 1 − 𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 5 we show the results when the total halo luminosity is fixed to 𝑀𝑉 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 (technically, an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='01 dex is adopted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, the ‘fiducial’ result using the Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) mass-metallicity relation is only slightly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For example, the most massive progenitor is shifted to a slightly lower luminosity (by ∼ 1 dex in 𝑀𝑉 ), and the total number of dwarfs with 𝑀𝑉 < −10 is reduced (𝑁 ∼ 300).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In general, the changes are within the predicted uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' When an 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex metallicity offset is applied to the mass-metallicity relation, fixing the halo luminosity has a greater effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is unsurprising given that allowing for a more flexible halo luminosity favours a higher value than the fiducial 1×109𝐿⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In this case, the most massive progenitor has 𝐿 ∼ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1𝐿⊙ (compared to 𝑁 ∼ 3 with 𝐿 ∼ 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5𝐿⊙ when the total luminosity is more flexible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The number of low mass dwarfs is also reduced, with 𝑁 ∼ 110 with 𝑀𝑉 < −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This exercise emphasizes how important the assumed total halo luminosity, as well as the adopted mass-metallicity relation are for this type of analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We also show the surviving dwarf satellite luminosity function for comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, we show the observed (solid purple) and completeness-corrected (dashed purple) cumulative number counts given by Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The numbers of low luminosity satellite systems are much lower than the predicted number of de- stroyed dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is perhaps unsurprising given that our estimates are likely overestimated at low luminosities, owing both to sample size and our assumption of Gaussian MDFs (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Inter- estingly, the (cumulative) number counts are similar at intermediate luminosities (−16 ≲ 𝑀𝑉 ≲ −12) but destroyed dwarfs as massive as the LMC are not favoured unless the adopted mass-metallicity relation is adjusted from the fiducial 𝑧 = 0 form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2020) show using the Auriga simulation suite that the number of destroyed dwarfs in MW-mass haloes is larger than MNRAS 000, 1–13 (2023) Destroyed dwarfs with the MDF 7 20 15 10 5 0 MV 10 1 100 101 102 103 N(merged) MW halo: MV = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 20 15 10 5 0 MV 10 1 100 101 102 103 N(merged < MV) Satellite dwarf LF Mass-Metallicity relation: Kirby+2011 Kirby+2011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex 20 15 10 5 0 MV 10 1 100 101 102 103 N(merged) MW halo: MV = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 (fixed) 20 15 10 5 0 MV 10 1 100 101 102 103 N(merged < MV) Satellite dwarf LF Mass-Metallicity relation: Kirby+2011 Kirby+2011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The estimated differential (left) and cumulative (right) number of destroyed dwarfs in the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The dark(light) shaded regions show the 16-84(1-99) percentiles, and the solid lines are the medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The dashed black line indicates the assumed total stellar halo luminosity (𝑀𝑉 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In the top panels, the total luminosity has a flexible uncertainty of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 dex, whereas in the bottom panel the total luminosity is kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The results in blue are for when the 𝑧 = 0 mass-metallicity relation for dwarfs is assumed (Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In orange, we show the results when an −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex offset is applied to the relation, which has been postulated to be more applicable to destroyed dwarfs (Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For comparison, we show the surviving dwarf satellite luminosity function in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The dashed line indicates the completeness-corrected LF derived by Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' the number of surviving satellites, at least down to 𝑀𝑉 ∼ −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is in agreement with our results, however, our estimated total number of destroyed dwarfs is far higher than these models (by a factor of ∼ 3 − 10, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This could be a genuine tension with the models, but it is worth stressing that our number estimates at low luminosities are likely biased high, and the numbers could be reduced if we had larger sample sizes and/or the metal-poor tails of higher mass systems are taken into account (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Finally, given the heterogeneous nature of our sample of halo stars, we consider how different cuts in surface gravity affect the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Namely, dwarf stars and giants can have different metallicity biases, and probe different volumes in magnitude-limited surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The MDF of our halo sample split by log(𝑔) was shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, we MNRAS 000, 1–13 (2023) 8 Deason, Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 22 20 18 16 14 12 10 8 6 4 2 0 MV 10 1 100 101 102 103 N(merged < MV) MV = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7 (fixed) All log(g) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 log(g) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The estimated cumulative number of destroyed dwarfs in the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 5, but split into two bins with low (log(g) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5) and high (log(g) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5) surface gravity stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The thick gray line indicates the overall sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The stellar halo luminosity is fixed (𝑀𝑉 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' can see there are slight differences for low and high log(𝑔), and now we consider how our inferred number counts of destroyed dwarfs are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The cumulative number of destroyed dwarfs is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 6 with two different bins of log(𝑔), appropriate for dwarf stars (log(𝑔) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5) and giants (log(𝑔) < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' It is worth bearing in mind that our overall sample is dominated by the high surface gravity dwarf stars (approximately ∼ 2/3 have log(𝑔) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that here we only use the Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2011) mass-metallicity relation, and the total halo luminosity is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Encouragingly, the total number of progenitors (for 𝑀𝑉 ≲ −10) is very similar for the two bins of log(𝑔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, massive progenitors (𝐿 ≳ 108𝐿⊙) are only favoured in the high log(𝑔) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is likely because the MDF is biased towards lower metallicities for the giant star sample (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Moreover, the giant and dwarfs are probing slightly different volumes, with the high surface gravity dwarfs more concentrated around the solar neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This exercise highlights the difficulty of using a ‘hodge-podge’ of halo stars for our analysis, and it will clearly be preferable for future work to have a more homogeneous sample, where the selection function is clearly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 4 AURIGA SIMULATIONS Our modeling procedure makes various assumptions and simplifica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For example, it assumes each progenitor galaxy is sampled in a representative way, and that their MDFs are adequately described by a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In reality, this may not be the case, particularly for volume-limited Galactic-sized stellar haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' To this end, we test our model on simulated MW stellar haloes, which are representative of ‘realistic’ accreted populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We apply our modeling procedure to halo stars in the Auriga simulations (Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' these cos- mological hydrodynamical simulations are a suite of 𝑁 ∼ 30 high resolution (𝑚 𝑝 ∼ 5×104𝑀⊙) MW-mass (1−2×1012𝑀⊙) haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In this work, we make use of the 𝑁 = 28 haloes studied in Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2019), which omits two haloes currently undergoing major mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We only consider accreted halo stars, which are identified in Fattahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2019) as those that formed in subhaloes other than the main progenitor galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For each halo, we construct a sample of halo star particles within 𝑟 < 20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This is chosen to roughly mimic the volume limit of current observations, and ensure large enough sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The in- put into the model is the [Fe/H] values of the stellar particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Of course, in the simulations, we also know the progenitor galaxy of each star particle, and can thus test the estimated mass spectrum of accreted dwarfs from our modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The final ingredient we need to define is the mass-metallicity relation for the Auriga sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' (2021) show that the mass-metallicity relation for dwarf galaxies in Auriga is in good agreement with low mass dwarfs (𝑀star ∼ 106𝑀⊙), but is too metal-rich by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='5 dex for more massive dwarfs (see Figure 13 in Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We use all the destroyed dwarf progenitors across the 𝑁 = 28 Auriga haloes to cali- brate this relation3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, we do exclude dwarfs that are accreted recently (less than 5 Gyr ago) as these can have significantly different metallicities due to ongoing star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The debris from these events is still included in the analysis, but our calibration is only based on the relatively old dwarf galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that we only consider dwarf progenitors with 𝑀𝑉 > −7, which corresponds to a stellar mass of 𝑀star > 105𝑀⊙ or 𝑁 > 2 star particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We use the ‘peak’ stellar mass of each dwarf, which corresponds to the maximum stellar mass the progenitor has reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that we get similar results if the stellar mass at infall is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The resulting mass-metallicity4 relation for Auriga is: [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='69 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='39 × (log10𝐿 − 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' To es- timate the scatter around this mean relation, we calculate the scatter for each individual halo, and use the median value across all haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This results in a scatter around the mean [Fe/H] relation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='3 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Finally, we consider the spread in [Fe/H] for individual dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Un- like the observations, we find no strong evidence for a variation with dwarf mass, so instead adopt a constant dispersion of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4 dex of the MDF for all dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Armed with the mass-metallicity relation appropriate for Auriga, we can now test our modeling procedure on these cosmological haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' When applying our method to the Auriga haloes, we assume the total luminosity of the halo is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' This of course results in addi- tional uncertainty in the real observations, but we particularly want to investigate the systematic influences present in the cosmological sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We consider accreted dwarfs in the range −7 > 𝑀𝑉 > −22, and estimate the number of dwarfs in 15 bins with bin size of 1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 7 shows the resulting cumulative number of destroyed dwarfs in the Auriga haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Each panel shows a different halo, and our estimated numbers are shown with the solid black lines (median), and blue/orange shaded regions (16-84/1-99 percentiles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The points with error bars are the true values, with Poisson noise adopted for the uncertainties in each 𝑀𝑉 bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Note that the ‘true’ values include all dwarfs that have deposited any material within 20 kpc of the host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Thus, there can be cases where only a small fraction of a destroyed dwarf is included in the sample (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The green values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 7 are for all progenitors, while the purple are only those accreted earlier than 5 Gyr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In many cases, there is little difference between the green and purple values, because most dwarfs are accreted at earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, we highlight the most recently accreted dwarfs because these are likely not fully phase-mixed, and can significantly deviate from the mass-metallicity relation for (old) dwarf galaxies in Auriga (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In reality, we find that these recently accreted dwarfs only cause a significant effect if the progen- itors are relatively massive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Halo 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 3 To clarify, all destroyed dwarfs are used, not just those that have debris within 20 kpc of the host halo 4 Note that we assume a stellar mass-to-light ratio of (𝑀/𝐿) = 2 to convert stellar mass to luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 1–13 (2023) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='Destroyed dwarfs with the MDF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='N(merged < MV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='N(merged < MV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='N(merged < MV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='N(merged < MV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='MV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='halo_10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The estimated cumulative number of destroyed dwarfs for the 𝑁 = 28 Auriga haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The solid black line shows the median, and the shaded blue(orange) regions the 16-84(1-99) percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The red dashed line indicates the assumed total luminosity of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For each halo, accreted star particles are selected within 𝑟 < 20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The points with (Poisson) error bars indicate the ‘truth’, with all progenitors shown in green, and only those accreted > 5 Gyr ago in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The latter are shown because recently accreted dwarfs are likely (i) not fully phase-mixed, and (ii) can significantly deviate from the mass-metallicity relation for (old) dwarf galaxies in Auriga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We discuss these results more quantitatively below, but first cast a qualitative eye on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In general, our estimates agree well with the true mass spectrum of accreted dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' However, in some cases, there can be notable differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' We find that the most significant deviations are due to the following: (1) relatively massive progenitors that lie off the mass-metallicity relation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Halo 2, 6) and/or (2) progenitors with a low fraction of their material within the given radial range (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Halo 15, 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' These systematics, and sometimes the combination of both, are most likely to cause our method to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' On the other hand, there are a significant number of haloes for which we recover the mass spectrum very well, which is encouraging given the complexity of these hydrodynamic simulations, and the cosmological nature of their assembly histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 8 we give a more quantitative summary of our tests of the Auriga haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Here, for each halo (identified in the x-axis) we show the fraction of 𝑀𝑉 bins that have number estimates that agree within the 16-84, 5-95, and 1-99 percentile confidence limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The median recovery fractions across all haloes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='61, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='77, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='83, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' These fractions are below the expected fractions for a ‘perfect’ procedure, but this is unsurprising given the various sys- tematic influences present in the simulations, such as deviations from the adopted mass-metallicity relation and the presence of stellar de- bris that does not fully occupy the available phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' These, of course, are realistic effects that could be present in the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 9 we explore the halo-to-halo scatter more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' In the left-hand panel, we show the difference between the estimated and 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Halo ID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content='0 Fraction 16-84 5-95 1-99 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' Quantifying the test with Auriga haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' For each halo, we show the fraction of 𝑀𝑉 bins (1 mag wide) that have estimated numbers that agree within the 16 − 84 (red-filled circles), 5 − 95 (blue-filled squares), and 1−99 (green-filled diamonds) percentage confidence limits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The median values are shown with the horizontal coloured lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' true cumulative numbers of destroyed dwarfs as a function of 𝑀𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The black line shows the median of the 𝑁 = 28 haloes, and the blue and orange shaded regions show the 16-84 and 1-99 percentiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' The deviation from the true numbers is fairly symmet- MNRAS 000, 1–13 (2023) 10 Deason, Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANE3T4oBgHgl3EQfsQvt/content/2301.04667v1.pdf'} +page_content=' 22 20 18 16 14 12 10 8 MV 40 20 0 20 40 N( 3000 m) was identical to +the incident wave signal (x = 0). This confirms that wave action is conserved in these simulations. +To verify the model results quantitatively, we compared the change in the wave height and wavelength +10 + +Figure 4: Snapshot of the modelled surface elevation (blue line, left axis) and ambient current velocity (red line, right axis) +in the numerical domain for three different current velocities (U = [−3, 0, 3] m/s) for a monochromatic wave with amplitude +a = 0.01 m and period T = 10 s). The dashed black line indicates the envelope of the wave elevation, and the title of each +panel indicates the respective current velocity. +11 + +opposing current (U= -3 m/s) +1.0 +a) +(-) + 0.5 +(s/w) n +H/2 +0.0 +0 +0.5 +1.0 +3 +no current +1.0 +b) +0.5 +(s/w) n +H/2 +0.0 +0 +0.5 +1.0 +following current (U= 3 m/s) +1.0 +C) +(-) H/2 +0.5 +(s/ +0.0 +0 +U (m/ +0.5 +1.0 +-3 +0 +500 +1000 +1500 +2000 +2500 +3000 +x (m)Figure 5: Normalized change to the wave height H (panel a) and wavelength L (panel b) as a function of the current velocity +U for small-amplitude monochromatic waves with T = [5, 10, 15] s. The wave height and wavelength were normalized by the +wave parameters in absence of a current (indicated by [...]0). Converged model results of simulations (with ∆t = T/1000 and +∆x = L/100) are indicated by colored lines (see legend) and results from linear wave theory are indicated by the thick black +line. In the left panel, the horizontal blue line with dotted markers indicates the average change to the simulated wave height +H in the current region and the vertical line with horizontal bars indicates the maximum and minimum simulated H inside the +current region. The dashed vertical black lines indicate the current velocity at which wave blocking occurs according to linear +wave theory. +inside the current region with the results from linear wave theory (Fig. 5). For all three wave periods, +linear wave theory predicted that the wave height and wavelength varied significantly for the considered +range of current velocities (using Eq. 28). For opposing currents, the wave height H increased and the +wavelength L decreased, and vice versa for following currents (as was visually observed in Fig. 4). Current +induced changes to the wave field were larger for shorter wave periods, with wave blocking occurring for +T = [5, 10, 15] s at U ≈ [−1.92, −3.74, −4.87] m/s (indicated by the vertical black dashed line). +SWASH captured the changes to the wave height and wavelength for the range of ambient current +velocities and the three wave periods (Fig. 5). This included the nonlinear dependence of H and L for +U < 0 m/s. Furthermore, the model captured blocking of waves for opposing currents that are stronger +than the critical flow velocity of linear wave theory (indicated by the dashed black lines in Fig. 5). For all +three wave periods, simulations with current velocities stronger than the theoretical blocking velocity showed +a strong decay of the wave height down-wave of the blocking point (not shown). For simulations with U close +to but just weaker than the theoretical blocking velocity, dissipation of wave energy occurred in the model +over the current region (visible as the difference between the vertical lines with horizontal bars in Fig. 5a, +which indicates the maximum and minimum H in the current region). In the absence of physical mechanisms +for dissipation, this is likely related to numerical diffusion when the waves (with shorter lengths) propagate +in the current region. For weaker U this dissipation becomes smaller and the model results were in good +agreement with linear theory. This numerical dissipation was found to be dependent on the horizontal grid +12 + +2.0 +4 - +a) +T=5 s +b)7 +T=10 s +1.5 +1 +T=15 s +1 +H/Ho +L/Lo +1.0 +2 +- +1 +0.5 +0 - +0.0 +-6 +-4 +-2 +0 +2 +4 +-6 +-4 +2 +2 +U (m/s) +U (m/s)resolution and time step, with improved agreement for strong U for finer spatial and temporal resolutions +(in accordance with the results in Appendix A). +4.2. Sheared current fields +In coastal regions, spatially varying current fields exist (e.g., tidal currents) that can induce wave re- +fraction and result in focal zones that give rise to wave interference patterns (e.g., Yoon and Liu, 1989; +Akrish et al., 2020). In this section, we verify the ability of the model to capture such wave patterns using +two classical examples of wave-current interactions: the interactions of waves with a jet-like current and a +vortex ring. Model results were compared with the spectral wave model SWAN (Booij et al., 1999) extended +with a quasi-coherent formulation that accounts for wave interference due to variable topography (Smit and +Janssen, 2013; Smit et al., 2015) and currents (Akrish et al., 2020). +4.2.1. Model set-up +The model set-up was based on the work of Akrish et al. (2020). +The region of interest spanned a +domain of 4 × 4 km. Two different simulations were considered, one with a jet-shaped and the other with +a vortex-shaped current field, positioned along the central axis of the domain. The maximum velocities for +the simulations were 0.38 m/s and 1.0 m/s, respectively (refer to Akrish et al., 2020, for a mathemetical +formulation of the current fields). At the wavemaker positioned along the western boundary, a Gaussian +shaped wave-spectrum in frequency and direction was forced with Hs = 1 m, Tp = 20 s and a standard +deviation of 0.0015 Hz in frequency space and 1.78◦ in directional space. The waves had a mean direction +of θ0=15◦ and 0◦ (in Cartesian coordinates) for the jet and vortex current, respectively. +In the SWAN model, the physical domain was discretised with ∆x = ∆y = 50 m. The spectral domain +was discretised with 45 discrete frequencies that were logarithmically spaced between 0.005 and 0.085 Hz, +and with a directional resolution of 2◦ between -90 and 90◦. +For the SWASH model, we extended the +domain with a 500 m wide sponge layer at the eastern side of the domain to prevent any wave reflections. +The domain was discretised with a resolution of ∆x = 2 m and ∆y =4 m (which resulted in ≈ 100 points +per wavelength throughout the domain). The time step was set at ∆t = 0.05 s, equalling 300 points per +wave period and resulting in CFL ≈ 0.6. +4.2.2. Results +Due to changes in wavelength induced by the current, waves were refracted by the vortex ring (Fig. +6a-c and 6g-i). This current-induced refraction resulted in considerable variations in the significant wave +height, with ridges of larger wave heights where waves focussed and depressions of lower wave heights where +waves diverged (Fig. +6a-f). +For this current field, quasi-coherent (QC) effects needed to be taken into +account in SWAN to resolve the constructive and de-constructive wave interference that altered the wave +field downstream of the vortex ring (e.g., Akrish et al., 2020). +The bulk wave heights and mean wave +13 + +Figure 6: Changes to the significant wave height Hm0 and mean wave direction θ due to a vortex ring current field. Panels +a-c show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows +indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b) +and default SWAN (panel c). Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore +transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines). +directions predicted by the extended SWASH model were in satisfactory agreement with the results from +the SWAN QC model throughout the domain. +Similarly, waves refract as they propagated into the jet-like current field, resulting in a change of +mean wave direction (Fig. +7g-i) and in regions with increased and decreased wave heights due to con- +vergence/divergence of wave energy (Fig. 7a-f). Similar to the vortex ring, quasi-coherent effects need to be +incorporated in SWAN to account for the constructive and de-constructive wave interference that altered the +wave field, although this effect was smaller compared to the vortex ring. In general, the SWASH predictions +were in good agreement with SWAN QC. The results of this test case, and the vortex ring, illustrate that +SWASH including the additional terms in the model equations is able to capture the effect of current-induced +14 + +SWASH +SWAN QC +SWAN +4000 + 1.5 +af +bi +3000 + 1.0 +(w) +2000 +1000 + 0.5 +0.0 +0 +1000 +2000 +3000 +0 +1000 +2000 +3000 +0 +1000 +2000 +3000 +x (m) +x (m) +x (m) +x=1000 m +x=2000 m +x=3000 m +1.5 +(p +el +1.0 +0.5 +0.0 +SWASH +SWAN QC +15 +(6 +15 +h) +15 +SWAN +(6ap) +0 +0 +0 +15 +15 +15 +2000 +4000 +2000 +4000 +0 +2000 +4000 +y (m) +y (m) +y (m)Figure 7: Changes to the significant wave height Hm0 and mean wave direction θ due to a jet-like current field. Panels a-c +show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows +indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b) +and default SWAN (panel c). Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore +transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines). +refraction on the wave propagation and the resulting spatial variability in the wave field. +4.3. Wave blocking, reflections and breaking on opposing currents +As a final test case, we compare model predictions with the laboratory experiment of Chawla and Kirby +(1999, 2002) that considered wave blocking on opposing currents. The flume had a length of 30 m, a width +of 0.6 m and still water depth of 0.5 m, with a pump system to generate a recirculating current (with a +discharge of 0.095 m3/s) and a perforated wavemaker to generate waves on the current. A spatially varying +current was generated by means of a false wall constricting the width of the flume, with a minimal width of +0.36 m (see black line in Fig. 8a). Blocking of waves occurred close to the start of this narrow part of the +flume. +15 + +SWASH +SWAN QC +SWAN +4000 + 1.5 +a +h) +3000 + 1.0 +(w) ^ +2000 +1000 + 0.5 +0- +0.0 +0 +10002000 3000 +0 +1000 +2000 +3000 +0 +1000 +20003000 +x (m) +x (m) +x (m) +x=1000 m +x=2000 m +x=3000 m +1.5 +e) +f) +1.0 +0.5 +0.0 +SWASH +30 +30 +30 +SWAN OC +(6 +h) +i) +SWAN +(deg) +15 +15 +15 ++0 ++0 +0 +2000 +4000 +0 +2000 +4000 +2000 +4000 +y (m) +y (m) +y (m)Figure 8: Overview of the numerical setup of the Chawla and Kirby (1999) flume experiment. The top panel (a) shows the +flume width (black line, left axis) and a snapshot of the modelled free-surface elevation for test case 1 (blue line, right axis). +The bottom panel (b) shows the modelled (red line) and measured (black markers) current velocity (in the absence of waves). +The experiments with monochromatic waves considered a total of 23 test conditions that included 3 +different incident wave periods (T = [1.2, 1.3, 1.4] s) for a range of wave heights (H = 0.012 − 0.14 m). +For the low amplitude and low period waves, waves reflected with negligible transmission of wave energy +beyond the blocking point. For increasing wave heights, waves started breaking at the blocking point of +linear theory combined with increased transmission of wave energy beyond this theoretical blocking point. +In this paper, we considered 4 out of the 23 test cases: the largest and smallest wave height of both the +smallest and largest wave period (see Table 1). For case R1 and R11 waves reflected at the blocking point, +whereas waves were breaking and wave energy was transmitted beyond the theoretical blocking point for +case B6 and B18. +We compared model predictions with these laboratory observations for these 4 test cases. Furthermore, +we also computed the wave height transformation based on conservation of wave action. +∂ +∂x +cgE +σ += 0, +(29) +Conservation of wave action is computed based on the linear dispersion relationship (similar to Sec. 4.1) and +also based on the nonlinear dispersion relationship from 2nd order Stokes theory (e.g., Dean and Dalrymple, +1991). This nonlinear dispersion relationship accounts for the effect of amplitude dispersion, which was +found to be important for these laboratory experiments (Chawla and Kirby, 2002). +16 + +0.6 +(w) +a) +width +0.3 +0.03 +(w) 2 +0.0 +0.00 +EO'O- +0.0 +b) +(s/w) n +0.2 +-0.4 +0.6 +15 +10 +-5 +0 +n +10 +x (m)H (m) +T (s) +Ub (m/s) +kd (U = 0 m/s) +kd (U = −0.32 m/s) +kd (U = Ub m/s) +R1 +0.012 +1.2 +-0.47 +1.53 +2.36 +5.59 +B6 +0.126 +1.2 +-0.47 +1.53 +2.36 +5.59 +R11 +0.015 +1.4 +-0.55 +1.22 +1.69 +4.14 +B18 +0.141 +1.4 +-0.55 +1.22 +1.69 +4.14 +Table 1: Experimental conditions (wave height H, wave period T, theoretical blocking velocity Ub, and normalized water depth +kd for three current velocities) of the four test cases of the Chawla and Kirby (1999) flume experiment that were considered +in this paper. The wave height and wave period were measured at the first wave gauge inside the flume, at a distance of 4.2 +m (case 1 and 3) and 5.2 m downstream (case 2 and 4) of the start of the narrow flume section. The blocking velocity was +computed based on the linear dispersion relationship. The normalized water depth based on linear wave theory is provided in +the absence of the current, for U = −0.32 m/s, and at the theoretical blocking velocity Ub. +4.3.1. Model setup +We used a curvilinear grid with a constant streamwise resolution but varying alongshore width and +resolution to replicate the flume in the numerical model. Based on the sensitivity study for linear waves +(Appendix A), the horizontal grid resolution in streamwise direction was set to ensure at least 100 points per +wavelength (in the absence of a current). This resulted in a total of 1500 cells in the streamwise direction. +We used 3 cells in the spanwise direction to reduce computational overhead. This implies that spanwise +effects were not included in the modelling, such as the sidewall boundary layers that were observed in the +flume (Chawla and Kirby, 2002). To investigate the influence of the vertical resolution, simulations were +run with [2, 4, 20] layers. The model time step was set at a value that corresponds to CFL ≈ 0.4 resulting in +about 250-500 points per wave period, which was found to be sufficiently fine for these test conditions. Waves +were generated based on linear wave theory at x = −15 m (in the absence of a current), with the incident +wave height calculated from conservation of wave action (Eq. (28)) based on the measured wave height at +the first wave gauge (located at x ≈ −5 m). A sponge layer with a width of at least three wavelengths was +positioned at the end of the flume to prevent wave reflections. +We conducted two sets of simulations to replicate the four test cases. In the first set, which serves as a +benchmark for the proposed model extension, the waves and current were modelled simultaneously through +the original set of model equations. A re-circulating current was generated through modifying the kinematic +boundary condition at the bottom (see Appendix B). The resulting discharge that is imposed at the bottom +replicates a pump system through which a volume of water is pumped into the domain at one end of the +flume and is taken out at the other end of the flume. In this manner, a current was generated inside the +numerical flume. In all simulations with this pump system, the discharge was set at Q = 0.095 m3/s based +on Chawla and Kirby (2002). With this model set-up, the modelled depth-averaged current field was in +good agreement with observations taken in the flume for a reference case excluding waves (Fig. 8b). In the +17 + +second set of simulations, we account for the current through the additional terms in the equations that +were derived in Section 2. The ambient current velocities were obtained from the simulation with the pump +system without waves (Fig. 8b). In the following, we refer to the simulations with the additional terms to +model the influence of the current on waves as an Ambient Current (AC) simulation, and we refer to the +benchmark simulations as a Pump simulation. +Non-hydrostatic models like SWASH inherently account for the dissipation by breaking waves but require +high vertical resolutions to capture the onset of wave breaking correctly (e.g,. Smit et al., 2013). To capture +the onset of breaking with coarse resolutions, Smit et al. (2013) introduced the Hydrostatic Front Approxi- +mation (HFA) that neglects the non-hydrostatic pressure locally to trigger wave breaking (i.e., switching to +the Non-Linear Shallow Water Equations, NSLWE). However, numerical instabilities developed in all 2-layer +simulations with HFA. We believe this is related to the normalized water depth of the waves at breaking. +For depth-induced wave breaking in the absence of currents (for which the HFA is normally applied), the +normalized water depth is relatively low at breaking (kd < 1), resulting in a relatively small non-hydrostatic +pressure contribution. In the wave-current simulations of this test case, the normalized water depth is rela- +tively large (kd > 4 near wave blocking, see Table 1). As a results, the contribution from the non-hydrostatic +pressure is relatively large at the location of incipient wave breaking. Excluding a relatively large contri- +bution from the non-hydrostatic pressure likely resulted in numerical instabilities and caused the model to +crash. As a result, the HFA approach cannot be used to improve the model predictions of the 2-layer model +in the case of breaking waves on a strong opposing current. In the following, we therefore only show results +for 2-layer simulations excluding HFA. +4.3.2. Results - wave reflections +For the wave condition with the smallest wave height and wave period (case R1, Table 1), waves reflected +at the blocking point, resulting in a nodal pattern in the wave height H and negligible transmission of wave +energy for x > 0 m (Fig. 9a). An energy balance based on conservation of wave action (eq. (28)) provided +a reasonable good description of the location of wave blocking. Differences between the energy balance with +the linear dispersion relationship and 2nd order Stokes dispersion relationship were generally small except +near the blocking location, where the blocking location is spatially shifted by approximately 0.25 m when +accounting for amplitude dispersion. +In Fig. 9a, we compare both results of the simulations with an Ambient Current (AC) and of a benchmark +simulation in which the current is included through a re-circulating pump. Both model setups (AC and +Pump) reproduced this blocking and reflection of waves as the simulations captured the nodal pattern in +the wave height for x < 0 m and wave energy was not transmitted beyond x = 0 m (Fig. 9a). Model +simulations were found to be sensitive to the number of layers, and were approximately converged for 4 +layers (as illustrated by the results of the AC simulations). The nodal structure in H was stronger and +18 + +Figure 9: Comparisons between the measured and modelled wave height for test cases R1 and R11 of the wave-current flume +experiment of Chawla and Kirby (1999). The black circle markers indicate the experimental observations, and the coloured +lines indicate the model predictions (light and dark blue, 2 and 4-layer simulations with AC (Ambient Current), respectively; +black, benchmark 4-layer simulation with pump system). The thin red lines show the results from an energy balance based +on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order Stokes dispersion +relationship (full red line). +spatially shifted towards the wavemaker using two vertical layers with both the AC (Fig. 9a) and Pump +setup (not shown). This indicates that reflections were stronger and blocking occurred at a weaker opposing +current velocity when using this coarsest vertical resolution. Increasing the vertical resolution improved the +results of the AC simulation, although H was over predicted at blocking compared to the measurements +and the benchmark simulation. Results of the 4-layer benchmark Pump simulation were in good agreement +with the measurements, apart from a slight spatial shift (approx. 0.15m) of the blocking location and nodal +pattern. +For test case R11, blocking was expected at x ≈ 0 m based on the energy balance with linear dispersion +(Fig. 9b). In contrast, the energy balance with 2nd order dispersion predicted no blocking but transmission +of energy for x > 0m. In the laboratory, partial reflections occurred at the blocking point with partial +transmission of energy for x > 0. Both the AC and Pump simulations captured these patterns. Similar to +case R1, simulations approximately converged when 4 layers were used. The 4-layer benchmark simulation +was in best agreement with the observations, and captured both the spatial variability and magnitude of +H. The 4-layer AC simulations overpredicted H near the linear blocking point for x > 0 m (similar to test +case R1) and predicted weaker reflections resulting in a less pronounced nodal pattern for x < 0 m. +These results show that the proposed extension of the model equations captured the overall patterns in +the wave height that was observed in the laboratory and simulated by the benchmark model. Model results +of both the AC and Pump simulations were found to be sensitive to the number of layers, indicating that +the dispersive properties of the model affected the location of blocking and controlled the magnitude of wave +reflections. Discrepancies in the blocking location at coarse vertical resolutions were larger for case R1 (with +19 + +case R1 +case R11 +0.10 +a) +b) +EBL +(w) +EBNL +0.05 +2V AC +4V AC +4V Pump +0.00 +2.0 +-1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +2.0 +-1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +x (m) +x (m)a smaller wave period and thus larger kd compared to R11). This is consistent with the expected response +based on the numerical dispersion relationship (Fig. 3): the relative absolute error in Ub compared to linear +theory was 1.49% and 0.42% for R1 and R11 when using 2 layers, respectively, and < 0.25% when using 4 +layers. +4.3.3. Results - wave breaking +For larger incident wave heights (case B6 and B18), wave breaking on the opposing current was observed +during the experiment in the narrow region of the flume (x = 0 − 5 m) and wave energy was transmitted +beyond the blocking point from linear theory (Fig. 10). For case B6, the 2-layer AC simulation did not +capture the transmission of wave energy beyond the blocking point, but showed signs of wave reflections +near x = 0 m, resulting in an over prediction of the wave height for −2 < x < 0 m (Fig. 10a). Similar +results were observed for the simulation with a Pump system (not shown). These results indicate that the +2-layer simulations failed to capture the breaking of waves and transmission of energy beyond the linear +blocking point for this particular test case. Increasing the number of vertical layers significantly improved +the model results (as indicated by the 4 and 20-layer AC simulations, Fig. 10a). In particular, the 20V +Pump benchmark simulation captured H throughout most of the domain, including the transmission of +energy beyond the linear blocking point and the gradual decay of H for x > 0 m. The 20V AC simulation +also captured part of this wave transmission but only up to x ≈ 1 m, and over predicted H near x = 0 m. +For case B18, with a larger incident wave height and period, wave energy was transmitted beyond the +linear blocking point for the 2-layer simulation with no sign of wave reflections (Fig. 10b). However, H was +Figure 10: Comparisons between the measured and modelled (20-layer simulations) wave height for test cases B6 and B18 of the +wave-current flume experiment of Chawla and Kirby (1999). The black circle markers indicate the experimental observations, +and the coloured lines indicate the model predictions (light to dark blue, 2, 4 and 20-layer simulations with AC (Ambient +Current); black, benchmark 20-layer simulations with pump system). +The thin red lines show the results from an energy +balance based on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order +Stokes dispersion relationship (full red line). +20 + +case B6 +case B18 +0.50 +a +b += +EB, +EBNL +(w) +0.25 +2V AC +4V AC +20V AC +20V Pump +0.00 +-4 +-2 +-4 +-2 +2 +4 +x (m) +x (m)over predicted for x > −2m. Simulations with the Pump provided similar results (not shown). Increasing +the number of vertical layers significantly improved the model results, and 20V AC simulations agreed well +with 20V Pump simulations apart from a slight over prediction for x > −1 m. Both 20V models were also +in satisfactory agreement with the observations, apart from an over prediction of H for x > 0 m. +For the test cases with breaking waves, the model predictions were found to be sensitive to the vertical +resolution. A relatively fine vertical resolution was found to be required to capture changes in the wave +height. For case B6, a fine vertical resolution was required to prevent wave-reflections at the blocking point +and to capture (part of) the transmission of energy for x > 0. In contrast, wave energy was transmitted +beyond the linear blocking point at coarse resolutions for case B18. +For this test case, higher vertical +resolutions were required to better capture the shoaling of waves on the opposing current. The shoaling in +2-layer simulations was similar to the linear energy balance, whereas shoaling in the case of more vertical +layers was comparable to the nonlinear energy balance and the measurements. This suggests that for B18 +a higher vertical resolution is required to capture the effect of (nonlinear) amplitude dispersion. +5. Discussion +The results of this work demonstrated that the extended SWASH model was able to capture the dominant +effects of currents on waves. Comparisons with linear theory and a spectral wave model showed that the +model captured current-induced changes to the wave amplitude and length, and current-induced refraction. +Comparisons with the laboratory experiment of Chawla and Kirby (1999, 2002) showed that the model +reproduced the (partial) reflection of monochromatic waves on an opposing current near the blocking point +in the case of small amplitude waves, and (partial) transmission and wave breaking in the case of larger +amplitude waves. For these challenging test cases, model results were found to be sensitive to the number +of vertical layers. In particular, a fine vertical resolution was required to capture the nonlinear shoaling, +breaking and (partial) transmission of the large amplitude waves on the opposing current. Importantly, the +results from the extended SWASH model were generally in good agreement with fully resolved benchmark +simulations that intrinsically accounted for the wave-current interactions. This indicates that additional +physics in the fully resolved SWASH model (e.g., vertical variations in the ambient flow, and the influence +of waves on the ambient currents) did not significantly affect the wave dynamics in these test cases. +Instead, this indicates that model-data discrepancies were largely inherited from the fully resolved model. +For example, these could be related to the exclusion of span-wise flow effects, and shortcomings in the +turbulence modelling (e.g., no wave breaking induced turbulence at the free-surface, incomplete description +of turbulent boundary layers). To our knowledge, current state-of-the-art CFD models such as RANS and +SPH-type models have not been widely used to simulate these nor similar laboratory experiments that +consider such complex wave-current interactions. Only a few authors have used CFD for selected cases of +21 + +laboratory experiments (e.g., Olabarrieta et al., 2010; Teles et al., 2013; Chen and Zou, 2018; Yao et al., +2023) but not for a wide variety of conditions such as the reflective and breaking cases that were considered +in this work. As such, we currently lack a clear benchmark that indicates how accurate fully resolved 3D +models including more sophisticated turbulence models can capture wave-current interactions. +6. Conclusions +This study has demonstrated that the non-hydrostatic modelling approach can be extended to account +for the effect of depth-uniform currents on the wave dynamics. By introducing a separation of scales and +assuming vertically uniform mean currents, additional terms were derived that account for changes in the +wave properties in the presence of spatially varying currents. These additional terms were included in the +open-source SWASH model. +A linear analysis of the model equations confirmed that the proposed model extension resolves the effect +of currents on the linear wave properties (e.g., change in wavelength and group velocity). Comparisons of +model predictions with linear wave theory further verified the numerical implementation. The extended +SWASH model captured changes in the wavelength and amplitude in the presence of opposing and following +currents for small amplitude waves. As a next step, we validated the model for more complex spatially +varying flow fields: a vortex ring and a jet-like current. +SWASH predictions were compared with the +spectral wave model SWAN, including the Quasi-Coherent formulation to account for constructive and de- +constructive wave interference effects. Comparisons of bulk wave parameters (significant wave height and +mean wave direction) showed that the extended SWASH model was able to account for the current-induced +refraction of both flow fields, and the resulting spatial variability in the wave height. +Finally, we compared model predictions with a flume experiment that considered blocking and breaking +of monochromatic waves on a strong opposing current. Although the model tended to overpredict the wave +height, it was able to reproduce reflections of small amplitude waves, and breaking of larger amplitude +waves. +For breaking waves, model results were improved by increasing the vertical resolution (from 2 +to 20 layers). Results of the newly derived model were generally consistent with fully resolved SWASH +simulations (in which a recirculating current was included through an inflow and outflow boundary at the +bottom). This indicates that model-data discrepancies were largely inherited from the fully-resolved model +and not introduced by missing physics in the extended model (e.g., no vertical variation of the ambient +current, and no effect of waves on the ambient current). +The findings of this work thereby demonstrated that phase-resolving models can be extended with +additional terms to account for the major effect of ambient depth-uniform currents on the wave dynamics. +This will allow models like SWASH to more accurately and efficiently simulate the wave dynamics in coastal +environments where tidal and/or wind-driven currents are present. +22 + +Figure A.1: Changes to the height (panel a and c) and length (panel b and d) of a monochromatic wave (T = 10 s) on an +opposing current (U = [−3, −1] m/s) as a function of the temporal resolution with a fixed grid resolution ∆x/L0 = 60 (panel +a-b) and as a function of the horizontal grid resolution with a fixed temporal resolution ∆t = T/1000 (panel c-d). The full +lines indicate the SWASH results and the dashed lines indicate the results according to linear wave theory. Results for U = −3 +m/s are printed in blue and results for U = −1 m/s in orange. For SWASH, the horizontal line with marker indicates the +average change to the simulated wave height H in the current region, and the vertical lines with horizontal endings indicate +the maximum and minimum H in the current region. The wave height and length are normalized by the incident wave height +and length, respectively. +Appendix A. Sensitivity study +The behaviour of the SWASH model was found to be sensitive to the horizontal grid resolution ∆x and +the time-step ∆t. To illustrate the sensitivity to the grid resolution, we consider a set of simulations of a +T = 10 s monochromatic wave on a U = [−3, −1] m/s current for a range of horizontal grid and temporal +resolutions. To study the influence of ∆t and ∆x separately, the first set considers simulations with fixed +∆x = L0/60 for a range of ∆t, and the second set corresponds to several simulations with fixed ∆t = T/1000 +but for a range of ∆x. +Changes to the wavelength L were not sensitive to either ∆x and ∆t. On the other hand, changes to +the wave height H were sensitive to the model settings. The sensitivity was larger for the stronger current +velocity. +Modelled changes to H were less sensitive to the horizontal grid resolution, except for coarse +resolutions (∆x/L0 < 40), with relatively weak improvement for ∆x/L0 ≤ 40 (Fig. A.1). Modelled changes +to H were sensitive to the time-step, especially for U = −3 m/s. For this current velocity at larger time- +steps, significant dissipation of wave energy occurred in the current region (as illustrated by the vertical +lines in Fig. A.1 at smaller ∆t/T). For finer temporal resolutions, this non-physical dissipation reduced +23 + +3 +1.5 +a) +b) +(w) +2 +U=3 m/s +(w) +1.0 +“H/H +U=1 m/s +L/L o +0.5 +0.0 +103 +104 +103 +104 +T/△t +T/△t3 +1.5 +( +(p +(w) +2 +1.0 +H/H +107/7 +0.5 +0.0 +50 +100 +150 +50 +100 +150 +Lo/Ax +Lo/Axand model results approximately converged to the solution of linear wave theory. This sensitivity to the +horizontal grid and temporal resolution was primarily significant for strong opposing currents relative to +the wave group velocity. For following currents and weak opposing currents the model results were not +sensitive to ∆x and ∆t (as illustrated by the results for U = −1 m/s). Based on this sensitivity study, the +optimal horizontal grid and temporal resolution for which model predictions were sufficiently converged was +concluded to be ∆x = L0/100 and ∆t = T/1000. +Appendix B. Re-circulating current +To generate a re-circulating current in the model, we impose an inward and outward flux at the bottom +at either side of the model domain. For this purpose, we have adopted the kinematic boundary condition +as follows, +wz=−d = −u∂d +∂x − v ∂d +∂y ± fs +P +W , +(B.1) +where P is a discharge and W is the width of the region where the discharge is specified. By introducing an +equal discharge of opposing sign in a region at either side of the numerical domain, a recirculating current +is generated inside the domain. To reduce the spin-up time, we use a smoothing function fs to gradually +ramp up the discharge from 0 to P. The smoothing function is defined as, +fs = 0.5 (1 + tanh( t +TS +− 3)), +(B.2) +where TS is the smoothing period of the pump (taken as TS = 15 s in the simulations of this work). +Appendix C. Linear semi-discrete analysis of the model equations +The numerical dispersion relationship can be derived from the linearized and semi-discretized set of +model equations (e.g., Cui et al., 2014; Bai and Cheung, 2013; Smit et al., 2014). Based on Smit et al. +(2014), the linearized and semi-discretized SWASH equations extended with the additional terms for the +wave-current interactions (on the right hand side) for N vertical layers reads, +∂u′ +n− 1 +2 +∂t ++ g ∂ζ′ +∂x + 1 +2 +∂pnh,n +∂x ++ 1 +2 +∂pnh,n−1 +∂x += −U +∂u′ +n− 1 +2 +∂x +, +for n = 1...N, +(C.1) +∂wn + wn−1 +∂t ++ 2pnh,n − pnh,n−1 +∆z += −U ∂wn + wn−1 +∂x +, +for n = 1...N, +(C.2) +∂u′ +n− 1 +2 +∂x ++ wn − wn−1 +∆z += 0, +for n = 1...N, +(C.3) +∂ζ′ +∂t + ∆z +N +� +n=1 +∂u′ +n− 1 +2 +∂x += −U ζ′ +∂x. +(C.4) +The flow variables in the above set of equations are located on a staggered grid, with u′ located in a cell center +(n − 1 +2) and w and pnh at a vertical cell face (n). Assuming a horizontal bottom (w0=0) and considering +24 + +the initial value problem in an infinite domain (with ∆z = d/N), we assume that the flow variables have +a solution of the form y = ˆyexp(ikx − iωt) (where ˆy is the complex amplitude of a flow variable, k is the +wavenumber and ω the absolute wave frequency). Substituting this into the above set of equations for each +variable results in a system of equations of the form Aˆy = 0. The numerical dispersion relationship can +subsequently found from Det(A) = 0 using symbolic algebra software. With the addition of an ambient +current U, the numerical dispersion relationship provides a relationship between ω and k in the presence of +a current with velocity U for N vertical layers. The relative group velocity can be found from the numerical +dispersion relationship as cg,r = ∂σ +∂k for an arbitrary current velocity U (with ω = σ + kU). +25 + +References +Akrish, G., Smit, P., Zijlema, M., Reniers, A., 2020. 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URL: http://onlinelibrary.wiley.com/doi/10.1002/ +fld.821/abstracthttp://doi.wiley.com/10.1002/fld.821, doi:10.1002/fld.821. +30 + diff --git a/DdAzT4oBgHgl3EQfwf7y/content/tmp_files/load_file.txt b/DdAzT4oBgHgl3EQfwf7y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e451d27855ce64935bc9c47b569d87058955444d --- /dev/null +++ b/DdAzT4oBgHgl3EQfwf7y/content/tmp_files/load_file.txt @@ -0,0 +1,1725 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf,len=1724 +page_content='Including the effect of depth-uniform ambient currents on waves in a non-hydrostatic wave-flow model Dirk P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Rijnsdorpa,∗, Arnold van Rooijenb, Ad Reniersa, Marion Tissiera, Floris de Wita,c, Marcel Zijlemaa aEnvironmental Fluid Mechanics section, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Netherlands bOceans Graduate School & UWA Oceans Institute, The University of Western Australia, Australia cSvasek Hydraulics, The Netherlands Abstract Currents can affect the evolution of waves in nearshore regions through altering their wavenumber and amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Including the effect of ambient currents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', tidal and wind-driven) on waves in phase-resolving wave models is not straightforward as it requires appropriate boundary conditions in combination with a large domain size and long simulation duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this paper, we extended the non-hydrostatic wave-flow model SWASH with additional terms that account for the influence of a depth-uniform ambient current on the wave dynamics, in which the current field can be taken from an external source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', from observations or a circulation model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We verified the model ability by comparing predictions to results from linear theory, laboratory experiments and a spectral wave model that accounts for wave interference effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' With this extension, the model was able to account for current-induced changes to the wave field (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', changes to the wave amplitude, length and direction) due to following and opposing currents, and two classical examples of sheared currents (a jet-like current and vortex ring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, the model captured the wave dynamics in the presence of strong opposing currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This includes reflections of relatively small amplitude waves at the theoretical blocking point, and transmission of breaking waves beyond the theoretical blocking point for larger wave amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The proposed model extension allows phase-resolving models to more accurately and efficiently simulate the wave dynamics in coastal regions with tidal and/or wind-driven flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Keywords: wave-current interactions, non-hydrostatic, SWASH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Introduction Complex coastal regions such as estuaries and tidal inlets often feature the joint occurrence of surface gravity waves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', swell and wind seas) and currents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', riverine, tidal, and wind-driven flows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These processes typically occur at different spatial and temporal length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Currents generally experience ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Email address: d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='rijnsdorp@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='nl (Dirk P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Rijnsdorp) Preprint submitted to Elsevier January 5, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='01725v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 variations at hour to day timescales and over O(km) length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To the contrary, waves have periods of several seconds and length scales of O(10 − 100 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Waves propagating over spatially varying currents conserve wave action (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Bretherton and Garret, 1968;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2005) but experience a change in their wavelength associated with the Doppler’ shift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Peregrine, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Holthuijsen, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As a result, the wave celerity and group velocity change, resulting in changes in wave amplitude and wave direction (current-induced shoaling and refraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In strong currents that oppose the direction of wave propagation, the group velocity cg approaches zero, resulting in significant increases of the wave height and wave-blocking when cg = 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Chawla and Kirby, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, waves steepen in opposing currents which may trigger wave breaking resulting in additional dissipation of wave energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Chawla and Kirby, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Current-induced changes in the wave shape can in turn impact the magnitude of wave-driven sediment transport (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Roelvink and Stive, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Hoefel and Elgar, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Including for the current effects on waves is thus important when predicting sediment transport and the resulting morphological changes in coastal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To date, modelling of combined wave-current actions in coastal regions has generally relied on the coupling of phase-averaged wave models and circulation models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Lesser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Roelvink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Uchiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Dodet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Olabarrieta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2014) through either the radiation stress (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Longuet-Higgins and Stewart, 1962, 1964) or vortex force formalism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Craik and Leibovich, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' McWilliams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Such coupled models have been successfully adopted to simulate the hydrodynamics in a variety of nearshore regions, ranging from sandy beaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Orzech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Luijendijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Rafati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2021) to tidal inlets and rivers where strong ambient currents can occur (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Dodet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Nienhuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' However, such a coupled approach relies on spectral wave models that do not intrinsically account for phase-dependent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', wave-interference and diffraction) and nonlinear wave processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', triad interactions and wave breaking) but rely on parametrizations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As an alternative to phase-averaged wave models, phase-resolving wave models have been developed to simulate the nearshore evolution of waves in the presence of ambient currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Linear phase-resolving wave models based on the mild-slope equations have been shown to capture changes to the wave kinematics associated with the Doppler shift (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Booij, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Kirby and Dalrymple, 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This has allowed such models to capture the effect of prescribed ambient currents on the nearshore wave evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Touboul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Models based on the mild-slope equations generally rely on assumptions of linear wave theory, although they can be extended to account for higher order wave effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Kaihatu and Kirby, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, they do not inherently account for wave-induced currents but require a coupling to a circulation model to capture such effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Alternatively, weakly to fully nonlinear phase-resolving wave-flow models based on Boussinesq-type for- mulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Peregrine, 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Madsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Nwogu, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Kirby, 2016) or the non-hydrostatic 2 approach (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Zijlema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Wei and Jia, 2014) can be used to simulate waves and wave-induced currents in coastal regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Feddersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Rijnsdorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Such models intrinsically account for phase-dependent wave effects, nonlinear wave interactions, and the generation of wave-induced currents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', longshore currents and rip currents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' However, directly including tidal and/or wind-driven currents in such models is not straightforward due to the range of spatial and temporal scales required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For example, including tidal currents in a phase-resolving model would typically require a significantly larger computational time to allow for spin-up of the tidal flow and a larger domain with appropriate boundary conditions to allow for the propagation of the tidal wave in and out of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Due to the excessive computational costs of such a model setup, this presently inhibits a direct inclusion of such currents in phase-resolving wave-flow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Several efforts have been made to account for the interactions between waves and a prescribed ambient current in nonlinear phase-resolving models based on the Boussinesq or non-hydrostatic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Most efforts focused on extending Boussinesq-type formulations to account for interactions between waves and an ambient current (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Son and Lynett, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Yang and Liu, 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Efforts to extend non-hydrostatic models have been limited to de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (2017), who added a spatially homogeneous pressure term in the alongshore momentum equation of a non-hydrostatic model to simulate the nearshore wave dynamics in the presence of alongshore tidal flows at a sandy beach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Despite this progress on including the effect of ambient currents on waves in nonlinear phase-resolving wave-flow models, their application at complex coastal sites have not yet been able to account for the effect of spatially varying current fields from tides and/or wind on the wave dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Risandi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Rijnsdorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Baker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this work, we extend the non-hydrostatic wave model SWASH (Zijlema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011) to account for the effect of a prescribed depth-uniform ambient current on the wave dynamics, in which the current field can be obtained from an external source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', observations or a circulation model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' By introducing a separation of scales and assuming vertically uniform mean flows, we derive additional terms to the governing equations that account for the effect of a spatially varying depth-uniform current on the waves (Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Comparisons with linear wave theory, a spectral wave model and flume experiments show that the proposed model is able to account for changes in the wave height and wavelength due to an ambient currents (Section 3-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In Section 5-6, we conclude that the proposed extension allows non-hydrostatic models to account for the effect of ambient currents on waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Numerical Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Governing equations The governing equations of the model are the Reynolds-Averaged Navier-Stokes (RANS) equations for an incompressible fluid that is bounded by the bottom d(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' y) and a free-surface ζ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' where (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' z) 3 are the Cartesian coordinates and t is time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' ∂u ∂x + ∂v ∂y + ∂w ∂z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (1) ∂u ∂t + u∂u ∂x + v ∂u ∂y + w∂u ∂z + g ∂ζ ∂x + ∂pnh ∂x = ∂τxx ∂x + ∂τxy ∂y + ∂τxz ∂z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (2) ∂v ∂t + u∂v ∂x + v ∂v ∂y + w∂v ∂z + g ∂ζ ∂y + ∂pnh ∂y = ∂τyx ∂x + ∂τyy ∂y + ∂τyz ∂z ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (3) ∂w ∂t + u∂w ∂x + v ∂w ∂y + w∂w ∂z + ∂pnh ∂z = ∂τzx ∂x + ∂τzy ∂y + ∂τzz ∂z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (4) In this set of equations, pnh is the non-hydrostatic pressure, (u, v, w) are the velocity components in (x, y, z) direction, respectively, τ represents the turbulent stress (estimated using an eddy viscosity approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The kinematic boundary conditions at the bottom and the free-surface follow from the assumption that the vertical boundaries of the fluid are single valued functions of the horizontal coordinates, wz=ζ = ∂ζ ∂t + u ∂ζ ∂x + v ∂ζ ∂y , (5) wz=−d = −u∂d ∂x − v ∂d ∂y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (6) Integrating the local continuity equation over the water column results in a global continuity equation that describes the temporal evolution of the free-surface, ∂ζ ∂t + ∂ ∂x ζ � −d udz + ∂ ∂y ζ � −d vdz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (7) Assuming a constant atmospheric pressure (equal to zero for convenience) and neglecting viscous stresses at the free-surface, the non-hydrostatic pressure is set to zero at the free-surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Stelling and Zijlema, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' At the bottom, the tangential stress is prescribed based on the quadratic friction law (in the case of a coarse vertical resolution) or the law of the wall (in the case of a fine vertical resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Turbulent stresses are modelled using the eddy-viscosity model and the k-ϵ turbulence closure model (See Rijnsdorp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2017, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Combined with boundary conditions at all horizontal edges of the physical domain, the above set of equations forms the basis of the SWASH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Including the effect of currents on waves In this work, we set out to decouple the modelling of the surface waves and the currents that are slowly-varying with respect to the wave timescale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', tidal currents and wind-driven currents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' With this approach, we aim to account for the current effect on waves through prescribing an ambient current field from an other model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', a circulation model) that alters the wave dynamics solved by the RANS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4 To this end, we separate the horizontal flow variables and surface elevation as, u(x, y, z, t) = U(x, y) + u′(x, y, z, t), (8) v(x, y, z, t) = V (x, y) + v′(x, y, z, t), (9) ζ(x, y, t) = η(x, y) + ζ′(x, y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (10) In these equations, [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=']′ denotes variables which we associate with wave-related motions and wave-induced currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Capital letters (U and V ) represent vertically uniform horizontal flow velocities and η a mean water level, which both vary over a timescale much larger than the wave motions and are considered to be constant over the wave-timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Substituting this separation of variables into the governing equations and neglecting the viscous contributions and tangential stress at the bottom yields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' ∂U + u′ ∂x + ∂V + v′ ∂y + ∂w ∂z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (11) ∂U + u′ ∂t + (U + u′)∂U + u′ ∂x + (V + v′)∂U + u′ ∂y + w∂U + u′ ∂z + g ∂η + ζ′ ∂x + ∂pnh ∂x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (12) ∂V + v′ ∂t + (U + u′)∂V + v′ ∂x + (V + v′)∂V + v′ ∂y + w∂V + v′ ∂z + g ∂η + ζ′ ∂y + ∂pnh ∂y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (13) ∂w ∂t + (U + u′)∂w ∂x + (V + v′)∂w ∂y + w∂w ∂z + ∂pnh ∂z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (14) ∂η + ζ′ ∂t + ∂ ∂x η+ζ′ � −d (U + u′)dz + ∂ ∂y η+ζ′ � −d (V + v′)dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (15) By taking the temporal average over the wave-motion scales and integrating the horizontal momentum equations over the vertical we obtain the following depth-averaged mean flow equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' ∂U ∂x + ∂V ∂y = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (16) ∂U ∂t + U ∂U ∂x + V ∂U ∂y + g ∂η ∂x = − η � −d (u′ ∂u′ ∂x + v′ ∂u′ ∂y )dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (17) ∂V ∂t + U ∂V ∂x + V ∂V ∂y + g ∂η ∂y = − η � −d (u′ ∂v′ ∂x + v′ ∂v′ ∂y )dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (18) ∂η ∂t + ∂ (d + η) U ∂x + ∂ (d + η) V ∂y = − ∂ ∂x η+ζ′ � −d u′dz − ∂ ∂y η+ζ′ � −d v′dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (19) In these equations, we can recognise the contribution to the radiation stress gradient from the orbital velocities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', u′ ∂u′ ∂x ) and contributions in the global continuity equation that are related to stokes drift (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', the part of the integral above the wave trough in the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the following we assume that waves do not influence the ambient currents, and neglect these contributions in the mean flow equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5 Subsequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' we derive a new set of wave equations by subtracting the mean equations (16)-(19) from the instantaneous equations (11)-(15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' ∂u′ ∂x + ∂v′ ∂y + ∂w ∂z = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (20) ∂u′ ∂t + u′ ∂u′ ∂x + v′ ∂u′ ∂y + w∂u′ ∂z + g ∂ζ′ ∂x + ∂pnh ∂x = −(U ∂u′ ∂x + u′ ∂U ∂x + V ∂u′ ∂y + v′ ∂U ∂y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (21) ∂v′ ∂t + u′ ∂v′ ∂x + v′ ∂v′ ∂y + w∂v′ ∂z + g ∂ζ′ ∂y + ∂pnh ∂y = −(U ∂v′ ∂x + u′ ∂V ∂x + V ∂v′ ∂y + v′ ∂V ∂y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (22) ∂w ∂t + u′ ∂w ∂x + v′ ∂w ∂y + w∂w ∂z + ∂pnh ∂z = −(U ∂w ∂x + V ∂w ∂y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (23) ∂ζ′ ∂t + ∂ ∂x η+ζ′ � −d u′dz + ∂ ∂y η+ζ′ � −d v′dz = −∂ζ′U ∂x − ∂ζ′V ∂y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (24) In the above set of equations, we can recognise the original set of equations (when dropping the prime superscripts) including several additional terms (on the right-hand-side) that account for the influence of a depth-uniform ambient current on the wave motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We note that the influence of changes in the mean water level associated with the ambient current in the global continuity equation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', the integral up to η+ζ′ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (24)) can be straightforwardly incorporated by incorporating η in the still water depth (d = d + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Numerical implementation In the numerical implementation of the governing set of equations, the continuous description of time and horizontal dimensions are replaced by discrete approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In SWASH, the equations are discretised on regular or curvilinear grid for the horizontal dimensions and a terrain-following layering system for the vertical coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A staggered grid arrangement is used to position the flow variables on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Further details regarding the numerical implementation of the original set of equations can be found in several previous papers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Stelling and Zijlema, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Zijlema and Stelling, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Zijlema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011), and will not be detailed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 𝑖!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 𝑖 + 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 𝑖" 𝑖 𝑖 + 1 𝜁#, 𝑈 𝑢# 𝑖 − 1" 𝑖 − 1 Figure 1: Illustration of the arrangement of the ambient velocity U and wave-related variables [ζ, u] on the computational grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 6 The flow velocities [U, V ] from the ambient current are positioned on the grid at the same location as the free-surface variable of the original set of equations ζ′ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', at horizontal cell centres, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Linear interpolation is used to define the ambient current on the SWASH grid in the case that the ambient current is provided on a coarser grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The numerical implementation of the additional terms is – where possible – based on the existing implementation of the advective terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The terms in the horizontal momentum equations are discretised using the MacCormack predictor-corrector technique (MacCormack, 1969) combined with flux limiters (See Zijlema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2011, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We use a flux limited first-order Euler scheme to discretise the terms in the vertical momentum equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Finally, the terms in the global continuity equation are discretised using central differences and the Crank-Nicholson method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Linear properties of the model equations We analysed the linear properties of the model equations by deriving the numerical linear dispersion relationship (see Appendix C) to verify that the model captures the effect of currents on waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The numerical dispersion relationship derived from the extended model equations (20)-(24) provides a polynomial relationship fN between the absolute wave frequency ω (in the reference frame of a stationary observer) and the wavenumber k for depth d and current velocity U depending on the number of layers N, ω = fN(k, d, U, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (25) We compared linear wave properties based on this numerical dispersion relationship with the Doppler shifted dispersion relationship from linear theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Holthuijsen, 2007), ω − kU = σ = � gk tanh kd, (26) in which σ is the intrinsic angular frequency (in the reference frame of an observer that is moving with the current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Based on this numerical and linear dispersion relationship, several wave properties can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The relative group velocity (in a reference frame moving with the current) is given by cg,r = ∂σ ∂k , and the absolute group velocity (in the reference frame of a fixed observer) is cg = cg,r + U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, we also compared the numerical dispersion relationship of the extended model equations to the Doppler shifted numerical relationship of the original model equations, ω − kU = σ = fN,U=0(k, d, N), (27) where fN,U=0 is the numerical dispersion relationship in the absence of a current (Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This Doppler shifted numerical dispersion relationship provides the influence of a current on waves when the current is simulated as part of the model equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', by means of a pump system as described in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Importantly, we found that all linear properties based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (25) (the numerical dispersion 7 Figure 2: Absolute relative error in the absolute wave frequency ω (panel a-c) and relative group velocity cg,r = ∂σ ∂k (panel d-f) as a function of the normalized water depth kd for U = [0, −2, −4] m/s (left to right panels, as indicated by the subplot titles) based on the numerical dispersion relationship of the N layer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results are shown for N = [1, 2, 4] layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The vertical dashed lines indicate where blocking occurs, with the colors indicating the number of layers of the numerical dispersion relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The vertical black line indicates where blocking occurs according to the linear dispersion relionship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' relationship of the extended model equations) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (27) (the Doppler shifted numerical dispersion relationship) were identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This confirms that the linear effect of current on waves can be captured by including additional terms in the model equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the remainder of this section, we therefore only compared linear wave properties based on the numerical dispersion relationship of the extended model equations (25) and the Doppler shifted dispersion relationship based on linear theory (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Assuming that the horizontal scales are sufficiently resolved, the dispersive property of the model depends on the number of vertical layers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Introducing a current does not significantly affect the error in wave dispersion, as ∆ω under currents is comparable to the case with U = 0 m/s (compare Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2a,c,d with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Discrepancies in cg,r similarly depend on the number of layers and are not significantly affected by introducing a current (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2e-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' When introducing an opposing current (U < 0), no wave solution exists beyond a certain kd as indicated by the vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 2c-d and 2g-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Here, waves are blocked as cg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The kd at which blocking occurs is sensitive to the number of layers, and is in better agreement with linear theory when a larger number of layers is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This is further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 3, which shows the current velocity at which blocking occurs (Ub) as a function of kd based on the linear and numerical dispersion relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' With coarse vertical resolutions, waves are blocked on weaker opposing currents compared to linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Increasing the number of vertical layers improves Ub, with errors in Ub < 10% for kd < [2, 7, 30] in the case of N = [1, 2, 4] layers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These findings show that the number of 8 U= 4m/s U=0m/s U=-2m/s U=-4m/s a) b) () (p 4 白4 32 0 100 100 101 10° 100 15 f) h) e g) 10 4 △5 0 10° 100 101 100 101 10°101 kd (rad) kd (rad) kd (rad) kd (rad)Figure 3: Panel a: Blocking current velocity Ub (panel a) as a function of kd based on the linear dispersion relationship (black line) and the numerical dispersion relationship for N = [1, 2, 4] (blue, red, and yellow line, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Panel b: Absolute relative error in Ub from the numerical dispersion relationship for N = [1, 2, 4] relative to the linear dispersion relationship as a function of kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' layers controls the accuracy with which the model recovers the linear wave properties in the presence of a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Test Cases 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Linear waves on opposing and following currents To verify the numerical implementation of the additional terms in the governing equations, we compared model predictions of changes in the wavelength and wave amplitude due to a gradient in the current velocity to linear wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As illustrated by the linear properties of the equations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 3), waves that travel over a current gradient experience a change in their kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The wavelength decreases and the amplitude increases for waves on an opposing current and vice-versa on a following current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this section, we verify if the developed model captures these changes to the wave field for linear waves that interact with opposing and following currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We considered monochromatic waves with a height of H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='01 m and wave periods T = [5, 10, 15] s in water of constant depth d = 10 m (corresponding to kd = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4] in the absence of a current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A range of current velocities was simulated with U ranging from -6 to 4 m/s with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='25 m/s increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We compared the influence of the current on the wave height and the wavelength with linear wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The change in wavelength and group velocity follows from the linear dispersion relationship (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The change in wave height follows from the conservation of wave action, ∂ ∂x cgE σ = 0, (28) with the wave energy density E of a monochromatic wave (E = 1/8H2) and the absolute group velocity cg taken from linear theory (with cg = cg,r +U, and cg,r = ∂σ ∂k obtained from the linear dispersion relationship).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9 0 20 a (s /u) 5 △Ub 10 °-10 15 0 10-1 100 101 100 101 10-1 kd (rad) kd4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model set-up To allow for the current effect on the waves to develop, the model setup included a transition region with a width of several wavelengths to gradually transition from no current to the respective maximum current velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The transition region had a width of 10L0, and the region with maximum flow had a width of 10L0 (with L0 the wavelength in the absence of a current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These widths were found to be sufficient to allow for a gradual change in the wave dynamics, and provided a sufficiently large domain to determine the wave parameters in the presence of the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Waves were generated at the left boundary with a wavemaker based on linear wave theory which was positioned 3L0 away from the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A sponge layer with a width of 5L0 was positioned in front of the right boundary to absorb the waves and prevent any wave reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The sponge layer was positioned at a distance of 3L0 from the transition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The model was set-up with two layers in the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The horizontal resolution and time-step were selected based on a sensitivity study (Appendix A): the horizontal grid resolution was set at ∆x = L0/100 and the time-step was set at ∆t = T/1000 (with T the incident wave period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The surface elevation ζ was outputted at all computational grid points for a duration of 5 wave periods after a spin-up time that ensured statistically stationary results inside the numerical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We used zero-crossing analysis in the maximum current region to determine the wavelength in presence of a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' First, the surface elevation ζ was interpolated to a fine horizontal grid in the current region to allow for an accurate estimation of the wavelength independent of the grid resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The wavelength was subsequently computed from the zero-crossing analysis as the average wavelength over the current region and the output duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We computed the wave height in the current region as H = 2√2m0 (with the zeroth order moment m0 computed as the standard deviation of the surface elevation ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To gain insight in the spatial variation of H, we computed the mean, the maximum and minimum value of H in the current region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results were excluded when wave-blocking occurred in the model simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Wave blocking was recognised when the wave energy at the down-wave end of the domain (behind the current region) was < 1% of the incident wave energy at the numerical wavemaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results To illustrate the impact of the current on the wave field, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4 shows an example of the surface elevation inside the model domain for three different current velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For these three cases, modelled changes to the surface elevation in opposing and following currents qualitatively agreed with the expected changes to the wave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In an opposing current, the wavelength decreased and the wave height increased (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In contrast, the wavelength increased and the wave height decreased for a following current (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In all three illustrative cases, the wave signal at the downwave end of the flume (x > 3000 m) was identical to the incident wave signal (x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This confirms that wave action is conserved in these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To verify the model results quantitatively, we compared the change in the wave height and wavelength 10 Figure 4: Snapshot of the modelled surface elevation (blue line, left axis) and ambient current velocity (red line, right axis) in the numerical domain for three different current velocities (U = [−3, 0, 3] m/s) for a monochromatic wave with amplitude a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='01 m and period T = 10 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The dashed black line indicates the envelope of the wave elevation, and the title of each panel indicates the respective current velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 11 opposing current (U= -3 m/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 a) (-) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 (s/w) n H/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 3 no current 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 (s/w) n H/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 following current (U= 3 m/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 C) (-) H/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 (s/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0 U (m/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 3 0 500 1000 1500 2000 2500 3000 x (m)Figure 5: Normalized change to the wave height H (panel a) and wavelength L (panel b) as a function of the current velocity U for small-amplitude monochromatic waves with T = [5, 10, 15] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The wave height and wavelength were normalized by the wave parameters in absence of a current (indicated by [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=']0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Converged model results of simulations (with ∆t = T/1000 and ∆x = L/100) are indicated by colored lines (see legend) and results from linear wave theory are indicated by the thick black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the left panel, the horizontal blue line with dotted markers indicates the average change to the simulated wave height H in the current region and the vertical line with horizontal bars indicates the maximum and minimum simulated H inside the current region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The dashed vertical black lines indicate the current velocity at which wave blocking occurs according to linear wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' inside the current region with the results from linear wave theory (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For all three wave periods, linear wave theory predicted that the wave height and wavelength varied significantly for the considered range of current velocities (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For opposing currents, the wave height H increased and the wavelength L decreased, and vice versa for following currents (as was visually observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Current induced changes to the wave field were larger for shorter wave periods, with wave blocking occurring for T = [5, 10, 15] s at U ≈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='92, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='74, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='87] m/s (indicated by the vertical black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' SWASH captured the changes to the wave height and wavelength for the range of ambient current velocities and the three wave periods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This included the nonlinear dependence of H and L for U < 0 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, the model captured blocking of waves for opposing currents that are stronger than the critical flow velocity of linear wave theory (indicated by the dashed black lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For all three wave periods, simulations with current velocities stronger than the theoretical blocking velocity showed a strong decay of the wave height down-wave of the blocking point (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For simulations with U close to but just weaker than the theoretical blocking velocity, dissipation of wave energy occurred in the model over the current region (visible as the difference between the vertical lines with horizontal bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5a, which indicates the maximum and minimum H in the current region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the absence of physical mechanisms for dissipation, this is likely related to numerical diffusion when the waves (with shorter lengths) propagate in the current region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For weaker U this dissipation becomes smaller and the model results were in good agreement with linear theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This numerical dissipation was found to be dependent on the horizontal grid 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 4 - a) T=5 s b)7 T=10 s 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1 T=15 s 1 H/Ho L/Lo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 6 4 2 0 2 4 6 4 2 2 U (m/s) U (m/s)resolution and time step, with improved agreement for strong U for finer spatial and temporal resolutions (in accordance with the results in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Sheared current fields In coastal regions, spatially varying current fields exist (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', tidal currents) that can induce wave re- fraction and result in focal zones that give rise to wave interference patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Yoon and Liu, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Akrish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this section, we verify the ability of the model to capture such wave patterns using two classical examples of wave-current interactions: the interactions of waves with a jet-like current and a vortex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model results were compared with the spectral wave model SWAN (Booij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 1999) extended with a quasi-coherent formulation that accounts for wave interference due to variable topography (Smit and Janssen, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2015) and currents (Akrish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model set-up The model set-up was based on the work of Akrish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The region of interest spanned a domain of 4 × 4 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Two different simulations were considered, one with a jet-shaped and the other with a vortex-shaped current field, positioned along the central axis of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The maximum velocities for the simulations were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='38 m/s and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 m/s, respectively (refer to Akrish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020, for a mathemetical formulation of the current fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' At the wavemaker positioned along the western boundary, a Gaussian shaped wave-spectrum in frequency and direction was forced with Hs = 1 m, Tp = 20 s and a standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0015 Hz in frequency space and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='78◦ in directional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The waves had a mean direction of θ0=15◦ and 0◦ (in Cartesian coordinates) for the jet and vortex current, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the SWAN model, the physical domain was discretised with ∆x = ∆y = 50 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The spectral domain was discretised with 45 discrete frequencies that were logarithmically spaced between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='005 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='085 Hz, and with a directional resolution of 2◦ between -90 and 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For the SWASH model, we extended the domain with a 500 m wide sponge layer at the eastern side of the domain to prevent any wave reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The domain was discretised with a resolution of ∆x = 2 m and ∆y =4 m (which resulted in ≈ 100 points per wavelength throughout the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The time step was set at ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='05 s, equalling 300 points per wave period and resulting in CFL ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results Due to changes in wavelength induced by the current, waves were refracted by the vortex ring (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 6a-c and 6g-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This current-induced refraction resulted in considerable variations in the significant wave height, with ridges of larger wave heights where waves focussed and depressions of lower wave heights where waves diverged (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 6a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For this current field, quasi-coherent (QC) effects needed to be taken into account in SWAN to resolve the constructive and de-constructive wave interference that altered the wave field downstream of the vortex ring (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Akrish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The bulk wave heights and mean wave 13 Figure 6: Changes to the significant wave height Hm0 and mean wave direction θ due to a vortex ring current field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Panels a-c show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b) and default SWAN (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' directions predicted by the extended SWASH model were in satisfactory agreement with the results from the SWAN QC model throughout the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Similarly, waves refract as they propagated into the jet-like current field, resulting in a change of mean wave direction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 7g-i) and in regions with increased and decreased wave heights due to con- vergence/divergence of wave energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 7a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Similar to the vortex ring, quasi-coherent effects need to be incorporated in SWAN to account for the constructive and de-constructive wave interference that altered the wave field, although this effect was smaller compared to the vortex ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In general, the SWASH predictions were in good agreement with SWAN QC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The results of this test case, and the vortex ring, illustrate that SWASH including the additional terms in the model equations is able to capture the effect of current-induced 14 SWASH SWAN QC SWAN 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 af bi 3000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 (w) 2000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0 1000 2000 3000 0 1000 2000 3000 0 1000 2000 3000 x (m) x (m) x (m) x=1000 m x=2000 m x=3000 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 (p el 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 SWASH SWAN QC 15 (6 15 h) 15 SWAN (6ap) 0 0 0 15 15 15 2000 4000 2000 4000 0 2000 4000 y (m) y (m) y (m)Figure 7: Changes to the significant wave height Hm0 and mean wave direction θ due to a jet-like current field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Panels a-c show a spatial overview of the significant wave height (colors) and mean wave direction (black arrows), with the red arrows indicating the ambient current field, for SWASH (panel a), SWAN including the Quasi-Coherent (QC) formulation (panel b) and default SWAN (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Panels d-i show the wave height (d-f) and mean wave direction (g-i) along three alongshore transects predicted by SWASH (black lines), SWAN QC (orange lines) and default SWAN (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' refraction on the wave propagation and the resulting spatial variability in the wave field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Wave blocking, reflections and breaking on opposing currents As a final test case, we compare model predictions with the laboratory experiment of Chawla and Kirby (1999, 2002) that considered wave blocking on opposing currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The flume had a length of 30 m, a width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='6 m and still water depth of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 m, with a pump system to generate a recirculating current (with a discharge of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='095 m3/s) and a perforated wavemaker to generate waves on the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A spatially varying current was generated by means of a false wall constricting the width of the flume, with a minimal width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='36 m (see black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Blocking of waves occurred close to the start of this narrow part of the flume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 15 SWASH SWAN QC SWAN 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 a h) 3000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 (w) ^ 2000 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0 10002000 3000 0 1000 2000 3000 0 1000 20003000 x (m) x (m) x (m) x=1000 m x=2000 m x=3000 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 e) f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 SWASH 30 30 30 SWAN OC (6 h) i) SWAN (deg) 15 15 15 +0 +0 0 2000 4000 0 2000 4000 2000 4000 y (m) y (m) y (m)Figure 8: Overview of the numerical setup of the Chawla and Kirby (1999) flume experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The top panel (a) shows the flume width (black line, left axis) and a snapshot of the modelled free-surface elevation for test case 1 (blue line, right axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The bottom panel (b) shows the modelled (red line) and measured (black markers) current velocity (in the absence of waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The experiments with monochromatic waves considered a total of 23 test conditions that included 3 different incident wave periods (T = [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4] s) for a range of wave heights (H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='012 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='14 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For the low amplitude and low period waves, waves reflected with negligible transmission of wave energy beyond the blocking point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For increasing wave heights, waves started breaking at the blocking point of linear theory combined with increased transmission of wave energy beyond this theoretical blocking point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this paper, we considered 4 out of the 23 test cases: the largest and smallest wave height of both the smallest and largest wave period (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For case R1 and R11 waves reflected at the blocking point, whereas waves were breaking and wave energy was transmitted beyond the theoretical blocking point for case B6 and B18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We compared model predictions with these laboratory observations for these 4 test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Furthermore, we also computed the wave height transformation based on conservation of wave action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' ∂ ∂x cgE σ = 0, (29) Conservation of wave action is computed based on the linear dispersion relationship (similar to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1) and also based on the nonlinear dispersion relationship from 2nd order Stokes theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Dean and Dalrymple, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This nonlinear dispersion relationship accounts for the effect of amplitude dispersion, which was found to be important for these laboratory experiments (Chawla and Kirby, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='6 (w) a) width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='03 (w) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content="00 EO'O- 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 b) (s/w) n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='6 15 10 5 0 n 10 x (m)H (m) T (s) Ub (m/s) kd (U = 0 m/s) kd (U = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='32 m/s) kd (U = Ub m/s) R1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='59 B6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='126 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='59 R11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='14 B18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='141 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='14 Table 1: Experimental conditions (wave height H, wave period T, theoretical blocking velocity Ub, and normalized water depth kd for three current velocities) of the four test cases of the Chawla and Kirby (1999) flume experiment that were considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The wave height and wave period were measured at the first wave gauge inside the flume, at a distance of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2 m (case 1 and 3) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2 m downstream (case 2 and 4) of the start of the narrow flume section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The blocking velocity was computed based on the linear dispersion relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The normalized water depth based on linear wave theory is provided in the absence of the current, for U = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='32 m/s, and at the theoretical blocking velocity Ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model setup We used a curvilinear grid with a constant streamwise resolution but varying alongshore width and resolution to replicate the flume in the numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Based on the sensitivity study for linear waves (Appendix A), the horizontal grid resolution in streamwise direction was set to ensure at least 100 points per wavelength (in the absence of a current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This resulted in a total of 1500 cells in the streamwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We used 3 cells in the spanwise direction to reduce computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This implies that spanwise effects were not included in the modelling, such as the sidewall boundary layers that were observed in the flume (Chawla and Kirby, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To investigate the influence of the vertical resolution, simulations were run with [2, 4, 20] layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The model time step was set at a value that corresponds to CFL ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4 resulting in about 250-500 points per wave period, which was found to be sufficiently fine for these test conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Waves were generated based on linear wave theory at x = −15 m (in the absence of a current), with the incident wave height calculated from conservation of wave action (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (28)) based on the measured wave height at the first wave gauge (located at x ≈ −5 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A sponge layer with a width of at least three wavelengths was positioned at the end of the flume to prevent wave reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We conducted two sets of simulations to replicate the four test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the first set, which serves as a benchmark for the proposed model extension, the waves and current were modelled simultaneously through the original set of model equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A re-circulating current was generated through modifying the kinematic boundary condition at the bottom (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The resulting discharge that is imposed at the bottom replicates a pump system through which a volume of water is pumped into the domain at one end of the flume and is taken out at the other end of the flume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In this manner, a current was generated inside the numerical flume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In all simulations with this pump system, the discharge was set at Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='095 m3/s based on Chawla and Kirby (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' With this model set-up, the modelled depth-averaged current field was in good agreement with observations taken in the flume for a reference case excluding waves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the 17 second set of simulations, we account for the current through the additional terms in the equations that were derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The ambient current velocities were obtained from the simulation with the pump system without waves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the following, we refer to the simulations with the additional terms to model the influence of the current on waves as an Ambient Current (AC) simulation, and we refer to the benchmark simulations as a Pump simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Non-hydrostatic models like SWASH inherently account for the dissipation by breaking waves but require high vertical resolutions to capture the onset of wave breaking correctly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To capture the onset of breaking with coarse resolutions, Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (2013) introduced the Hydrostatic Front Approxi- mation (HFA) that neglects the non-hydrostatic pressure locally to trigger wave breaking (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', switching to the Non-Linear Shallow Water Equations, NSLWE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' However, numerical instabilities developed in all 2-layer simulations with HFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' We believe this is related to the normalized water depth of the waves at breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For depth-induced wave breaking in the absence of currents (for which the HFA is normally applied), the normalized water depth is relatively low at breaking (kd < 1), resulting in a relatively small non-hydrostatic pressure contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the wave-current simulations of this test case, the normalized water depth is rela- tively large (kd > 4 near wave blocking, see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As a results, the contribution from the non-hydrostatic pressure is relatively large at the location of incipient wave breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Excluding a relatively large contri- bution from the non-hydrostatic pressure likely resulted in numerical instabilities and caused the model to crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As a result, the HFA approach cannot be used to improve the model predictions of the 2-layer model in the case of breaking waves on a strong opposing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the following, we therefore only show results for 2-layer simulations excluding HFA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results - wave reflections For the wave condition with the smallest wave height and wave period (case R1, Table 1), waves reflected at the blocking point, resulting in a nodal pattern in the wave height H and negligible transmission of wave energy for x > 0 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' An energy balance based on conservation of wave action (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (28)) provided a reasonable good description of the location of wave blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Differences between the energy balance with the linear dispersion relationship and 2nd order Stokes dispersion relationship were generally small except near the blocking location, where the blocking location is spatially shifted by approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='25 m when accounting for amplitude dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9a, we compare both results of the simulations with an Ambient Current (AC) and of a benchmark simulation in which the current is included through a re-circulating pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Both model setups (AC and Pump) reproduced this blocking and reflection of waves as the simulations captured the nodal pattern in the wave height for x < 0 m and wave energy was not transmitted beyond x = 0 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model simulations were found to be sensitive to the number of layers, and were approximately converged for 4 layers (as illustrated by the results of the AC simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The nodal structure in H was stronger and 18 Figure 9: Comparisons between the measured and modelled wave height for test cases R1 and R11 of the wave-current flume experiment of Chawla and Kirby (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The black circle markers indicate the experimental observations, and the coloured lines indicate the model predictions (light and dark blue, 2 and 4-layer simulations with AC (Ambient Current), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' black, benchmark 4-layer simulation with pump system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The thin red lines show the results from an energy balance based on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order Stokes dispersion relationship (full red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' spatially shifted towards the wavemaker using two vertical layers with both the AC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9a) and Pump setup (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This indicates that reflections were stronger and blocking occurred at a weaker opposing current velocity when using this coarsest vertical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Increasing the vertical resolution improved the results of the AC simulation, although H was over predicted at blocking compared to the measurements and the benchmark simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results of the 4-layer benchmark Pump simulation were in good agreement with the measurements, apart from a slight spatial shift (approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='15m) of the blocking location and nodal pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For test case R11, blocking was expected at x ≈ 0 m based on the energy balance with linear dispersion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In contrast, the energy balance with 2nd order dispersion predicted no blocking but transmission of energy for x > 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In the laboratory, partial reflections occurred at the blocking point with partial transmission of energy for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Both the AC and Pump simulations captured these patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Similar to case R1, simulations approximately converged when 4 layers were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The 4-layer benchmark simulation was in best agreement with the observations, and captured both the spatial variability and magnitude of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The 4-layer AC simulations overpredicted H near the linear blocking point for x > 0 m (similar to test case R1) and predicted weaker reflections resulting in a less pronounced nodal pattern for x < 0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These results show that the proposed extension of the model equations captured the overall patterns in the wave height that was observed in the laboratory and simulated by the benchmark model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Model results of both the AC and Pump simulations were found to be sensitive to the number of layers, indicating that the dispersive properties of the model affected the location of blocking and controlled the magnitude of wave reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Discrepancies in the blocking location at coarse vertical resolutions were larger for case R1 (with 19 case R1 case R11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='10 a) b) EBL (w) EBNL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='05 2V AC 4V AC 4V Pump 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 x (m) x (m)a smaller wave period and thus larger kd compared to R11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This is consistent with the expected response based on the numerical dispersion relationship (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 3): the relative absolute error in Ub compared to linear theory was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='49% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='42% for R1 and R11 when using 2 layers, respectively, and < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='25% when using 4 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results - wave breaking For larger incident wave heights (case B6 and B18), wave breaking on the opposing current was observed during the experiment in the narrow region of the flume (x = 0 − 5 m) and wave energy was transmitted beyond the blocking point from linear theory (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For case B6, the 2-layer AC simulation did not capture the transmission of wave energy beyond the blocking point, but showed signs of wave reflections near x = 0 m, resulting in an over prediction of the wave height for −2 < x < 0 m (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Similar results were observed for the simulation with a Pump system (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These results indicate that the 2-layer simulations failed to capture the breaking of waves and transmission of energy beyond the linear blocking point for this particular test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Increasing the number of vertical layers significantly improved the model results (as indicated by the 4 and 20-layer AC simulations, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In particular, the 20V Pump benchmark simulation captured H throughout most of the domain, including the transmission of energy beyond the linear blocking point and the gradual decay of H for x > 0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The 20V AC simulation also captured part of this wave transmission but only up to x ≈ 1 m, and over predicted H near x = 0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For case B18, with a larger incident wave height and period, wave energy was transmitted beyond the linear blocking point for the 2-layer simulation with no sign of wave reflections (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' However, H was Figure 10: Comparisons between the measured and modelled (20-layer simulations) wave height for test cases B6 and B18 of the wave-current flume experiment of Chawla and Kirby (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The black circle markers indicate the experimental observations, and the coloured lines indicate the model predictions (light to dark blue, 2, 4 and 20-layer simulations with AC (Ambient Current);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' black, benchmark 20-layer simulations with pump system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The thin red lines show the results from an energy balance based on conservation of wave action using the linear dispersion relationship (dashed red line) and the 2nd order Stokes dispersion relationship (full red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 20 case B6 case B18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='50 a b = EB, EBNL (w) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='25 2V AC 4V AC 20V AC 20V Pump 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='00 4 2 4 2 2 4 x (m) x (m)over predicted for x > −2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Simulations with the Pump provided similar results (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Increasing the number of vertical layers significantly improved the model results, and 20V AC simulations agreed well with 20V Pump simulations apart from a slight over prediction for x > −1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Both 20V models were also in satisfactory agreement with the observations, apart from an over prediction of H for x > 0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For the test cases with breaking waves, the model predictions were found to be sensitive to the vertical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A relatively fine vertical resolution was found to be required to capture changes in the wave height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For case B6, a fine vertical resolution was required to prevent wave-reflections at the blocking point and to capture (part of) the transmission of energy for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In contrast, wave energy was transmitted beyond the linear blocking point at coarse resolutions for case B18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For this test case, higher vertical resolutions were required to better capture the shoaling of waves on the opposing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The shoaling in 2-layer simulations was similar to the linear energy balance, whereas shoaling in the case of more vertical layers was comparable to the nonlinear energy balance and the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This suggests that for B18 a higher vertical resolution is required to capture the effect of (nonlinear) amplitude dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Discussion The results of this work demonstrated that the extended SWASH model was able to capture the dominant effects of currents on waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Comparisons with linear theory and a spectral wave model showed that the model captured current-induced changes to the wave amplitude and length, and current-induced refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Comparisons with the laboratory experiment of Chawla and Kirby (1999, 2002) showed that the model reproduced the (partial) reflection of monochromatic waves on an opposing current near the blocking point in the case of small amplitude waves, and (partial) transmission and wave breaking in the case of larger amplitude waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For these challenging test cases, model results were found to be sensitive to the number of vertical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' In particular, a fine vertical resolution was required to capture the nonlinear shoaling, breaking and (partial) transmission of the large amplitude waves on the opposing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Importantly, the results from the extended SWASH model were generally in good agreement with fully resolved benchmark simulations that intrinsically accounted for the wave-current interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This indicates that additional physics in the fully resolved SWASH model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', vertical variations in the ambient flow, and the influence of waves on the ambient currents) did not significantly affect the wave dynamics in these test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Instead, this indicates that model-data discrepancies were largely inherited from the fully resolved model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For example, these could be related to the exclusion of span-wise flow effects, and shortcomings in the turbulence modelling (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', no wave breaking induced turbulence at the free-surface, incomplete description of turbulent boundary layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To our knowledge, current state-of-the-art CFD models such as RANS and SPH-type models have not been widely used to simulate these nor similar laboratory experiments that consider such complex wave-current interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Only a few authors have used CFD for selected cases of 21 laboratory experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Olabarrieta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Teles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Chen and Zou, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2023) but not for a wide variety of conditions such as the reflective and breaking cases that were considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As such, we currently lack a clear benchmark that indicates how accurate fully resolved 3D models including more sophisticated turbulence models can capture wave-current interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Conclusions This study has demonstrated that the non-hydrostatic modelling approach can be extended to account for the effect of depth-uniform currents on the wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' By introducing a separation of scales and assuming vertically uniform mean currents, additional terms were derived that account for changes in the wave properties in the presence of spatially varying currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' These additional terms were included in the open-source SWASH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A linear analysis of the model equations confirmed that the proposed model extension resolves the effect of currents on the linear wave properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', change in wavelength and group velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Comparisons of model predictions with linear wave theory further verified the numerical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The extended SWASH model captured changes in the wavelength and amplitude in the presence of opposing and following currents for small amplitude waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' As a next step, we validated the model for more complex spatially varying flow fields: a vortex ring and a jet-like current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' SWASH predictions were compared with the spectral wave model SWAN, including the Quasi-Coherent formulation to account for constructive and de- constructive wave interference effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Comparisons of bulk wave parameters (significant wave height and mean wave direction) showed that the extended SWASH model was able to account for the current-induced refraction of both flow fields, and the resulting spatial variability in the wave height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Finally, we compared model predictions with a flume experiment that considered blocking and breaking of monochromatic waves on a strong opposing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Although the model tended to overpredict the wave height, it was able to reproduce reflections of small amplitude waves, and breaking of larger amplitude waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For breaking waves, model results were improved by increasing the vertical resolution (from 2 to 20 layers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results of the newly derived model were generally consistent with fully resolved SWASH simulations (in which a recirculating current was included through an inflow and outflow boundary at the bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This indicates that model-data discrepancies were largely inherited from the fully-resolved model and not introduced by missing physics in the extended model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', no vertical variation of the ambient current, and no effect of waves on the ambient current).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The findings of this work thereby demonstrated that phase-resolving models can be extended with additional terms to account for the major effect of ambient depth-uniform currents on the wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This will allow models like SWASH to more accurately and efficiently simulate the wave dynamics in coastal environments where tidal and/or wind-driven currents are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 22 Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1: Changes to the height (panel a and c) and length (panel b and d) of a monochromatic wave (T = 10 s) on an opposing current (U = [−3, −1] m/s) as a function of the temporal resolution with a fixed grid resolution ∆x/L0 = 60 (panel a-b) and as a function of the horizontal grid resolution with a fixed temporal resolution ∆t = T/1000 (panel c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The full lines indicate the SWASH results and the dashed lines indicate the results according to linear wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Results for U = −3 m/s are printed in blue and results for U = −1 m/s in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For SWASH, the horizontal line with marker indicates the average change to the simulated wave height H in the current region, and the vertical lines with horizontal endings indicate the maximum and minimum H in the current region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The wave height and length are normalized by the incident wave height and length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Sensitivity study The behaviour of the SWASH model was found to be sensitive to the horizontal grid resolution ∆x and the time-step ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To illustrate the sensitivity to the grid resolution, we consider a set of simulations of a T = 10 s monochromatic wave on a U = [−3, −1] m/s current for a range of horizontal grid and temporal resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To study the influence of ∆t and ∆x separately, the first set considers simulations with fixed ∆x = L0/60 for a range of ∆t, and the second set corresponds to several simulations with fixed ∆t = T/1000 but for a range of ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Changes to the wavelength L were not sensitive to either ∆x and ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' On the other hand, changes to the wave height H were sensitive to the model settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The sensitivity was larger for the stronger current velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Modelled changes to H were less sensitive to the horizontal grid resolution, except for coarse resolutions (∆x/L0 < 40), with relatively weak improvement for ∆x/L0 ≤ 40 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Modelled changes to H were sensitive to the time-step, especially for U = −3 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For this current velocity at larger time- steps, significant dissipation of wave energy occurred in the current region (as illustrated by the vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1 at smaller ∆t/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For finer temporal resolutions, this non-physical dissipation reduced 23 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 a) b) (w) 2 U=3 m/s (w) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 “H/H U=1 m/s L/L o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 103 104 103 104 T/△t T/△t3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 ( (p (w) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 H/H 107/7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='0 50 100 150 50 100 150 Lo/Ax Lo/Axand model results approximately converged to the solution of linear wave theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' This sensitivity to the horizontal grid and temporal resolution was primarily significant for strong opposing currents relative to the wave group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For following currents and weak opposing currents the model results were not sensitive to ∆x and ∆t (as illustrated by the results for U = −1 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Based on this sensitivity study, the optimal horizontal grid and temporal resolution for which model predictions were sufficiently converged was concluded to be ∆x = L0/100 and ∆t = T/1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Re-circulating current To generate a re-circulating current in the model, we impose an inward and outward flux at the bottom at either side of the model domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' For this purpose, we have adopted the kinematic boundary condition as follows, wz=−d = −u∂d ∂x − v ∂d ∂y ± fs P W , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1) where P is a discharge and W is the width of the region where the discharge is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' By introducing an equal discharge of opposing sign in a region at either side of the numerical domain, a recirculating current is generated inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' To reduce the spin-up time, we use a smoothing function fs to gradually ramp up the discharge from 0 to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The smoothing function is defined as, fs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='5 (1 + tanh( t TS − 3)), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2) where TS is the smoothing period of the pump (taken as TS = 15 s in the simulations of this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Linear semi-discrete analysis of the model equations The numerical dispersion relationship can be derived from the linearized and semi-discretized set of model equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Bai and Cheung, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Based on Smit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (2014), the linearized and semi-discretized SWASH equations extended with the additional terms for the wave-current interactions (on the right hand side) for N vertical layers reads, ∂u′ n− 1 2 ∂t + g ∂ζ′ ∂x + 1 2 ∂pnh,n ∂x + 1 2 ∂pnh,n−1 ∂x = −U ∂u′ n− 1 2 ∂x , for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='N, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1) ∂wn + wn−1 ∂t + 2pnh,n − pnh,n−1 ∆z = −U ∂wn + wn−1 ∂x , for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='N, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2) ∂u′ n− 1 2 ∂x + wn − wn−1 ∆z = 0, for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='N, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='3) ∂ζ′ ∂t + ∆z N � n=1 ∂u′ n− 1 2 ∂x = −U ζ′ ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='4) The flow variables in the above set of equations are located on a staggered grid, with u′ located in a cell center (n − 1 2) and w and pnh at a vertical cell face (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Assuming a horizontal bottom (w0=0) and considering 24 the initial value problem in an infinite domain (with ∆z = d/N), we assume that the flow variables have a solution of the form y = ˆyexp(ikx − iωt) (where ˆy is the complex amplitude of a flow variable, k is the wavenumber and ω the absolute wave frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Substituting this into the above set of equations for each variable results in a system of equations of the form Aˆy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The numerical dispersion relationship can subsequently found from Det(A) = 0 using symbolic algebra software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' With the addition of an ambient current U, the numerical dispersion relationship provides a relationship between ω and k in the presence of a current with velocity U for N vertical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' The relative group velocity can be found from the numerical dispersion relationship as cg,r = ∂σ ∂k for an arbitrary current velocity U (with ω = σ + kU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' 25 References Akrish, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Smit, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Zijlema, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Reniers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Modelling statistical wave interferences over shear currents.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Reniers, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Including tidal currents in a wave-resolving model, in: Coastal Dynamics Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Depth-integrated wave – current models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Part 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Two-dimensional formulation and applications doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='1017/jfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content='831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=' Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf'} +page_content=', Li, Z.' metadata={'source': 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The configuration space of k points on a manifold carries an action of +its diffeomorphism group. The homotopy quotient of this action is equivalent to +the classifying space of diffeomorphisms of a punctured manifold, and therefore ad- +mits results about homological stability. Inspired by the works of Segal, McDuff, +Bodigheimer, and Salvatore, we look at generalised configuration spaces where par- +ticles have labels and even partially summable labels, in which points are allowed to +collide whenever their labels are summable. These generalised configuration spaces +also admit actions of the diffeomorphism group and we look at their homotopy quo- +tients. Our main result is a decoupling theorem for these homotopy quotients on +surfaces: in a range, their homology is completely described by the product of the +moduli space of surfaces and a generalised configuration space of points in R∞. +Using this result, we show these spaces admit homological stability with respect to +increasing the genus, and we identify the stable homology. This can be interpreted +as an Diff-equivariant homological stability for factorization homology. In addition, +we use this result to study the group completion of the monoid of moduli spaces of +configurations on surfaces. +1. Introduction +The ordered configuration space of k points on a smooth manifold M without bound- +ary is defined as +�Ck(M) := {(m1, . . . , mk) ∈ M k | mi ̸= mj if i ̸= j}. +When M is a smooth manifold with boundary, we denote by �Ck(M) the space of ordered +configurations of k points in its interior. The symmetric group Σk acts on this space +by permuting the order of the k points. The configuration space of k points on M, +denoted Ck(M), is the quotient �Ck(M)/Σk. +We denote by Diff∂(M) the group of +diffeomorphisms of a manifold M which fix a collar of its boundary. +In this paper, we will focus on 2-dimensional manifolds and we denote by F k +g,b an +orientable surface of genus g, k punctures, and b ≥ 1 boundary components. +The +spaces Ck(Fg,b) admit an action of the group Diff∂(Fg,b), where a diffeomorphism φ +acts by taking a collection of k points to its image via φ. Of interest here, is the Borel +construction (homotopy quotient) of this action, denoted Ck(Fg,b)//Diff∂(Fg,b), to which +we refer to as a moduli of configurations of k points in Fg,b. It is simple to show that +Ck(Fg,b)//Diff∂(Fg,b) ≃ BDiff∂(F k +g,b) +(1.1) +and in fact this relation is not only true for surfaces, but for any manifold with k +punctures. In particular, this allows us to deduce homological stability results for these +moduli of configurations of k points, directly from the known theorems for classifying +spaces of punctured surfaces. For instance, when b ≥ 1 these spaces admit homological +stability when increasing the genus and when increasing then number of points [Har85, +Date: January 3, 2023. +2020 Mathematics Subject Classification. 57R19, 55R80, 55R40, 55P47. +This material is based upon work supported by CNPq (201780/2017-8) and by the NSF Grant No. +DMS-1928930 while the author participated in a program hosted by the MSRI in 2022. +1 +arXiv:2301.00093v1 [math.AT] 31 Dec 2022 + +2 +LUCIANA BASUALDO BONATTO +Har90, RW16, RW14]. Moreover, it was shown in [BT01] that the stable homology of this +classifying space can be computed from the homology of BDiff∂(Fg,b)×B(Σk ≀GL+ +2 (R)), +what is known as a decoupling theorem. +In this paper, we study the analogue of this moduli space for generalised configuration +spaces. As a first case, we look at labelled configurations: given a pointed space Z, the +space of configurations in M with labels in Z is the quotient +C(M; Z) := +� +�� +k≥0 +�Ck(M) ×Σk Zk +� +� +� +∼ +under the relation (m1, . . . , mk; z1, . . . , zk) ∼ (m1, . . . , mk−1; z1, . . . , zk−1) if zk is the +basepoint of Z. We can interpret this space geometrically by considering it as the space +of particles in M where each particle is labelled by an element of Z, and a particle is +allowed to disappear if labelled by the basepoint. +This space has been of interest since the 70’s, appearing on the seminal works of +May [May72] and Segal [Seg73]. It was noted that the space C(Rn; Z) can be given the +structure of an (A∞-)monoid by taking multiplication to be roughly given by stacking +configurations side by side [Seg73]. One of the main results about this space is what +today is called a scanning map C(Rn; Z) → ΩnΣnZ which was shown by Segal to induce +a weak-homotopy equivalence on group-completions. This idea has been generalised in +many directions. For instance, B¨odigheimer [B¨od87] proved an analogous statement for +configurations on general manifolds. In addition, similar results were proven for the +case where the spaces of labels has extra structure, such as (partial) multiplications +[McD75, Seg79, Gue95, Kal01, Sal01]. We discuss the later case in Section 1.1. +More recently this labelled configuration space and scanning map argument have been +expanded to sophisticated constructions in factorization homology [AFT17] on the one +hand and, on the other, in the form of configuration spaces of manifolds, has been used +to compute the stable homology of the moduli spaces of Riemann surfaces [MW07] and +higher dimensional manifolds [GRW18, GRW17]. +Labelled configuration spaces also inherit an action of the diffeomorphism group. +Even more, if Z is a pointed GL+ +2 (R)-space, we can define an action of Diff∂(Fg,b) on +C(M; Z) where a diffeomorphism φ acts by taking a collection of k points to its image +via φ, and the label z of a point w is taken to the label dwφ·z of the point φ(w). Unlike +the case of configurations with a fixed number of points, C(Fg,b; Z)//Diff∂(Fg,b) is not +in general equivalent to the classifying space of a diffeomorphism group. Hence we ask +if it still has homological stability and if it admits an analogous decoupling theorem. +For a surface Fg,b, with b ≥ 1, taking the boundary connected sum with the surface +F1,1 induces a homomorphism Diff∂(Fg,b) → Diff∂(Fg+1,b), given by extending a map +on Fg,b by the identity. We call this the stabilisation map and study what it induces on +the moduli of configuration spaces: +Theorem A. Let Z be a pointed GL+ +2 (R)-space and b ≥ 1. The stabilisation map on +the Borel constructions +s∗ : C(Fg,b; Z)//Diff∂(Fg,b) → C(Fg+1,b; Z)//Diff∂(Fg+1,b) +induces a homology isomorphism in degrees ≤ 2 +3g. +Moreover, we can determine precisely what the stable homology is: +Theorem B. Let Z is a pointed connected GL+ +2 (R)-space. There is a map +C(Fg,b; Z)//Diff∂(Fg,b) → Ω∞MTSO(2) × Ω∞Σ∞ +� +EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z +� +which is compatible with the stabilisation maps and induces a homology isomorphism in +degrees ≤ 2 +3g. + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +3 +In the above, EGL+ +2 (R) denotes the total space of a universal fibration for BGL+ +2 (R), +we use (−)+ to denote adjoining a disjoint basepoint to a space, and − +∧ +GL+ +2 (R) +− denotes +the quotient of the smash product of pointed topological GL+ +2 (R)-spaces by the diagonal +action of GL+ +2 (R). +Both of the results above are consequences of Theorem C below, which is an ana- +logue of the decoupling theorem in [BT01]. It implies that the stable homology of this +moduli of configuration spaces can be understood through a decoupling map τ × ε, +which separates the points in the configurations from the underlying surfaces. The map +τ : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) forgets the data of the configurations, and ε +forgets the underlying surface, but still remembers the points in the configuration and +some local tangential data around them (for a detailed description of these maps see +Section 3). +Theorem C (Decoupling Theorem for Labelled Configurations). Let τ and ε be the +maps described above. Then +τ × ε : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞; EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z) +induces a homology isomorphism in degrees ≤ 2 +3g. +This result may be interpreted in physical terms: in the high genus limit, the con- +straints for the particles to stay on the underlying surface are lifted and the particles +are now free. +As a final application of Theorem C, we look at monoids of moduli of configurations +on surfaces. Boundary connected sum induces a multiplication on the level of classifying +spaces making +� +g BDiff∂(Fg,1) into a topological monoid. This important construction +and its group completion have been central in the study of the stable homology of +mapping class groups of surfaces [Mil86, Til00, MW07, GRW10, GRW18, GRW17]. +This gluing of surfaces induces also a multiplication on the Borel constructions +C(Fg,1; Z)//Diff∂(Fg,1) × C(Fh,1; Z)//Diff∂(Fh,1) → C(Fg+h,1; Z)//Diff∂(Fg+h,1) +making +� +g C(Fg,1; Z)//Diff∂(Fg,1) into a topological monoid. We study its group com- +pletion. +Corollary D. For any pointed GL+ +2 (R)-space Z, the decoupling map induces a weak +equivalences on group completions +ΩB +�� +g C(Fg,1; Z)//Diff∂(Fg,1) +� +≃ ΩB +�� +g BDiff∂(Fg,1) +� +×ΩBC(R∞; EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z). +1.1. Configuration spaces with partially summable labels. The main result of +this paper considers a more general type of configuration spaces with labels in a framed +partial 2-monoid, where particles are allowed to collide if their labels are summable. +This space has been explored in works such as [McD75, Seg79, Gue95, Kal01, Sal01] +and, when the labels are E2-algebras (not partial), is equivalent to factorization ho- +mology [AF15] and topological chiral homology [Lur09]. The first example of this con- +struction can be seen in McDuff’s configuration spaces of positive and negative particles +[McD75], where particles are labelled by “charges” and are allowed to collide whenever +their charges are opposite. More generally, Salvatore [Sal01] defines partial 2-monoids, +which are, in essence, spaces with a multiplication similar to an E2-algebra structure, +but with the restriction that this multiplication does not need to be defined for every +tuple of elements (see Definition 4.4). Whenever P is equipped with a compatible ac- +tion of GL+ +2 (R), we call it a framed partial monoid, and we can define the space of +configurations in Fg,b with partially summable labels in P, denoted CΣ(Fg,b; P). The +definition CΣ(Fg,b; P) requires much more machinery then the case for non-summable + +4 +LUCIANA BASUALDO BONATTO +labels, such as the Fulton-MacPherson operad, and yields more complicated models for +configuration spaces. We discuss these constructions in Section 4.1. As before, these +generalised configuration spaces admit an action of the diffeomorphism group and we +study its Borel construction. +Theorem E. Let P be a framed partial 2-monoid and b ≥ 1. The stabilisation map on +the Borel constructions +s∗ : CΣ(Fg,b; P)//Diff∂(Fg,b) → CΣ(Fg+1,b; P)//Diff∂(Fg+1,b) +induces a homology isomorphism in degrees ≤ 2 +3g. +While homological stability for configurations with summable labels with respect to +increasing the number of points had been studied in [KM16], the above result is the first +to look at stability with respect to increasing the genus. Theorem E can be interpreted +as a Diff∂-equivariant homological stability result for factorisation homology. +As in the case for labelled configurations, this result is a consequence of a decou- +pling theorem for the space CΣ(Fg,b; P)//Diff∂(Fg,b), which is the main result of this +paper. This is much more intricate than the decoupling for labelled configurations. The +proof uses a semi-simplicial resolution of CΣ(Fg,b; P) developed in section 4.2, which +we refer to as the disc model for configurations, denoted |DΣ(Fg,b; P)•| (Proposition +4.13). This model makes explicit the connection between these spaces and factorization +homology. In the decoupling context, we naturally encounter an analogue of this space +with 2-dimensional discs with configurations embedded in R∞, we denote this space +|D2 +Σ(R∞; P)•| (Definition 4.17). Using the Decoupling Theorem for Labelled Configu- +rations (Theorem D) we then prove: +Theorem F (Decoupling Theorem for Configurations with Summable Labels). For +P a framed partial 2-monoid, there is a weak equivalence CΣ(Fg,b; P)//Diff∂(Fg,b) ≃ +|DΣ(Fg,b; P)•|//Diff∂(Fg,b) and the decoupling map +|DΣ(Fg,b; P)•|//Diff∂(Fg,b) → BDiff∂(Fg,b) × |D2 +Σ(R∞; P)•|. +induces a homology isomorphism in degrees ≤ 2 +3g. +In future work we will discuss the homotopy type of the space |D2 +Σ(R∞; P)•| and +its description as an infinite loop space. We conjecture that it is also equivalent to a +configuration in R∞ with partially summable labels. +Analogous to the case of labelled configurations, the spaces CΣ(Fg,1; P)//Diff∂(Fg,1) +assemble into a topological monoid, and the decoupling theorem descends into its group +completion. +Corollary G. For any path-connected framed partial 2-monoid with unit P, the decou- +pling map induce a homotopy equivalence +ΩB( +� +g CΣ(Fg,1; P)//Diff∂(Fg,1)) ≃ ΩB( +� +g BDiff∂(Fg,1)) × ΩB(|D2 +Σ(R∞; P)•|). +1.2. Outline of the paper. We start by recalling in Section 2 background results +which will be used throughout the paper, especially on Section 4. This can be skipped +and referred back to when necessary. +In Section 3 we introduce labelled configuration spaces and prove Theorem C. Using +this, we deduce Theorems A and B, and Corollary D. +In Section 4 we discuss the case of configurations with summable labels, and prove +the main results of the paper. We start by recalling in 4.1 the definitions of framed +partial d-monoids and configuration spaces with partially summable labels. We then +construct semi-simplicial resolutions for these spaces in Section 4.2. In Section 4.3, we +use this disc model together with the Decoupling Theorem for Labelled Configurations +(Theorem D) to prove Theorem F. Finally, we use this result to deduce Theorem E and +Corollary G. + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +5 +Acknowledgements. I would like to thank Ulrike Tillmann for suggesting the prob- +lem and for the many insightful conversations. In addition, I would like to thank David +Ayala, Christopher Douglas, Jan Steinebrunner, and Nathalie Wahl for the helpful dis- +cussions and comments. +2. Preliminaries +In this section we recall techniques and results on semi-simplicial spaces, which will +be used in Section 4. The reader may skip this part and refer back to when necessary. +For a detailed exposition of the concepts in this section see [ERW19]. +A semi-simplicial space is a functor ∆op +inj → Top, where ∆inj is the category with ob- +ject the linearly ordered sets [p] = {0 < · · · < p} and morphisms the injective monotone +maps. We denote such a functor by X• and write Xp = X•({1, . . . , p}). The datum of +a semi-simplicial space is equivalent to the collection of spaces Xp, p ≥ 0, together with +face maps di : Xp → Xp−1 for i = 0, . . . , p, satisfying didj = dj−1di if i < j. +Denote by ∆p the standard p-simplex +∆p = +� +(t0, . . . , tp) ∈ Rp+1��� +p +� +i=1 +ti = 1 and ti ≥ 0 for all i +� +. +To each morphism φ : [p] → [q] in ∆inj, there is a continuous map φ∗ : ∆p → ∆q such +that φ∗(t0, . . . , tp) = (s0, . . . , sq) with sj = � +i∈φ−1(j) ti. The geometric realisation of a +semi-simplicial space X• is the quotient space +|X•| := +�� +p +Xp × ∆p +� � +∼ +where (x, φ∗t) ∼ (φ∗x, t), and φ is a morphism of ∆inj. +2.1. Semi-simplicial nerve of a poset. Any topological poset (Q, <) defines a semi- +simplicial space Q• by setting Qp to be the subspace of tuples (q0 < · · · < qp) ∈ Qp+1, +and face maps +di : Qp −→ Qp−1 +for 0 ≤ i ≤ p +(q0 < · · · < qi < · · · < qp) �−→ (q0 < · · · < qi−1 < qi+1 < · · · < qp). +We refer to Q• as the semi-simplicial nerve of the poset Q. +Given a topological poset (Q, <), the space Q×Q can be equipped with a partial order +where (q1, q2) < (q′ +1, q′ +2) if qi < q′ +i and qj ≤ q′ +j, for {i, j} = {1, 2}. We say that a pointed +such Q is a partially ordered topological monoid if it is equipped with a multiplication +− · − : Q × Q → Q +which is strictly associative, unital and order preserving. In this case, the geometric +realisation of the semi-simplicial nerve Q• is naturally endowed with a multiplication · +defined by +� +(q0 < · · · < qm; t0, . . . , tm) · (q0 < · · · < qk; t0, . . . , tk) +� += += (q0 · q0 < · · · < q0 · qk < · · · < qm · q0 · · · < qm · qk; t0 · t, . . . , tm · t) +where ti · t = tit0, . . . , titk, for all i = 0, . . . , m. It is straightforward to verify that this +is well-defined. +Lemma 2.1. For (Q, <, µ) a partially ordered topological monoid, (|Q•|, |µ|) is a topo- +logical monoid. Moreover, any map of partially ordered topological monoids f : Q → Q′ +induces a map of topological monoids +f∗ : |Q•| → |Q′ +•|. +The proof is a straightforward computation and follows directly from the definitions. + +6 +LUCIANA BASUALDO BONATTO +2.2. Spectral sequence. We quickly recall below the spectral sequence defined in +[Seg68, Proposition 5.1] associated to a semi-simplicial space, which is the key for the +homology argument used in the proof of Theorem 4.18 (see [ERW19, Section 1.4] for a +detailed discussion). +For any semi-simplicial space X•, the geometric realisation |X•| admits a filtration +by its skeleta, with |X•|(0) = X0 and +|X•|(q) = |X•|(q−1) ∪Xq×∂∆q Xq × ∆q. +This filtration yields a spectral sequence +E1 +p,q = Hp+q(|X•|(q), |X•|(q−1)) =⇒ Hp+q(|X•|) +and by excision and the Kunneth isomorphism, the left-hand term can be re-written to +give a spectral sequence with +E1 +p,q ∼= Hp(Xq) =⇒ Hp+q(|X•|). +Therefore a map of semi-simplicial spaces inducing a level-wise homology isomorphism +gives an isomorphism of the first pages of the respective spectral sequences, and therefore +a homology isomorphism between the geometric realisations. +2.3. Semi-simplicial Resolutions. In the proof of the decoupling we will use a semi- +simplicial resolution of the spaces of configurations with summable labels. Showing that +we indeed have a resolution will be a direct application of [GRW14, Theorem 6.2]. For +completeness, we quickly recall the statement of this result to clarify the conditions that +will be checked in the proof of Proposition 4.13. We follow the notation and definitions +of [GRW14]. +An augmented semi-simplicial space is a triple (X•, X−1, ε•), where X• is a semi- +simplicial space, X−1 is a space and ε• is a collection of continuous maps εp : Xp → +X−1 satisfying diεp = εp−1 for all p ≥ 0 and all face maps di. +We also say that +ε• : X• → X−1 is an augmentation for X•. It is simple to verify that an augmentation +induces a continuous map |ε•| : |X•| → X−1. +Definition 2.2. An augmented topological flag complex [GRW14, Definition 6.1] is an +augmented semi-simplicial space ε : X• → X−1 such that +(i) The map Xn → X0 ×X−1 · · ·×X−1 X0 taking an n-simplex to its (n+1) vertices +is a homeomorphism onto its image, which is an open subset. +(ii) A tuple (v0, . . . , vn) ∈ X0 ×X−1 · · ·×X−1 X0 lies in Xn if and only if (vi, vj) ∈ X1 +for all i < j. +In other words, in an augmented topological flag complex, the space of n-simplices +can be described as an open subspace of the (n + 1)-tuples of vertices with the same +image under ε, and such a tuple forms an n-simplex if and only if the pairs of vertices +are all 1-simplices. The result below is a criterion to determine when an augmented +topological flag complex X• → X−1 induces a weak equivalence |X•| → X−1. +Theorem 2.3 ([GRW14], Theorem 6.2). Let X• → X−1 be an augmented topological +flag complex. Suppose that +(i) The map ε : X0 → X−1 has local lifts of any map from a disc, i.e. given a map +f : Dn → X−1, a point p ∈ ε−1(f(x)), there is an open neighbourhood U ⊂ Dn +of x and a map F : U → X0 such that ε ◦ F = f|U and F(x) = p. +(ii) ε : X0 → X−1 is surjective. +(iii) For any p ∈ X−1 and any non-empty finite set {v1, . . . , vn} ∈ ε−1(p) there exists +a v ∈ ε−1(p) with (v1, v) ∈ X1 for all i. +Then |X•| → X−1 is a weak homotopy equivalence. + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +7 +3. Decoupling Labelled Configuration Spaces +In this section, we recall the definition of configuration spaces with labels and in- +troduce a model for the homotopy quotients C(Fg,b; Z)//Diff∂(Fg,b). +With this, we +construct the decoupling map of Theorem C. We then prove this result as Theorem +3.2 and use it to prove Theorems A and B, and Corollary D. These are, respectively, +Corollary 3.4, Corollary 3.5, and Corollary 3.7 below. +Definition 3.1. Let M be a manifold and Z be a well-pointed space. The configuration +space of M with labels in Z, denoted C(M; Z), is the quotient +� +k≥0 �Ck(M) × +Σk +Zk/ ∼ +where �Ck(M) denotes the ordered configuration space of k points in M, and +(m1, . . . , mk; z1, . . . , zk) ∼ (m1, . . . , mk−1; z1, . . . , zk−1) +whenever zk is the basepoint of Z. +If Z is a pointed GL+ +2 (R)-space, and M is a surface Fg,b, with b ≥ 1, the space +C(Fg,b; Z) carries a natural action by the diffeomorphism group of Fg,b: for φ ∈ Diff∂(Fg,b) +φ · (m1, . . . , mk; z1, . . . , zk) := (φ(m1), . . . , φ(mk); Dm1φ · z1, . . . , Dm1φ · zk). +The basepoint relation is preserved by this action as Z is a pointed GL+ +2 (R)-space. +The decoupling result is about the homotopy quotient of this action, that is, the Borel +construction C(Fg,b; Z)//Diff∂(Fg,b). +From now on, we denote by Fg,b a fixed orientable surface of genus g, b ≥ 1 bound- +ary components, and pick once and for all a framing on Fg,b, that is, a section s of +the frame bundle Fr(TFg,b) → Fg,b. We fix now a model for EDiff∂(Fg,b) that will +be used to construct the decoupling map of Theorem 3.2. Let Emb(Fg,b, R∞) denote +colimn→∞ Emb(Fg,b, Rn). This space has a free action of Diff∂(Fg,b) by precomposition +and by [BF81] the quotient map Emb(Fg,b, R∞) → Emb(Fg,b, R∞)/Diff∂(Fg,b) has slices, +hence it is a principal Diff∂(Fg,b)-bundle. Moreover, by Whitney’s embedding theorem, +Emb(Fg,b, R∞) is weakly contractible and therefore it is a model for EDiff∂(Fg,b). We +then let +C(Fg,b; Z)//Diff∂(Fg,b) ≃ Emb(Fg,b, R∞) +× +Diff∂(Fg,b) C(Fg,b; Z). +Analogously, we will take the model for BDiff∂(Fg,b) given by +BDiff∂(Fg,b) ≃ Emb(Fg,b, R∞)/Diff∂(Fg,b). +This can be interpreted as the space of abstract submanifolds of R∞ which are diffeo- +morphic to Fg,b, but without a fixed diffeomorphism. Analogously, a point in C(Fg,b; Z)// +Diff∂(Fg,b) consists of one such abstract manifold, together with a labelled configuration. +The decoupling map will be the product of two maps +τ : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) +(3.1) +ε : C(Fg,b; Z)//Diff∂(Fg,b) → C(R∞; EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z). +(3.2) +Recall that EGL+ +2 (R) is the total space of a universal fibration for BGL+ +2 (R), we use +(−)+ to denote adjoining a disjoint basepoint to a space, and − +∧ +GL+ +2 (R) +− denotes the +quotient of the smash product of pointed topological GL+ +2 (R)-spaces by the diagonal +action of GL+ +2 (R). +Intuitively, the map τ forgets the configuration, while ε forgets the underlying surface, +but remembers the labelled configuration together with data on their tangent space on +the submanifold. + +8 +LUCIANA BASUALDO BONATTO +In details, τ is the classifying map for the homotopy quotient, and it is simply the +one induced by the projection Emb(Fg,b, R∞)×Ck(Fg,b) → Emb(Fg,b, R∞). +To define ε, we take as model for BGL+ +2 (R) the oriented Grassmanian manifold of +2-dimensional oriented subsbaces of R∞, Gr+(2, ∞), and let EGL+ +2 (R) denote the total +space of the universal GL+ +2 (R)-bundle over it. Then using the identification Fr(TR∞) ∼= +R∞ ×EGL+ +2 (R), an embedding e : Fg,b �→ R∞ induces a map e∗ : Fr(TFg,b) → ESO(2) +from the bundle of framings on TFg,b, taking a basis of TpFg,b to its image via Dpe. The +map ε takes a point represented by a labelled configuration [m1, . . . , mk; z1, . . . , zk] and +an embedding e : Fg,b �→ R∞ to the labelled configuration in R∞ given by +[e(m1), . . . , e(mk); [e∗(s(m1)), z1], . . . , [e∗(s(mk)), zk]] +where s(p) denotes the chosen oriented frame on p ∈ Fg,b. It is simple to verify that this +indeed defines a continuous function to the configuration space C(R∞; (EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z). +Theorem 3.2. Let τ and ε be the maps in (3.1). Then the decoupling map +τ × ε : C(Fg,b; Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞; (EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z) +induces a homology isomorphism in degrees ≤ 2 +3g. +The proof of the result above will build upon a decoupling result for unlabelled +configurations with a fixed number of points, which was first proved by [BT01] and +generalised in [Han09, Bon22]. We show here a slight generalisation of the result which +we will need in the proof. The space C(M; Z) is constructed as a quotient of the union +of spaces �Ck(M) ×Σk Zk, and the Lemma below is about the decoupling map in each of +these components. +In fact, we will work on a slightly more general context which will be more convenient +for the proof: let X be a well-pointed space with an action of the wreath product +Σk ≀ GL+ +2 (R) (in the context above we were using X = Zk). The space �Ck(Fg,1) × X +comes equipped with two actions: Σk acts diagonally by permuting the points in the +configuration and by the action on X, and Diff∂(Fg,b) acts by +φ · (m1, . . . , mk; x) = (φ(m1), . . . , φ(mk); (dm1φ, . . . , dmkφ)(x)). +Note that the actions of Σk and Diff∂(Fg,1) on this space commute. +As before, we have maps +τk : ( �Ck(Fg,1) × +Σk +X)//Diff∂(Fg,b) → BDiff∂(Fg,b) +(3.3) +εk : ( �Ck(Fg,1) × +Σk +X)//Diff∂(Fg,b) → ( �Ck(R∞) × (EGL+ +2 (R))k) +× +Σk≀GL+ +2 (R) +X. +(3.4) +Here τk is again simply the classifying map for the homotopy quotient, and εk is the +map taking a point represented by [m1, . . . , mk; x] and an embedding e : Fg,b �→ R∞ to +the class +� +[e(m1), . . . , e(mk); e∗(s(m1)), . . . , e∗(s(mk))], x +� +where s(p) denotes the chosen oriented frame on p ∈ Fg,b. We remark that replacing X +by Zk one recovers precisely the definition of the maps τ and ε in (3.1). +Lemma 3.3 ([BT01, Bon22]). For any Σk ≀GL+ +2 (R)-space X, let τk and εk be the maps +defined in (3.3). Then +τk×εk : ( �Ck(Fg,b)× +Σk +X)//Diff∂(Fg,b) → BDiff∂(Fg,b)×( �Ck(R∞)×(EGL+ +2 (R))k) +× +Σk≀GL+ +2 (R) +X. +induces a homology isomorphism in degrees ≤ 2 +3g. + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +9 +For completeness, we include a short proof of the above result. For details see [BT01, +Bon22]. +Proof of Lemma 3.3. We start by reducing the proof to the case when X is a point. The +projections +( �Ck(Fg,b) × +Σk +X)//Diff∂(Fg,b) → Ck(Fg,b)//Diff∂(Fg,b) +BDiff∂(Fg,b) × ( �Ck(R∞) × (EGL+ +2 (R))k) +× +Σk≀GL+ +2 (R) +X → BDiff∂(Fg,b) × Ck(R∞, BGL+ +2 (R)) +are both fibrations with fibre X, and the map τk × εk induces a map between the +corresponding fibre sequences, which is the identity on the fibers. If the map between +the base spaces induces a homology isomorphism in degrees ≤ 2 +3g, then by Zeeman’s +Comparison Theorem [Zee57] applied to the Serre spectral sequences associated to these +fibrations, so does τk × εk. Hence it is enough to show that the map between the base +spaces +τk × εk : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) × Ck(R∞, BGL+ +2 (R)) +induces a homology isomorphism in degrees ≤ 2 +3g. +By Palais’ Theorem [Pal60], the map εk is a fibration with fiber BDiff∂(Fg,b+n). +Moreover, the map τk × εk induces a map of fibre sequences +BDiff∂(Fg,b+n) +Ck(Fg,b)//Diff∂(Fg,b) +Ck(R∞, BGL+ +2 (R)) +BDiff∂(Fg,b) +BDiff∂(Fg,b) × Ck(R∞, BGL+ +2 (R)) +Ck(R∞, BGL+ +2 (R)) +proj +τk×εk +εk +where the leftmost map is the one induced by capping off the n extra boundary compo- +nents by gluing discs. This map was shown to induce a homology isomorphism in the +range ≤ 2 +3g [Har85, Iva87, Iva89, Iva93, Bol12, RW16]. Hence, by Zeeman’s Compar- +ison Theorem [Zee57] applied to the Serre spectral sequences associated to these fibre +sequences, the map between the total spaces also induces homology isomorphisms in the +range ≤ 2 +3g. +□ +Equipped with Lemma 3.3, we are now ready to prove Theorem 3.2. +Proof of Theorem 3.2. The spaces C(Fg,b; Z) and C(R∞; (EGL+ +2 (R))+ ∧GL+ +2 (R) Z) con- +sist of configuration with an arbitrary number of particles. However they have natural +filtrations C≤k(−) by the subspaces of configurations with at most k points. These +induce filtrations +Xk := C≤k(Fg,b; Z)//Diff∂(Fg,b) +Yk := BDiff∂(Fg,b) × C≤k(R∞; (EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z). +that are preserved under the map τ × δ. +We will inductively show the restrictions +Xk → Yk are homology isomorphisms for all k. +Since Z is a space with a good basepoint, the inclusions Xk−1 �→ Xk and Yk−1 �→ Yk +are cofibrations. Their subquotients are +Xk/Xk−1 = (EDiff∂(Fg,b))+ +∧ +Diff∂(Fg,b) +�Ck(Fg,b)+ ∧ +Σk Z∧k +and +Yk/Yk−1 = (BDiff∂(Fg,b))+ ∧ �Ck(R∞)+ ∧ +Σk ((EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z)∧k. + +10 +LUCIANA BASUALDO BONATTO +Comparing the spectral sequences associated to these filtrations, it is enough to show +that the induced map on these subquotients is a homology isomorphism. Consider the +map of cofibrations: +EDiff∂(Fg,b) +× +Diff∂(Fg,b) Ck(Fg,b) +EDiff∂(Fg,b) +× +Diff∂(Fg,b) ( �Ck(Fg,b) × +Σk +Z∧k) +Xk/Xk−1 +BDiff∂(Fg,b) × Ck(R∞, BGL+ +2 (R)) +BDiff∂(Fg,b) × ( �Ck(R∞) × +Σk +((EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z)∧k) +Yk/Yk−1 +τk×εk +τk×εk +By Lemma 3.3 with X = ∗ the left-hand map induces a homology isomorphism in +degrees ≤ 2 +3g, and by the same result with X = Z∧n, so does the middle map. Then +by the five lemma, the right-hand map also induces a homology isomorphism in degrees +≤ 2 +3g, as required. +□ +3.1. Homological Stability. Let Fg+1,b be a surface of genus g+1 and b ≥ 1 boundary +components, which is obtained from Fg,b by a boundary connected sum with F1,1. Then +extending diffeomorphisms by the identity on F1,1 gives a map of topological groups +s : Diff∂(Fg,b) �→ Diff∂(Fg+1,b) +which we refer to as the stabilisation map. +Moreover, the inclusion Fg,1 �→ Fg+1,b +induces a continuous map of labelled configuration spaces C(Fg,b; Z) → C(Fg+1,b; Z) +which is s-equivariant. Together this implies we have an induced map on the Borel +constructions: +Corollary 3.4. For b ≥ 1, the stabilisation map on the Borel constructions +s∗ : C(Fg,b; Z)//Diff∂(Fg,b) → C(Fg+1,b; Z)//Diff∂(Fg+1,b) +induces a homology isomorphism in degrees ≤ 2 +3g. +The above result is a corollary of Theorem 3.2, however care has to be taken with +respect to the model we have used for the Borel constructions and classifying spaces. +Namely, it is not clear how to define a stabilisation map on the level of embedding +spaces Emb(Fg,b, R∞) → Emb(Fg+1,b, R∞) which induces the desired map on Borel +constructions. +This can be remedied by taking as model for EDiff∂(Fg,b) a weakly +contractible subspace of Emb(Fg,b, R∞) that still has a free and has proper action of +Diff∂(Fg,b) and in which the stabilisation is clear. +Fix a boundary component of Fg,b and an embedding S1 �→ {0}×R∞. We denote by +Emb∂(Fg,b, (−∞, 0]×R∞) the space of all extensions to an embedding of Fg,b which are +standard on a collar neighbourhood of the marked boundary. By the same arguments +as above, Emb∂(Fg,b, (−∞, 0] × R∞) is a model for EDiff∂(Fg,b), and the inclusion map +Emb∂(Fg,b, (−∞, 0] × R∞) �→ Emb(Fg,b, R∞) +is a Diff∂(Fg,b)-equivariant weak homotopy equivalence. Fixing an embedding e : F1,2 �→ +R∞ which restricts to the chosen embedding on a collar of the boundary, we get an +inclusion +Emb∂(Fg,b, (−∞, 0] × R∞) → Emb∂(Fg+1,b, (−∞, 0] × R∞) +given by extending any embedding of Fg,b by e. This is clearly compatible with the +stabilisation map. +Proof of Corollary 3.4. Using as model for EDiff∂(Fg,b) the space Emb∂(Fg,b, (−∞, 0]× +R∞) described above, we get a commutative diagram + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +11 +C(Fg,b; Z)//Diff∂(Fg,b) +C(Fg+1,b; Z)//Diff∂(Fg+1,b) +BDiff∂(Fg,b) × C(R∞; (EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z) +BDiff∂(Fg+1,b) × C(R∞; (EGL+ +2 (R))+ +∧ +GL+ +2 (R) +Z). +s +τ×ε +τ×ε +s×id +By Theorem 3.2 the vertical maps induce homology isomorphisms in degrees ≤ 2 +3g, +and by Harer’ Stability Theorem [Har85, Iva87, Iva89, Iva93, Bol12, RW16] and Kun- +neth’s Theorem, so does the bottom map. Therefore the top map must also induce +homology isomorphisms in degrees ≤ 2 +3g. +□ +The Decoupling Theorem also allows us to identify what the homology stabilises to. +Let C(F∞; Z)//Diff∂(F∞) and BDiff∂(F∞) be respectively +C(F∞; Z)//Diff∂(F∞) := colim(C(F1,1; Z)//Diff∂(F1,1) +s−→ C(F2,1; Z)//Diff∂(F2,1) +s−→ . . . ) +BDiff∂(F∞) := colim(BDiff∂(F1,1 +s−→ BDiff∂(F2,1) +s−→ . . . ). +Corollary 3.5. For Z a pointed connected GL+ +2 (R)-space, the decoupling map induces +τ × ε : C(F∞; Z)//Diff∂(F∞) → Ω∞MTSO(2) × Ω∞Σ∞ +� +EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z +� +which is a homology isomorphism in all degrees. +Proof. Theorem 3.2 and Corollary 3.4 imply that the decoupling map on the colimits +τ × ε : C(F∞; Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞, EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z) +(3.5) +induces a homology isomorphism. By [GTMW09, MW07], BDiff∂(F∞) admits a map to +Ω∞MTSO(2) which is a homology isomorphism, and by [Seg73], the right-most config- +uration space in (3.5) is homotopy equivalent to Ω∞Σ∞ � +EGL+ +2 (R)+ ∧GL+ +2 (R) Z +� +. +□ +3.2. Monoid of Moduli of Labelled Configuration Spaces. Gluing two surfaces +Fg,1 and Fh,1 along part of their boundary defines a map of topological groups +Diff∂(Fg,1) × Diff∂(Fh,1) → Diff∂(Fg+h,1). +This can be made into an associative operation if we fix once and for all oriented surfaces +Fg,b of genus g and one boundary component, and compatible with the stabilisation (see +Section 3.1). Using these choices for our surfaces, we can see that the above map gives an +associative operation in the collection of diffeomorphism groups of all Fg,1. An example +of such surfaces and multiplication is depicted in Figure 1. +µ +, += +Figure 1. Example of the map µ : MC(C)2 × MC(C)3 → MC(C)5 +where C is the space of colours, with white as the basepoint. +Up to homotopy, the above gluing process is equivalent to gluing the boundary circles +of surfaces Fg,1 and Fh,1, to two of the three boundary circles of F0,3, what is called the +pair of pants multiplication. We choose to think of this multiplication in terms of the +first description because in this way the product is strictly associative. +This operation also induces a multiplication on the classifying spaces BDiff∂(Fg,1), +which can be made associative by picking a convenient model. As in Section 3.1, we +will use as EDiff∂(Fg,1) a certain subspace of Emb(Fg,1, R∞). +Namely, we can fix + +12 +LUCIANA BASUALDO BONATTO +embeddings δg : S1 �→ [0, g] × R∞ such that δg(S1) ∩ ({0} × R∞) = {0} × (−1, 1) × {0} +and δg(S1) ∩ ({g} × R∞) = {g} × (−1, 1) × {0}. We denote by Embδg(Fg,1, [0, g] × R∞) +the space of all extensions to an embedding of Fg,1 which are standard on a collar +neighbourhood of the boundary. By the same arguments as above, Embδg(Fg,1, [0, g] × +R∞) is a model for EDiff∂(Fg,1), and the inclusion map +Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,1, R∞) +is a Diff∂(Fg,1)-equivariant weak homotopy equivalence. +By picking embeddings δg +which are compatible with the stabilisation map, we can define a multiplication +Embδg(Fg,1, [0, g] × R∞) × Embδh(Fh,1, [0, h] × R∞) → Embδg+h(Fg+h,1, [0, g + h] × R∞) +by extending an embedding of Fg,1 by the translation of the embedding of Fh,1 in the +first coordinate by g. This is clearly associative and it is compatible with the associative +multiplication on the diffeomorphism groups. +Taking as model for EDiff∂(Fg,1) the spaces Embδg(Fg,1, [0, g] × R∞) we obtain +an associative multiplication on classifying spaces. +This structure equips the space +� +g≥0BDiff∂(Fg,1) with the structure of a topological monoid, which we refer to as the +surface monoid. This is equivalent to the one studied by Tillmann in [Til00], which was +essential in the proof of the Madsen-Weiss Theorem [MW07]. +The operation on diffeomorphism groups and spaces EDiff∂(Fg,1) described above, +together with the fixed identifications Fg+h,1 = Fg,1#∂Fh,1, induce an associative mul- +tiplication also on the Borel constructions (see Figure 1) +µ : C(Fg,1; Z)//Diff∂(Fg,1) × C(Fh,1; Z)//Diff∂(Fh,1) → C(Fg+h,1; Z)//Diff∂(Fg+h,1). +Definition 3.6. We denote by MC(Z)g := C(Fg,1; Z)//Diff∂(Fg,1). The monoid of +moduli of configurations labelled by Z is the topological monoid given by the disjoint +union MC(Z) := +� +g≥0MC(Z)g together with the operation µ. +Theorem 3.7. For any pointed GL+ +2 (R)+-space Z, the decoupling map induces a weak +equivalences on group completions +ΩB +�� +g C(Fg,1; Z)//Diff∂(Fg,1) +� +≃ ΩB +�� +g BDiff∂(Fg,1) +� +×ΩB C(R∞; EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z). +Segal showed in [Seg73] that the space Ω∞Σ∞(X) is the group completion of the con- +figuration space C(R∞; X) seen as a topological monoid where the operation is roughly +given by transposition. Using the inclusion Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,b, R∞), +we see that the decoupling map of Theorem 3.2 induces a map between the spaces using +the current model. The proof of the above theorem will consist of showing that the +decoupling induces a monoidal map between MC(Z) and the monoids Tillmann and +Segal, and to show that this induces a homotopy equivalence on group completions. +Lemma 3.8. The maps τ and ε defined in 3.1 are compatible with these monoidal +structures. +For the map τ, this result follows directly from the definition. For ε, some care has +to be taken into making the configuration spaces into actual topological monoids (see +[Seg73]). Namely, instead of C(R∞; X), we use the homotopy equivalent space +C′(R∞, X) = {(c, t) ∈ C(R∞, X) × R : t ≥ 0, c ⊂ (0, t) × R∞}. +The monoidal structure is given by juxtaposition, ie. (c, t), (c′, t′) �→ (c ∪ Tt(c′), t + t′) +where Tt(−) is the map that translates a configuration by t on the first direction. The +map τ of the decoupling theorem is then equivalent to the monoidal map MC(Z) → +C′(R∞, X) taking an element of [e, c] ∈ MC(Z)g to (τ([e, c]), g). + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +13 +Proof of Theorem 3.7. It is enough to show that the map of monoids given by the decou- +pling induces a homotopy equivalence on group completions. As the group completions +are loop spaces, they are in particular simple and, by the Whitehead theorem for simple +spaces, it suffices to show that it induces a homology equivalence on the group comple- +tions. Both monoids are homotopy commutative, hence the group completion theorem +[MS76] can be applied. Therefore it is enough to prove that the induced map on the +limit spaces (defined in Section 3.1) +τ × ε : C(F∞; Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞; EGL+ +2 (R)+ +∧ +GL+ +2 (R) +Z) +(3.6) +is a homology equivalence. This holds by Theorem 3.2 and Corollary 3.4. +□ +4. Decoupling Configuration Spaces with Partially Summable Labels +In this section we prove the main result of this paper, which is a decoupling result for +configuration spaces with partially summable labels. In this case, the labelling space is +equipped with a partial multiplication and the particles are allowed to collide whenever +their labels can be multiplied. The space CΣ(M; P) of configurations in M with par- +tially summable labels in P has been defined in [Sal01] and its definition requires more +sophisticated tools such as the Fulton-MacPherson configuration spaces and operad. In +section 4.1 we recall these definitions and the concept of a partial d-monoids. +To prove the decoupling theorem for CΣ(M; P), we develop a semi-simplicial resolu- +tion for this space in section 4.2, denoted |DΣ(M; P)•|. In Proposition 4.13 we show +that indeed this space is weakly equivalent to CΣ(M; P). In the decoupling context, we +naturally encounter another space of discs with configurations which is constructed in +Definition 4.17. +With these semi-simplicial spaces, we prove Theorem F, combining Corollary 4.14 +and Theorem 4.18. We then use this to deduce Theorem E and Corollary G, which are, +respectively Corollaries 4.19 and 4.20. +4.1. Fulton-MacPherson configuration space and partially summable labels. +In this section we recall the concept of a configuration space with summable labels in a +partial monoid as described in [Sal01]. +A partial d-monoid P is, in essence, a space with a continuous operation that is +not defined on all collections of elements, but only on a subset of composable elements. +The data defining such a partial monoid consists of the subset of composable elements, +together with an operation on this subset which is associative. We are interested in +partial monoids that moreover have the structure of an Ed-algebra. Essential to this +construction is Fd, the Fulton-MacPherson operad in dimension d. This is a cofibrant +replacement for the little d-discs operad and therefore will be used to make precise the +notion of a Ed-partial monoid. Crucially for us, the Fulton-MacPherson operad is defined +in terms of configuration spaces of points, in which the particles in the configuration are +allowed to collide, but keeping track of the relative position of the particles before they +collided and the order in which collided. +We follow the definition of the Fulton-MacPherson configuration space Ck(M) of a +manifold M as described in [Sin04]. To make it precise, we consider M as embedded in +some Rm and we denote by k the set {1, . . . , k}. We record the directions of particles +in a collision, by defining for (i, j) ∈ �C2(k), a map πi,j : �Ck(Rm) → Sm−1 sending +a configuration (x1, . . . , xk) to the unit vector in the direction xi − xj. The order of +collision is recorded by defining for (i, j, ℓ) ∈ �C3(k) the map si,j,ℓ : �Ck(Rm) → [0, ∞] +sending (x1, . . . , xk) to |xi − xj|/|xi − xℓ|. + +14 +LUCIANA BASUALDO BONATTO +Definition 4.1 ([Sin04], Definition 1.3). The Fulton-MacPherson configuration space +Ck(M) of the manifold M is the closure of the image of the map +i × (πi,j| � +Ck(M)) × (si,j,k| � +Ck(M)) : �Ck(M) −→ M k × (Sm−1)k(k−1) × [0, ∞]k(k−1)(k−2). +The space Ck(M) is homotopy equivalent to the ordered configuration space �Ck(M) +[Sin04, Corollary 4.5] and whenever M is compact, Ck(M) is a compactification of +�Ck(M). Moreover, this construction is functorial with respect to embeddings [Sin04, +Corollary 4.8], i.e. any embedding f : M �→ N induces an embedding f∗ : Ck(M) → +Ck(N). +The following result gives a convenient way to represent elements of Ck(M), which +will be used throughout the chapter. +Proposition 4.2 ([Sin04], Theorem 3.8). Each element in Ck(M) is uniquely deter- +mined by: +(1) A configuration of points P1, . . . , Pl in the interior of M, with 1 ≤ l ≤ n (we +refer to these as the infinitesimal configuration), +(2) For each 1 ≤ i ≤ l, a tree Ti with fi leaves (twigs), no bivalent vertices, +so that �l +i=1 fi = n, and for each vertex in Ti of valence m an element in +Cm(TPiM)/G(d), where G(d) is the group of affine transformations of Rd gen- +erated by translations and positive dilations. +(3) A global ordering of the k leaves of the trees. +This interpretation of the elements of Ck(M) also provides a way of describing the +map f∗ : Ck(M) → Ck(N) induced by an embedding f : M �→ N into a d′-manifold N: +it takes a point with infinitesimal configurations P1, . . . , Pl ∈ M to one with infinitesimal +configurations f(P1), . . . , f(Pl) ∈ N, it preserves the trees and ordering of the leaves, +but changes the labels of the vertices of the trees by taking a label ξ ∈ Cm(TPiM)/G(d) +to the label DPif(ξ) ∈ Cm(Tf(Pi)N)/G(d′). +The Fulton-MacPherson operad Fd is built out of subspaces of these configurations +in Rd. +Intuitively, for each k ≥ 0 the space Fd(k) is the subspace of the ordered +configurations of k points in Rd, in which the points have collided at the origin. +Definition 4.3. The Fulton-MacPherson operad in dimension d, denoted Fd, is defined +by taking Fd(k) to be the pullback +Fd(k) +Ck(Rd) +0 +(Rd)k. +⌟ +In other words, it is the subspace of Ck(Rd) with infinitesimal configuration given by a +single point at the origin. +With the description of this space as in Proposition 4.2, the composition of this operad +is given by grafting of trees. Pictorially, we will often represent elements of Fd(k) as +trees of configurations such as in the rightmost picture of Figure 2. +As shown in [Sal01], there exists a model structure on the category of topological op- +erads in which Fd is a cofibrant replacement of the little d-discs operad. An algebra over +Fd, called by Salvatore a d-monoid, consists of a space A together with Σk-equivariant +maps +Fd(k) × Ak → A +which commute with the structure maps of Fd. + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +15 +10 +9 +8 +5 +1 +7 +6 +4 +3 +2 +11 +12 +10 +9 +8 +5 +1 +7 +6 +4 +3 +2 +11 +12 +8 +5 +1 +3 +9 +7 +2 +12 +10 +6 +4 +11 +Figure 2. Representation of an element in the space F2(12) in terms +of a tree of infinitesimal configurations. +Definition 4.4 ([Sal01], Definition 1.7 and 2.6). A partial d-monoid is a space P to- +gether with monomorphisms of Σk-spaces +i : Compk �→ Fd(k) ×Σk P k. +and composition maps ρ : Compk → P such that +(1) The unit η × P : P → Fd(1) × P factors uniquely through �η : P → Comp1 and +the composition ρ ◦ �η is the identity idP ; +(2) For f ∈ Fd(k) and [fj; pj] ⊂ Compmj, for j = 1, . . . , k, the element +[f; ρ(f1; p1), . . . , ρ(fk; pk)] ∈ Fd(k) ×Σk P k +belongs to Compk if and only if +[µ(f; f1, . . . , fk); p1, . . . , pk] +belongs to Compm1+···+mk. Moreover, if that is the case, then their image by ρ +coincide. +A pointed space (P, 0) is a partial d-monoid with unit if in addition it satisfies +3. For every k there is an inclusion u : Fd(k) ×Σk +� +k P �→ Compk such that the +composition with i◦u is the subspace inclusion, and ρ◦i : Fd(k)×Σk +� +k P → P +is the projection onto the P coordinate. +The definition above is better understood when in comparison to Ed-algebras, which +Salvatore calls d-monoids. Given a d-monoid M and a infinitesimal configuration f ∈ +Fd(k), we can always compose any k elements of M via the composition rule described by +f. In a partial d-monoid P, that is not the case. We instead are given a subset of the k- +tuples of P which are composable via the operation described by a given f ∈ Fd(k). The +space Compk should be then thought of as the pairs of possible composition rules in Fd +together with the tuples of elements of P which can be composed with this composition +rule. The map ρ is computes the results of compositions, whenever they are defined. +Example 4.5. +(a) Any space X admits the structure of a trivial partial d-monoid, +by defining Comp1 = {[1, x] : x ∈ X} and Compk = ∅, for all k ̸= 1. In this +case, the only composition rule that can be performed is the identity and no +other collection of points in X is composable in any way. +(b) When X is a space equipped with a basepoint ∗, we can define a unital partial +d-monoid by setting Compk = Fd(k) × ∨kX and ρ(f, x) = xi, where xi is the +unique non-basepoint coordinate, or ∗ otherwise. In this case, the basepoint +acts as a unit and compositions are only defined when done with the unit. + +16 +LUCIANA BASUALDO BONATTO +(c) Every d-monoid is trivially a partial d-monoid. In particular, every Ωd-space is +a partial d-monoid. +(d) The canonical inclusion id,n : Rd �→ Rd+n also allows us to construct a partial +(d + n)-monoid P from a partial d-monoid by adding n trivial composition +directions. We call this the naive upgrade of a partial d-monoid P and denote +it Td,nP. The underlying space of Td,nP is P, and we define CompTd,nP +k +to be +the image of +CompP +k �→ Fd(k) ×Σk P k +(in,k)∗ +�−−−−→ Fd+n(k) ×Σk P k = Fd+n(k) ×Σk (Td,nP)k. +In [Sal01], it was always assumed that the partial d-monoids were good, in the sense +described below. +Definition 4.6. A partial d-monoid P is good if for every k the inclusion CompP +k �→ +Fd(k) ×Σk P k is a cofibration. +From now on, we always assume partial d-monoids to be good and to have a unit. +For future applications, we are further interested in partial d-monoids with compatible +actions of GLd(R), so we introduce this concept here and perform our constructions in +this more general setting. +Recall from Proposition 4.2 that an element of Fd(k) is described by a tree with k +ordered leaves and a decoration of the vertices by elements of �C|v|(Rd)/G(d). The action +of GLd(R) on Rd induces an action on �C|v|(Rd)/G(d). This gives an action of this group +on Fd(k) for every k, and it is simple to check that the operad maps µ are all GLd(R) +equivariant. +Definition 4.7 ([Sal01], Definition 4.3). The framed Fulton-MacPherson operad de- +noted fFd, is the operad defined by fFd(k) = Fd(k) × GLd(R)k, with structure map +�µ((x, g1, . . . , gk);(x1, g1 +1, . . . , gm1 +1 ), . . . , (xk, g1 +k, . . . , gmk +k +)) += (µ(x; g1x1, . . . , gkxk), g1g1 +1, . . . , gkgmk +k +). +The construction above is an instance of the construction A ⋊ G, the semi-direct +product of an operad A and group G. This construction, and the proof that the above +indeed defines an operad can be found in [SW03, Definition 2.1]. +Definition 4.8 ([Sal01], Definition 4.8). A framed partial d-monoid with unit is a +pointed GLd(R)-space P together with monomorphisms of Σk ≀ GLd(R)-spaces +i : fCompk �→ fFd(k) ×Σk P k. +and GLd(R)-equivariant composition maps ρ : fCompk → P satisfying properties 1-3 +of Definition 4.4. +The GLd(R)-bundle of frames on M induces a (GLd(R))k-bundle fCk(M) on Ck(M), +acted on by Σk. This is called the framed configuration space of k points in M. Using +the description of Proposition 4.2, an element of the space fCk(M) can be uniquely +determined by an infinitesimal configuration in M with labelled trees, together with +additional k frames of the tangent planes associated to the k leaves of the trees. +Then the space of framed configurations fC(M) = +� +k fCk(M) is a right fFd-module, +with GLd(R)-equivariant multiplication maps +m : fCk(M) ×Σk (fFd(n1) × · · · × fFd(nk)) → fCn1+···+nk(M). +defined by grafting the element of fFd(ni) on the i-th leaf of the element of fCk(M), +for all i = 1, . . . , k and using the frame on the leaves to identify Cm(Rd)/G(d) with a +configuration on the tangent space of M ([Sal01, Proposition 4.5]). It is simple to verify +that any co-dimension zero embedding e : M �→ N induces a right fFd-homomorphism +e∗ : fC(M) �→ fC(N). + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +17 +Definition 4.9 ([Sal01], Definition 4.14). Let P be a framed partial d-monoid, and let +M be a manifold of dimension d. Then the space of configurations in M with partially +summable labels in P, denoted CΣ(M; P) is defined as the co-equalizer of the following +� +k +� +fCk(M) ×Σk +� +� +π∈Map(n,k) +k� +i=1 +fCompP +π−1(i) +�� +� +k +fCk(M) ×Σk P k +(m×id)◦(id ×i) +id ×ρk +An element of CΣ(M; P) is then an equivalence class of elements in +� +k fCk(M) ×Σk +P k. From the description of fCk(M) above, we can see that an element in CΣ(M; P) +can be represented by an infinitesimal configuration w1, . . . , wl of ℓ < k points in M +together with trees Ti, for i = 1, . . . , ℓ, where the vertices of Ti are labelled by elements +in xi +j ∈ fFd, and the leaves of the trees are labelled by elements of pi +k ∈ P. +The +equivalence relation defining CΣ(M; P) implies that if some leaves labelled by p1, . . . , pk +are departing from a vertex labelled by x ∈ fFd(k) and the composition ρ(x; p1, . . . , pk) +is defined, then we identify this configuration with the one in which such leaves are +removed and their vertex is replaces by a leaf labelled by ρ(x; p1, . . . , pk). +For a framed partial d-monoid P and a co-dimension zero embedding f : M �→ N, +we get an induced map +f∗ : +� +k +fCk(M) ×Σk P k → +� +k +fCk(N) ×Σk P k. +With the description above, a point in the domain with infinitesimal configurations +w1, . . . , wℓ in M, with trees Ti, for i = 1, . . . , ℓ, where the vertices of Ti are labelled by +elements xi +j ∈ fFd, and the leaves of the trees are labelled by elements of pi +k ∈ P, is +taken to the point with infinitesimal configuration φ(x1), . . . , φ(xℓ), trees Ti, i = 1, . . . , ℓ, +and corresponding labels +dxiφ · xi +j and dxiφ · pi +k. +Here we are using the standard actions of GLd(R) on fFd and P. By the equivariance +condition in the definition of a framed partial monoid, the map preserves the equivalence +classes described above. Therefore any such co-dimension zero embedding induces a map +f∗ : CΣ(M; P) → CΣ(N; P). +(4.1) +Seeing the group Diff∂(M) as a subspace of Emb(M, M), the above construction +shows that CΣ(M; P) admits an action of Diff∂(M). We will be interested in a decou- +pling theorem for the space CΣ(M; P)//Diff∂(M). +4.2. Disc models for configuration spaces with partially summable labels. Let +M be a smooth compact d-manifold, possibly with boundary, and P a framed partial d- +monoid. The group Σk ≀GLd(R) has a canonical inclusion into Diff∂( +� +k Rd). We consider +Emb( +� +k Rd, M) to be a Σk ≀ GLd(R)-space with action given by pre-composition by the +inverse, and CΣ( +� +k Rd; P) to be a Σk ≀ GLd(R)-space with action induced by the action +of Diff∂( +� +k Rd), as described in the previous section. +Definition 4.10 ([MT14]). Let Z be a pointed GLd(R)-space, The space of tubular +configurations in M with labels in Z, denoted D(M; Z), is the quotient +� +�� +k≥0 +Emb( +� +k Rd, M) +× +Σk≀GLd(R) Zk +� +� +� +∼ +where (e1, . . . , ek; z1, . . . , zk) ∼ (e1, . . . , ek−1; z1, . . . , zk−1) whenever zk is the basepoint +of Z, for ei : Rd �→ M and zi ∈ Z. + +18 +LUCIANA BASUALDO BONATTO +The space D(M; Z) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an +embedding e : +� +k Rd �→ M, and z = (z1, . . . , zk) ∈ Zk, we define +φ · [e, z] = [φ ◦ e; De(01)φ · z1, . . . , De(0k)φ · zk] +where 0i denotes the origin of the i-th component of +� +k Rd. +Lemma 4.11 ([MT14], Propositions 2.7, 2.8, and 3.6). The inclusion of the origin +i : ∗ �→ Rd induces a Diff∂(M)-equivariant weak equivalence +i∗ : D(M; Z) +≃ +−→ C(M; Z). +One can think of D(M; Z) as disc models for configuration spaces with labels. To +construct a disc model for summable labels, we need much more structure: +Definition 4.12. The space of surrounded configurations in M with summable labels +in P, denoted DΣ(M; P), is the quotient +� +�� +k≥0 +Emb( +� +k Rd, M) +× +Σk≀GLd(R) CΣ( +� +k Rd; P) +� +� +� +∼ +where (e : +� +mRd → M, ξ) ∼ (e′ : +� +n Rd → M, ξ′) if there are injections k �→ m and k �→ n +such that the induced inclusions i1 : +� +k Rd → +� +mRd and i2 : +� +k Rd → +� +n Rd satisfy +ξ ⊂ Im i1, ξ′ ⊂ Im i2 +e ◦ i1 = e′ ◦ i2, +and e∗(ξ) = e′ +∗(ξ′). +Here e∗ denotes the map of configurations with summable labels induced by a codimen- +sion zero embedding, as described in (4.1). Since the Fulton-MacPherson configuration +spaces are functorial for co-dimension zero embeddings, we get a map +p : DΣ(M; P) → CΣ(M; P) +(4.2) +taking a class (e, ξ) to the configuration e∗(ξ). By definition, this map does not depend +on the choice of representative (e, ξ). The space DΣ(M; P) admits a partial ordering +by declaring (e, ξ) < (e′, ξ′) if, for some representative of the classes, e(ξ) = e′(ξ′) +and Im e ⊃ Im e′. We denote by DΣ(M; P)• the semi-simplicial nerve of the poset +DΣ(M; P). +By the definition of the partial order, the map p induces an augmentation DΣ(M; P)• → +CΣ(M; P). In particular, DΣ(M; P)• is an augmented topological flag complex (Defini- +tion 2.2), since the space of n-simplices is indeed an open subspace of the (n + 1)-tuples +of vertices with the same image under the augmentation map, and the condition for +a tuple to form an n-simplex is just given by checking the pairwise order relation. It +might be helpful to keep in mind the following visualisation for elements in the space +|DΣ(M; P)•|: any point can be expressed by a tuple +((e0, ξ0) < · · · < (ek, ξk); t0, . . . , tk) +t0 · · · + tk = 1. +The poset construction implies that for such a tuple e0(ξ0) = · · · = ek(ξk) and Im e0 ⊃ +· · · ⊃ Im ek. We can visualise it as a collection of descending embedded discs around +the configuration ei(ξi) in M. Each collection of embedded discs has weights adding +up to 1 and when the weight associated to a collection of embeddings goes to zero, +that collection disappears. The equivalence relation guarantees that we can always get +a representative for the collection of discs such that each component has at least one +particle of the configuration inside it. See Figure 3. +Proposition 4.13. The map p : DΣ(M; P) → CΣ(M; P) induces a weak homotopy +equivalence +|DΣ(M; P)•| → CΣ(M; P). + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +19 +t0 +t2 +t1 +Figure 3. Element in a 2-simplex of |DΣ(F3)|. +The proof of the above will be a direct application of Theorem 2.3 [GRW14, Theorem +6.2]. +Proof. As discussed before, the augmented semi-simplicial space p : DΣ(M; P)• → +CΣ(M; P) is an augmented topological flag complex. We need to verify that the hy- +potheses of Theorem 2.3 are satisfied: +(i) To see that p : DΣ(M; P) → CΣ(M; P) has local sections, take f : Dn → +CΣ(M; P) and a point (e, ξ) ∈ p−1(f(x)). Then the image of e is an open subset +of M containing f(x), and therefore the subspace V of configurations contained +in Im e is an open subset of CΣ(M; P) containing f(x). Let U = f −1(V ), which +is an open neighbourhood of x in Dn. Then the map F : U → DΣ(M; P) taking +y to (e, e−1(f(y))) satisfies p ◦ F = f|U and F(x) = (e, ξ). +(ii) p : DΣ(M; P) → CΣ(M; P) is surjective, since any configuration ξ admits a +tubular neighbourhood e. Then (e, e−1(ξ)) is an element of DΣ(M; P) in the +pre-image of ξ. +(iii) For a configuration ξ ∈ CΣ(M; P) and a non-empty finite subset {(e1, ξ1), . . . , (ek, ξk)} +in its pre-image, we can always find e an embedding of Rd’s containing ei(ξi) +and contained in all ei. For instance by taking e to small open discs around the +points in the configuration. +Then by Theorem 2.3, the map |DΣ(M; P)•| → CΣ(M; P) is a weak homotopy equiva- +lence. +□ +The space DΣ(M; P) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an +embedding e : +� +k Rd �→ M, and ξ = (x1, . . . , xk; p1, . . . , pk) a configuration of points in +� +k Rd with labels in P, we define φ · [e, ξ] = [φ ◦ e, ξ]. It is simple to verify this action +is well-defined and preserves the partial order in DΣ(M; P). It follows directly from +the definition that the augmentation map p : DΣ(M; P) → CΣ(M; P) is Diff∂(M)- +equivariant. +Since the partial order in DΣ(M; P) is compatible with the Diff∂(M)-action, it in- +duces a fibrewise partial order on DΣ(M; P) ×Diff∂(M) Emb(M, R∞) over BDiff∂(M). +Then the semi-simplicial nerve of the poset DΣ(M; P) ×Diff∂(M) Emb(M, R∞) is simply +is the fibrewise semi-simplicial nerve DΣ(M; P)• ×Diff∂(M) Emb(M, R∞). +Corollary 4.14. The map DΣ(M; P) → CΣ(M; P) induces a weak homotopy equiva- +lence +|DΣ(M; P)• +× +Diff∂(M) Emb(M, R∞)| −→ CΣ(M; P)//Diff∂(M). +The above follows directly from the definition of the definition of the partial order on +DΣ(M; P) ×Diff∂(M) Emb(M, R∞) and Proposition 4.13. +Corollary 4.14 allows us to use a disc model for the space of configurations with +partially summable labels when proving the decoupling in this setting. We finish this +section by introducing another space of discs and configurations which will be used in +the decoupling. +Definition 4.15. Let Z be a pointed GLd(R)-space and let M be a smooth compact +manifold of dimension n > d. The space of d-tubular configurations in M with labels in + +20 +LUCIANA BASUALDO BONATTO +Z, denoted Dd(M; Z), is the quotient +� +�� +k≥0 +Emb( +� +k Rd, M) +× +Σk≀GL+ +d (R) +Zk +� +� +� +∼ +where (e1, . . . , ek; z1, . . . , zk) ∼ (e1, . . . , ek−1; z1, . . . , zk−1) whenever zk is the basepoint +of Z, for ei : Rd �→ M and zi ∈ Z. +The difference between the spaces Dd(M; Z) and D(M; Z) (Definition 4.10) is that on +the former we look at embedded discs of a lower dimension than the ambient manifold +M. +Lemma 4.16. Let n > d, and denote by Ed,n denote the total space of the canonical +GL+ +d (R)-bundle over the oriented Grassmanian Gr+(d, n). The inclusion of the origin +i : ∗ �→ Rd induces a weak equivalence +i∗ : Dd(Rn; Z) +≃ +−→ C(M; (Ed,n)+ +∧ +GL+ +d (R) +Z). +This result should be seen as an analogue of Lemma 4.11 in the setting of d-tubular +configurations in Rn. +Proof. Recall that Emb( +� +k Rd, Rn) ≃ �Ck(Rn) × (Ed,n)k, where the map to �Ck(Rn) is +induced by the inclusion of the origins. Then we get weak equivalences +Emb( +� +k Rd, M) +× +(Σk≀GL+ +d (R)) +Zk +≃ +−→ ( �Ck(Rn) × (Ed,n)k) +× +(Σk≀GL+ +d (R)) +Zk. +These respect the equivalence relations and therefore induce a map +i∗ : Dd +Σ(Rn; P) +≃ +−→ C(M; (Ed,n)+ +∧ +GL+ +d (R) +Z). +The proof then follows from the same arguments of [MT14, Propositions 2.7 and 2.8]. +□ +We also need an analogue of Definition 4.12 for the case of d-tubular configurations. +Definition 4.17. Let M be a smooth compact manifold of dimension n > d. The space +of d-surrounded configurations in M with summable labels in P, denoted Dd +Σ(M; P), is +the quotient +� +�� +k≥0 +Emb( +� +k Rd, M) +× +Σk≀GL+ +d (R) +CΣ( +� +k Rd; P) +� +� +� +∼ +where (e : +� +mRd → M, ξ) ∼ (e′ : +� +n Rd → M, ξ′) if there are injections k �→ m and k �→ n +such that the induced inclusions i1 : +� +k Rd → +� +mRd and i2 : +� +k Rd → +� +n Rd satisfy +ξ ⊂ Im i1, ξ′ ⊂ Im i2 and e ◦ i1 = e′ ◦ i2. +We equip this space with a partial ordering by declaring (e, ξ) < (e′, ξ′) if, for some +representative of the classes, e(ξ) = e′(ξ′) and Im e ⊃ Im e′. +We denote the semi- +simplicial nerve of this poset by Dd +Σ(M; P)•. +4.3. The decoupling theorem. In this section we prove the main theorem of this +paper, which is a decoupling result for CΣ(Fg,b; P)//Diff∂(Fg,b). As in Section 3, we take +as a model for EDiff∂(Fg,b) the space Emb(Fg,b, R∞). For every g and b ≥ 0 we have +τΣ × εΣ : DΣ(Fg,b; P) × Emb(Fg,b, R∞) −→ Emb(Fg,b, R∞) × D2 +Σ(R∞; P) +((e : +� +k R2 �→ Fg,b, ξ), f : Fg,b �→ R∞) �−→ (f, (f ◦ e, ξ)). + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +21 +This map is Diff∂(Fg,b)-equivariant with respect to the diagonal action on the domain +and the action on Emb(Fg,b, R∞) on the target, and it preserves the poset structures. +Hence it induces a map τΣ × εΣ fitting into the following diagram +|DΣ(Fg,b; P)• +× +Diff∂(Fg,b) Emb(Fg,b, R∞)| +|BDiff∂(Fg,b) × D2 +Σ(R∞; P)•| +CΣ(Fg,b; P) +× +Diff∂(Fg,b) Emb(Fg,b, R∞) +BDiff∂(Fg,b) × |D2 +Σ(R∞; P)•| +≃ +τΣ×εΣ +≃ +Theorem 4.18. The map +τΣ × εΣ : |DΣ(Fg,b; P)• +× +Diff∂(Fg,b) Emb(Fg,b, R∞)| → |BDiff∂(Fg,b) × D2 +Σ(R∞; P)•| +induces a homology isomorphism in degrees ≤ 2 +3g. +Theorem 4.18 and Corollary 4.14 imply Theorem F. +The proof of the above, will consist on showing that τΣ × εΣ is a level-wise homology +isomorphism of semi-simplicial spaces in degrees ≤ 2 +3g and then show that this implies +that the same holds on the geometric realisations, as done in [GRW17, Section 4]. To do +this, we will use the spectral sequence recalled in Section 2.2, Lemma 4.11, and Theorem +3.2, the decoupling result for the space of non-colliding configurations with labels. +Throughout the proof, it will be helpful to keep in mind Figure 4. +Figure 4. Correspondence of (4.3) between an element of DΣ(F3; P)2 +(top) and D(F3, Z2) (bottom). The dotted regions represent the em- +bedded R2’s and the arrows indicate their labels. +Proof of Theorem 4.18. We start by showing that +DΣ(Fg,b; P)• +× +Diff∂(Fg,b) Emb(Fg,b, R∞) → BDiff∂(Fg,b) × D2 +Σ(R∞; P)• +induces level-wise homology isomorphisms in degrees ≤ 2 +3g. +Let Zp be the subspace of DΣ(R2; P)p consisting of those tuples e = ((e0, ξ0), . . . , (ep, ξp)) +with e0 = id. This space is pointed by the unique class on the empty configuration. We + +% +%22 +LUCIANA BASUALDO BONATTO +show there is a Diff∂(Fg,b)-equivariant homeomorphism +DΣ(Fg,b; P)p ∼= D(Fg,b; Zp) +(4.3) +where D(Fg,b; Zp) is the space of tubular configurations with labels in Zp (see Definition +4.10). Let [(e0, ξ0) < · · · < (ep, ξp)] denote the an element in DΣ(Fg,b; P)p. Then, by +definition e0(ξ0) = · · · = ep(ξp) and Im e0 ⊃ · · · ⊃ Im ep. Denote by ij : R2 �→ +� +k R2 the +map induced by the inclusion {j} �→ {1, . . . , k}, for 1 ≤ j ≤ k. The map DΣ(Fg,b; P)p → +DΣ(Fg,b; Zp) takes a sequence ((e0, ξ0), . . . , (ep, ξp)), to the class represented by the +embedding e0 and and the label associated to the jth component R2 ⊂ +� +k R2 given by +((id, ξ0), ((e0 ◦ ij)−1 ◦ e1, ξ1) . . . , ((e0 ◦ ij)−1 ◦ ep, ξp)) ∈ Zp. +For intuition behind this homeomorphism, see Figure 4. It is simple to explicitly con- +struct an inverse for this map and check this is a Diff∂(Fg,b)-equivariant homeomor- +phism. +By Lemma 4.11, the inclusion of the origins defines Diff∂(Fg,b)-equivariant map +D(Fg,b; Zp) → C(Fg,b; Zp) which is a weak-equivalence of Diff∂(Fg,b)-spaces. Together +with the homeomorphism (4.3) this implies that +DΣ(Fg,b; P)p +× +Diff∂(Fg,b) Emb(Fg,b, R∞) +≃ +−→ C(Fg,b; Zp) +× +Diff∂(Fg,b) Emb(Fg,b, R∞). +(4.4) +Similarly, we now show that +D2 +Σ(R∞; P)p +≃ +−→ C(R∞; (ESO(2))+ +∧ +SO(2) Zp). +(4.5) +By the same argument as above, we get a homeomorphism +D2 +Σ(R∞; P)p ∼= D2(R∞; Zp) +where D2(R∞; Zp) is the space of d-tubular configurations with labels in Zp as in Def- +inition 4.15. By Lemma 4.16, the inclusion of the origins induces a weak homotopy +equivalence D2(R∞; Zp) ≃ C(R∞; (ESO(2))+ ∧SO(2) Zp) which implies the homotopy +equivalence (4.5). +Then we have a commutative diagram +DΣ(Fg,b; P)p +× +Diff∂(Fg,b) Emb(Fg,b, R∞) +C(Fg,b; Zp) +× +Diff∂(Fg,b) Emb(Fg,b, R∞) +BDiff∂(Fg,b) × D2 +Σ(R∞; P)p +BDiff∂(Fg,b) × C(R∞; ESO(2)+ +∧ +SO(2) Zp) +≃ +(τΣ×εΣ)p +τ×ε +≃ +where the top and bottom maps are weak equivalences by (4.4) and (4.5), respectively. +The right-hand map induces homology isomorphisms in degrees ≤ 2 +3g by Theorem 3.2 +with Z = Zp. Therefore so does the map (τΣ × εΣ)p. +This implies that the map between the spectral sequences associated to the semi- +simplicial spaces X• = DΣ(Fg,b; P)•×Diff∂(Fg,b)Emb(Fg,b, R∞), and Y• = BDiff∂(Fg,b)× +D2 +Σ(R∞; P)• +E1 +p,q = Hq(Xp) +Hp+q(|X•|) +E′1 +p,q = Hq(Yp) +Hp+q(|Y•|). +τΣ×εΣ +induces an isomorphism on the E1-pages for all q ≤ 2 +3g, and therefore the right-hand +map is also an isomorphism in such degrees. +□ + +DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES +23 +As in the case for labelled configuration spaces, the Decoupling Theorem for Sum- +mable Labels, allows us to deduce homological stability results. +Corollary 4.19. For b ≥ 1, the map induced by gluing F1,1 along the boundary +CΣ(Fg,b; Z)//Diff∂(Fg,b) → CΣ(Fg+1,b; Z)//Diff∂(Fg+1,b) +induces a homology isomorphism in degrees ≤ 2 +3g. +The proof follows from the same arguments as in the proof of Corollary 3.4. +4.4. Monoids of configurations on surfaces with partially summable labels. +In this section, we show that Theorem 4.18 implies a splitting for the group completion +of the monoid of configuration on surfaces with partially summable labels, analogous to +Corollary 3.7. +As in Section 3.2, gluing two surfaces Fg,1 and Fh,1 along part of their boundary +defines an associative multiplication +BDiff∂(Fg,1) × BDiff∂(Fh,1) → BDiff∂(Fg+h,1). +For (P, 0) a framed partial 2-monoid with unit, the operation on diffeomorphism +groups and spaces EDiff∂(Fg,1) described above, together with the fixed identifications +Fg+h,1 = Fg,1#∂Fh,1, induce an associative multiplication also on the Borel construc- +tions +µ : CΣ(Fg,1; P)//Diff∂(Fg,1)×CΣ(Fh,1; P)//Diff∂(Fh,1) → CΣ(Fg+h,1; P)//Diff∂(Fg+h,1). +Analogous to the construction of Chapter 3, we denote the associated Borel construction +by +MCΣ(P)g = CΣ(Fg,1; P)//Diff∂(Fg,1). +This multiplication makes MCΣ(P) = +� +g≥0MCΣ(P)g into a topological monoid, which +we refer to as the monoid of configurations with summable labels in P. +On the other hand, the poset D2 +Σ(R∞; P) can be made into a partially ordered topo- +logical monoid, using the same strategy Segal used to define a topological monoid equiv- +alent to C(R∞, X), as we recalled in Section 3.2. +Corollary 4.20. For any path-connected framed partial 2-monoid with unit P, there is +a homotopy equivalence +ΩB(MCΣ(P)) ≃ ΩB +�� +g BDiff∂(Fg,1) +� +× ΩB |D2 +Σ(R∞; P)•|. +The proof of the above result consists of constructing a zig-zag of monoids +(4.6) +� +g≥0 +|DΣ(Fg,1; P)•//Diff∂(Fg,1)| +� +g≥0 +BDiff∂(Fg,1) × |D2 +Σ(R∞; P)•| +MCΣ(P) +≃ +τΣ×εΣ +The vertical arrow is induced by Corollary 4.14 while the horizontal arrow is induced +by the decoupling map. We describe the monoidal structures on the top spaces using a +general construction for posets recalled in Section 2.1. +Proof of Corollary 4.20. The space � +g≥0 +|DΣ(Fg,1; P)•//Diff∂(Fg,1)| is homotopy equiva- +lent to the geometric realisation of the semi-simplicial nerve of the poset +� +g≥0 +DΣ(Fg,1; P)//Diff∂(Fg,1) + +24 +LUCIANA BASUALDO BONATTO +with partial order defined component-wise. As before, gluing surfaces along part of their +boundary makes +� +g≥0DΣ(Fg,1; P)//Diff∂(Fg,1) into a partially ordered topological monoid. +Moreover, one can verify that the augmentations DΣ(Fg,1; P)• → CΣ(Fg,1; P) induce a +map of monoids. Together with Lemma 2.1, this gives a map of monoids +� +g≥0 +|DΣ(Fg,1; P)•//Diff∂(Fg,1)| → MCΣ(P). +(4.7) +On the other hand, the decoupling map gives a map of topological monoids +� +g≥0 +|DΣ(Fg,1; P)•//Diff∂(Fg,1)| → +� +g≥0 +BDiff∂(Fg,1) × |D2 +Σ(R∞; P)•| +(4.8) +just as in Lemma 3.8. +All that remains is to verify that the maps (4.7) and (4.8) induce homotopy equiva- +lences on group completions. As the group completions are loop spaces, they are simple +and, by Whitehead’s theorem, it suffices to show the maps induce homology equivalences +on the group completions. All monoids are homotopy commutative, hence the group +completion theorem [MS76] can be applied. 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MR 84769 +Email address: basualdo@mpim-bonn.mpg.de +Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany + diff --git a/GNAyT4oBgHgl3EQfSvfA/content/tmp_files/load_file.txt b/GNAyT4oBgHgl3EQfSvfA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a82de37703923f92705f6c1630e0ba3440e2f835 --- /dev/null +++ b/GNAyT4oBgHgl3EQfSvfA/content/tmp_files/load_file.txt @@ -0,0 +1,1356 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf,len=1355 +page_content='DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES LUCIANA BASUALDO BONATTO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The configuration space of k points on a manifold carries an action of its diffeomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The homotopy quotient of this action is equivalent to the classifying space of diffeomorphisms of a punctured manifold, and therefore ad- mits results about homological stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Inspired by the works of Segal, McDuff, Bodigheimer, and Salvatore, we look at generalised configuration spaces where par- ticles have labels and even partially summable labels, in which points are allowed to collide whenever their labels are summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' These generalised configuration spaces also admit actions of the diffeomorphism group and we look at their homotopy quo- tients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Our main result is a decoupling theorem for these homotopy quotients on surfaces: in a range, their homology is completely described by the product of the moduli space of surfaces and a generalised configuration space of points in R∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using this result, we show these spaces admit homological stability with respect to increasing the genus, and we identify the stable homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This can be interpreted as an Diff-equivariant homological stability for factorization homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In addition, we use this result to study the group completion of the monoid of moduli spaces of configurations on surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Introduction The ordered configuration space of k points on a smooth manifold M without bound- ary is defined as �Ck(M) := {(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk) ∈ M k | mi ̸= mj if i ̸= j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' When M is a smooth manifold with boundary, we denote by �Ck(M) the space of ordered configurations of k points in its interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The symmetric group Σk acts on this space by permuting the order of the k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The configuration space of k points on M, denoted Ck(M), is the quotient �Ck(M)/Σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote by Diff∂(M) the group of diffeomorphisms of a manifold M which fix a collar of its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this paper, we will focus on 2-dimensional manifolds and we denote by F k g,b an orientable surface of genus g, k punctures, and b ≥ 1 boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The spaces Ck(Fg,b) admit an action of the group Diff∂(Fg,b), where a diffeomorphism φ acts by taking a collection of k points to its image via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Of interest here, is the Borel construction (homotopy quotient) of this action, denoted Ck(Fg,b)//Diff∂(Fg,b), to which we refer to as a moduli of configurations of k points in Fg,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to show that Ck(Fg,b)//Diff∂(Fg,b) ≃ BDiff∂(F k g,b) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1) and in fact this relation is not only true for surfaces, but for any manifold with k punctures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In particular, this allows us to deduce homological stability results for these moduli of configurations of k points, directly from the known theorems for classifying spaces of punctured surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For instance, when b ≥ 1 these spaces admit homological stability when increasing the genus and when increasing then number of points [Har85, Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 57R19, 55R80, 55R40, 55P47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This material is based upon work supported by CNPq (201780/2017-8) and by the NSF Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DMS-1928930 while the author participated in a program hosted by the MSRI in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='00093v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='AT] 31 Dec 2022 2 LUCIANA BASUALDO BONATTO Har90, RW16, RW14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, it was shown in [BT01] that the stable homology of this classifying space can be computed from the homology of BDiff∂(Fg,b)×B(Σk ≀GL+ 2 (R)), what is known as a decoupling theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this paper, we study the analogue of this moduli space for generalised configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As a first case, we look at labelled configurations: given a pointed space Z, the space of configurations in M with labels in Z is the quotient C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) := � �� k≥0 �Ck(M) ×Σk Zk � � � ∼ under the relation (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) ∼ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk−1) if zk is the basepoint of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We can interpret this space geometrically by considering it as the space of particles in M where each particle is labelled by an element of Z, and a particle is allowed to disappear if labelled by the basepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This space has been of interest since the 70’s, appearing on the seminal works of May [May72] and Segal [Seg73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It was noted that the space C(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) can be given the structure of an (A∞-)monoid by taking multiplication to be roughly given by stacking configurations side by side [Seg73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' One of the main results about this space is what today is called a scanning map C(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) → ΩnΣnZ which was shown by Segal to induce a weak-homotopy equivalence on group-completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This idea has been generalised in many directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For instance, B¨odigheimer [B¨od87] proved an analogous statement for configurations on general manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In addition, similar results were proven for the case where the spaces of labels has extra structure, such as (partial) multiplications [McD75, Seg79, Gue95, Kal01, Sal01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We discuss the later case in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' More recently this labelled configuration space and scanning map argument have been expanded to sophisticated constructions in factorization homology [AFT17] on the one hand and, on the other, in the form of configuration spaces of manifolds, has been used to compute the stable homology of the moduli spaces of Riemann surfaces [MW07] and higher dimensional manifolds [GRW18, GRW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Labelled configuration spaces also inherit an action of the diffeomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Even more, if Z is a pointed GL+ 2 (R)-space, we can define an action of Diff∂(Fg,b) on C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) where a diffeomorphism φ acts by taking a collection of k points to its image via φ, and the label z of a point w is taken to the label dwφ·z of the point φ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Unlike the case of configurations with a fixed number of points, C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) is not in general equivalent to the classifying space of a diffeomorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Hence we ask if it still has homological stability and if it admits an analogous decoupling theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For a surface Fg,b, with b ≥ 1, taking the boundary connected sum with the surface F1,1 induces a homomorphism Diff∂(Fg,b) → Diff∂(Fg+1,b), given by extending a map on Fg,b by the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We call this the stabilisation map and study what it induces on the moduli of configuration spaces: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Z be a pointed GL+ 2 (R)-space and b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The stabilisation map on the Borel constructions s∗ : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → C(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+1,b) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, we can determine precisely what the stable homology is: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Z is a pointed connected GL+ 2 (R)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' There is a map C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → Ω∞MTSO(2) × Ω∞Σ∞ � EGL+ 2 (R)+ ∧ GL+ 2 (R) Z � which is compatible with the stabilisation maps and induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 3 In the above, EGL+ 2 (R) denotes the total space of a universal fibration for BGL+ 2 (R), we use (−)+ to denote adjoining a disjoint basepoint to a space, and − ∧ GL+ 2 (R) − denotes the quotient of the smash product of pointed topological GL+ 2 (R)-spaces by the diagonal action of GL+ 2 (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Both of the results above are consequences of Theorem C below, which is an ana- logue of the decoupling theorem in [BT01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It implies that the stable homology of this moduli of configuration spaces can be understood through a decoupling map τ × ε, which separates the points in the configurations from the underlying surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map τ : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) forgets the data of the configurations, and ε forgets the underlying surface, but still remembers the points in the configuration and some local tangential data around them (for a detailed description of these maps see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem C (Decoupling Theorem for Labelled Configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let τ and ε be the maps described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then τ × ε : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' EGL+ 2 (R)+ ∧ GL+ 2 (R) Z) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This result may be interpreted in physical terms: in the high genus limit, the con- straints for the particles to stay on the underlying surface are lifted and the particles are now free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As a final application of Theorem C, we look at monoids of moduli of configurations on surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Boundary connected sum induces a multiplication on the level of classifying spaces making � g BDiff∂(Fg,1) into a topological monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This important construction and its group completion have been central in the study of the stable homology of mapping class groups of surfaces [Mil86, Til00, MW07, GRW10, GRW18, GRW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This gluing of surfaces induces also a multiplication on the Borel constructions C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1) × C(Fh,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fh,1) → C(Fg+h,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+h,1) making � g C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1) into a topological monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We study its group com- pletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any pointed GL+ 2 (R)-space Z, the decoupling map induces a weak equivalences on group completions ΩB �� g C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1) � ≃ ΩB �� g BDiff∂(Fg,1) � ×ΩBC(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' EGL+ 2 (R)+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Configuration spaces with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The main result of this paper considers a more general type of configuration spaces with labels in a framed partial 2-monoid, where particles are allowed to collide if their labels are summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This space has been explored in works such as [McD75, Seg79, Gue95, Kal01, Sal01] and, when the labels are E2-algebras (not partial), is equivalent to factorization ho- mology [AF15] and topological chiral homology [Lur09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The first example of this con- struction can be seen in McDuff’s configuration spaces of positive and negative particles [McD75], where particles are labelled by “charges” and are allowed to collide whenever their charges are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' More generally, Salvatore [Sal01] defines partial 2-monoids, which are, in essence, spaces with a multiplication similar to an E2-algebra structure, but with the restriction that this multiplication does not need to be defined for every tuple of elements (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Whenever P is equipped with a compatible ac- tion of GL+ 2 (R), we call it a framed partial monoid, and we can define the space of configurations in Fg,b with partially summable labels in P, denoted CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The definition CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) requires much more machinery then the case for non-summable 4 LUCIANA BASUALDO BONATTO labels, such as the Fulton-MacPherson operad, and yields more complicated models for configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We discuss these constructions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As before, these generalised configuration spaces admit an action of the diffeomorphism group and we study its Borel construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let P be a framed partial 2-monoid and b ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The stabilisation map on the Borel constructions s∗ : CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,b) → CΣ(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg+1,b) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' While homological stability for configurations with summable labels with respect to increasing the number of points had been studied in [KM16], the above result is the first to look at stability with respect to increasing the genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem E can be interpreted as a Diff∂-equivariant homological stability result for factorisation homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As in the case for labelled configurations, this result is a consequence of a decou- pling theorem for the space CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,b), which is the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is much more intricate than the decoupling for labelled configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof uses a semi-simplicial resolution of CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) developed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, which we refer to as the disc model for configurations, denoted |DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This model makes explicit the connection between these spaces and factorization homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In the decoupling context, we naturally encounter an analogue of this space with 2-dimensional discs with configurations embedded in R∞, we denote this space |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using the Decoupling Theorem for Labelled Configu- rations (Theorem D) we then prove: Theorem F (Decoupling Theorem for Configurations with Summable Labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For P a framed partial 2-monoid, there is a weak equivalence CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,b) ≃ |DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|//Diff∂(Fg,b) and the decoupling map |DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|//Diff∂(Fg,b) → BDiff∂(Fg,b) × |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In future work we will discuss the homotopy type of the space |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| and its description as an infinite loop space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We conjecture that it is also equivalent to a configuration in R∞ with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Analogous to the case of labelled configurations, the spaces CΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1) assemble into a topological monoid, and the decoupling theorem descends into its group completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any path-connected framed partial 2-monoid with unit P, the decou- pling map induce a homotopy equivalence ΩB( � g CΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1)) ≃ ΩB( � g BDiff∂(Fg,1)) × ΩB(|D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We start by recalling in Section 2 background results which will be used throughout the paper, especially on Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This can be skipped and referred back to when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In Section 3 we introduce labelled configuration spaces and prove Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using this, we deduce Theorems A and B, and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In Section 4 we discuss the case of configurations with summable labels, and prove the main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We start by recalling in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1 the definitions of framed partial d-monoids and configuration spaces with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We then construct semi-simplicial resolutions for these spaces in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3, we use this disc model together with the Decoupling Theorem for Labelled Configurations (Theorem D) to prove Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Finally, we use this result to deduce Theorem E and Corollary G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 5 Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' I would like to thank Ulrike Tillmann for suggesting the prob- lem and for the many insightful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In addition, I would like to thank David Ayala, Christopher Douglas, Jan Steinebrunner, and Nathalie Wahl for the helpful dis- cussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Preliminaries In this section we recall techniques and results on semi-simplicial spaces, which will be used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The reader may skip this part and refer back to when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For a detailed exposition of the concepts in this section see [ERW19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A semi-simplicial space is a functor ∆op inj → Top, where ∆inj is the category with ob- ject the linearly ordered sets [p] = {0 < · · · < p} and morphisms the injective monotone maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote such a functor by X• and write Xp = X•({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , p}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The datum of a semi-simplicial space is equivalent to the collection of spaces Xp, p ≥ 0, together with face maps di : Xp → Xp−1 for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , p, satisfying didj = dj−1di if i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Denote by ∆p the standard p-simplex ∆p = � (t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tp) ∈ Rp+1��� p � i=1 ti = 1 and ti ≥ 0 for all i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To each morphism φ : [p] → [q] in ∆inj, there is a continuous map φ∗ : ∆p → ∆q such that φ∗(t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tp) = (s0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , sq) with sj = � i∈φ−1(j) ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The geometric realisation of a semi-simplicial space X• is the quotient space |X•| := �� p Xp × ∆p � � ∼ where (x, φ∗t) ∼ (φ∗x, t), and φ is a morphism of ∆inj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Semi-simplicial nerve of a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Any topological poset (Q, <) defines a semi- simplicial space Q• by setting Qp to be the subspace of tuples (q0 < · · · < qp) ∈ Qp+1, and face maps di : Qp −→ Qp−1 for 0 ≤ i ≤ p (q0 < · · · < qi < · · · < qp) �−→ (q0 < · · · < qi−1 < qi+1 < · · · < qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We refer to Q• as the semi-simplicial nerve of the poset Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Given a topological poset (Q, <), the space Q×Q can be equipped with a partial order where (q1, q2) < (q′ 1, q′ 2) if qi < q′ i and qj ≤ q′ j, for {i, j} = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We say that a pointed such Q is a partially ordered topological monoid if it is equipped with a multiplication − · − : Q × Q → Q which is strictly associative, unital and order preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this case, the geometric realisation of the semi-simplicial nerve Q• is naturally endowed with a multiplication · defined by � (q0 < · · · < qm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tm) · (q0 < · · · < qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tk) � = = (q0 · q0 < · · · < q0 · qk < · · · < qm · q0 · · · < qm · qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' t0 · t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tm · t) where ti · t = tit0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , titk, for all i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is straightforward to verify that this is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For (Q, <, µ) a partially ordered topological monoid, (|Q•|, |µ|) is a topo- logical monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, any map of partially ordered topological monoids f : Q → Q′ induces a map of topological monoids f∗ : |Q•| → |Q′ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof is a straightforward computation and follows directly from the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 6 LUCIANA BASUALDO BONATTO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We quickly recall below the spectral sequence defined in [Seg68, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1] associated to a semi-simplicial space, which is the key for the homology argument used in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18 (see [ERW19, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4] for a detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any semi-simplicial space X•, the geometric realisation |X•| admits a filtration by its skeleta, with |X•|(0) = X0 and |X•|(q) = |X•|(q−1) ∪Xq×∂∆q Xq × ∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This filtration yields a spectral sequence E1 p,q = Hp+q(|X•|(q), |X•|(q−1)) =⇒ Hp+q(|X•|) and by excision and the Kunneth isomorphism, the left-hand term can be re-written to give a spectral sequence with E1 p,q ∼= Hp(Xq) =⇒ Hp+q(|X•|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Therefore a map of semi-simplicial spaces inducing a level-wise homology isomorphism gives an isomorphism of the first pages of the respective spectral sequences, and therefore a homology isomorphism between the geometric realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Semi-simplicial Resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In the proof of the decoupling we will use a semi- simplicial resolution of the spaces of configurations with summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Showing that we indeed have a resolution will be a direct application of [GRW14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For completeness, we quickly recall the statement of this result to clarify the conditions that will be checked in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We follow the notation and definitions of [GRW14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' An augmented semi-simplicial space is a triple (X•, X−1, ε•), where X• is a semi- simplicial space, X−1 is a space and ε• is a collection of continuous maps εp : Xp → X−1 satisfying diεp = εp−1 for all p ≥ 0 and all face maps di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We also say that ε• : X• → X−1 is an augmentation for X•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to verify that an augmentation induces a continuous map |ε•| : |X•| → X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' An augmented topological flag complex [GRW14, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1] is an augmented semi-simplicial space ε : X• → X−1 such that (i) The map Xn → X0 ×X−1 · · ·×X−1 X0 taking an n-simplex to its (n+1) vertices is a homeomorphism onto its image, which is an open subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (ii) A tuple (v0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , vn) ∈ X0 ×X−1 · · ·×X−1 X0 lies in Xn if and only if (vi, vj) ∈ X1 for all i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In other words, in an augmented topological flag complex, the space of n-simplices can be described as an open subspace of the (n + 1)-tuples of vertices with the same image under ε, and such a tuple forms an n-simplex if and only if the pairs of vertices are all 1-simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The result below is a criterion to determine when an augmented topological flag complex X• → X−1 induces a weak equivalence |X•| → X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3 ([GRW14], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let X• → X−1 be an augmented topological flag complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Suppose that (i) The map ε : X0 → X−1 has local lifts of any map from a disc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' given a map f : Dn → X−1, a point p ∈ ε−1(f(x)), there is an open neighbourhood U ⊂ Dn of x and a map F : U → X0 such that ε ◦ F = f|U and F(x) = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (ii) ε : X0 → X−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (iii) For any p ∈ X−1 and any non-empty finite set {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , vn} ∈ ε−1(p) there exists a v ∈ ε−1(p) with (v1, v) ∈ X1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then |X•| → X−1 is a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Decoupling Labelled Configuration Spaces In this section, we recall the definition of configuration spaces with labels and in- troduce a model for the homotopy quotients C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' With this, we construct the decoupling map of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We then prove this result as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 and use it to prove Theorems A and B, and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' These are, respectively, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5, and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let M be a manifold and Z be a well-pointed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The configuration space of M with labels in Z, denoted C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z), is the quotient � k≥0 �Ck(M) × Σk Zk/ ∼ where �Ck(M) denotes the ordered configuration space of k points in M, and (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) ∼ (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk−1) whenever zk is the basepoint of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' If Z is a pointed GL+ 2 (R)-space, and M is a surface Fg,b, with b ≥ 1, the space C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) carries a natural action by the diffeomorphism group of Fg,b: for φ ∈ Diff∂(Fg,b) φ · (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) := (φ(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , φ(mk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Dm1φ · z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , Dm1φ · zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The basepoint relation is preserved by this action as Z is a pointed GL+ 2 (R)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The decoupling result is about the homotopy quotient of this action, that is, the Borel construction C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' From now on, we denote by Fg,b a fixed orientable surface of genus g, b ≥ 1 bound- ary components, and pick once and for all a framing on Fg,b, that is, a section s of the frame bundle Fr(TFg,b) → Fg,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We fix now a model for EDiff∂(Fg,b) that will be used to construct the decoupling map of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Emb(Fg,b, R∞) denote colimn→∞ Emb(Fg,b, Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This space has a free action of Diff∂(Fg,b) by precomposition and by [BF81] the quotient map Emb(Fg,b, R∞) → Emb(Fg,b, R∞)/Diff∂(Fg,b) has slices, hence it is a principal Diff∂(Fg,b)-bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, by Whitney’s embedding theorem, Emb(Fg,b, R∞) is weakly contractible and therefore it is a model for EDiff∂(Fg,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We then let C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) ≃ Emb(Fg,b, R∞) × Diff∂(Fg,b) C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Analogously, we will take the model for BDiff∂(Fg,b) given by BDiff∂(Fg,b) ≃ Emb(Fg,b, R∞)/Diff∂(Fg,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This can be interpreted as the space of abstract submanifolds of R∞ which are diffeo- morphic to Fg,b, but without a fixed diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Analogously, a point in C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)// Diff∂(Fg,b) consists of one such abstract manifold, together with a labelled configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The decoupling map will be the product of two maps τ : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1) ε : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' EGL+ 2 (R)+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2) Recall that EGL+ 2 (R) is the total space of a universal fibration for BGL+ 2 (R), we use (−)+ to denote adjoining a disjoint basepoint to a space, and − ∧ GL+ 2 (R) − denotes the quotient of the smash product of pointed topological GL+ 2 (R)-spaces by the diagonal action of GL+ 2 (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Intuitively, the map τ forgets the configuration, while ε forgets the underlying surface, but remembers the labelled configuration together with data on their tangent space on the submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 8 LUCIANA BASUALDO BONATTO In details, τ is the classifying map for the homotopy quotient, and it is simply the one induced by the projection Emb(Fg,b, R∞)×Ck(Fg,b) → Emb(Fg,b, R∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To define ε, we take as model for BGL+ 2 (R) the oriented Grassmanian manifold of 2-dimensional oriented subsbaces of R∞, Gr+(2, ∞), and let EGL+ 2 (R) denote the total space of the universal GL+ 2 (R)-bundle over it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then using the identification Fr(TR∞) ∼= R∞ ×EGL+ 2 (R), an embedding e : Fg,b �→ R∞ induces a map e∗ : Fr(TFg,b) → ESO(2) from the bundle of framings on TFg,b, taking a basis of TpFg,b to its image via Dpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map ε takes a point represented by a labelled configuration [m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk] and an embedding e : Fg,b �→ R∞ to the labelled configuration in R∞ given by [e(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , e(mk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' [e∗(s(m1)), z1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , [e∗(s(mk)), zk]] where s(p) denotes the chosen oriented frame on p ∈ Fg,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to verify that this indeed defines a continuous function to the configuration space C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let τ and ε be the maps in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the decoupling map τ × ε : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → BDiff∂(Fg,b) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧ GL+ 2 (R) Z) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof of the result above will build upon a decoupling result for unlabelled configurations with a fixed number of points, which was first proved by [BT01] and generalised in [Han09, Bon22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We show here a slight generalisation of the result which we will need in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) is constructed as a quotient of the union of spaces �Ck(M) ×Σk Zk, and the Lemma below is about the decoupling map in each of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In fact, we will work on a slightly more general context which will be more convenient for the proof: let X be a well-pointed space with an action of the wreath product Σk ≀ GL+ 2 (R) (in the context above we were using X = Zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space �Ck(Fg,1) × X comes equipped with two actions: Σk acts diagonally by permuting the points in the configuration and by the action on X, and Diff∂(Fg,b) acts by φ · (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' x) = (φ(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , φ(mk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (dm1φ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , dmkφ)(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Note that the actions of Σk and Diff∂(Fg,1) on this space commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As before, we have maps τk : ( �Ck(Fg,1) × Σk X)//Diff∂(Fg,b) → BDiff∂(Fg,b) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3) εk : ( �Ck(Fg,1) × Σk X)//Diff∂(Fg,b) → ( �Ck(R∞) × (EGL+ 2 (R))k) × Σk≀GL+ 2 (R) X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4) Here τk is again simply the classifying map for the homotopy quotient, and εk is the map taking a point represented by [m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , mk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' x] and an embedding e : Fg,b �→ R∞ to the class � [e(m1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , e(mk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' e∗(s(m1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , e∗(s(mk))], x � where s(p) denotes the chosen oriented frame on p ∈ Fg,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We remark that replacing X by Zk one recovers precisely the definition of the maps τ and ε in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3 ([BT01, Bon22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any Σk ≀GL+ 2 (R)-space X, let τk and εk be the maps defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then τk×εk : ( �Ck(Fg,b)× Σk X)//Diff∂(Fg,b) → BDiff∂(Fg,b)×( �Ck(R∞)×(EGL+ 2 (R))k) × Σk≀GL+ 2 (R) X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 9 For completeness, we include a short proof of the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For details see [BT01, Bon22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We start by reducing the proof to the case when X is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The projections ( �Ck(Fg,b) × Σk X)//Diff∂(Fg,b) → Ck(Fg,b)//Diff∂(Fg,b) BDiff∂(Fg,b) × ( �Ck(R∞) × (EGL+ 2 (R))k) × Σk≀GL+ 2 (R) X → BDiff∂(Fg,b) × Ck(R∞, BGL+ 2 (R)) are both fibrations with fibre X, and the map τk × εk induces a map between the corresponding fibre sequences, which is the identity on the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' If the map between the base spaces induces a homology isomorphism in degrees ≤ 2 3g, then by Zeeman’s Comparison Theorem [Zee57] applied to the Serre spectral sequences associated to these fibrations, so does τk × εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Hence it is enough to show that the map between the base spaces τk × εk : Ck(Fg,b)//Diff∂(Fg,b) → BDiff∂(Fg,b) × Ck(R∞, BGL+ 2 (R)) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By Palais’ Theorem [Pal60], the map εk is a fibration with fiber BDiff∂(Fg,b+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, the map τk × εk induces a map of fibre sequences BDiff∂(Fg,b+n) Ck(Fg,b)//Diff∂(Fg,b) Ck(R∞, BGL+ 2 (R)) BDiff∂(Fg,b) BDiff∂(Fg,b) × Ck(R∞, BGL+ 2 (R)) Ck(R∞, BGL+ 2 (R)) proj τk×εk εk where the leftmost map is the one induced by capping off the n extra boundary compo- nents by gluing discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This map was shown to induce a homology isomorphism in the range ≤ 2 3g [Har85, Iva87, Iva89, Iva93, Bol12, RW16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Hence, by Zeeman’s Compar- ison Theorem [Zee57] applied to the Serre spectral sequences associated to these fibre sequences, the map between the total spaces also induces homology isomorphisms in the range ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ Equipped with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3, we are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The spaces C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) and C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧GL+ 2 (R) Z) con- sist of configuration with an arbitrary number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' However they have natural filtrations C≤k(−) by the subspaces of configurations with at most k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' These induce filtrations Xk := C≤k(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) Yk := BDiff∂(Fg,b) × C≤k(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' that are preserved under the map τ × δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We will inductively show the restrictions Xk → Yk are homology isomorphisms for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Since Z is a space with a good basepoint, the inclusions Xk−1 �→ Xk and Yk−1 �→ Yk are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Their subquotients are Xk/Xk−1 = (EDiff∂(Fg,b))+ ∧ Diff∂(Fg,b) �Ck(Fg,b)+ ∧ Σk Z∧k and Yk/Yk−1 = (BDiff∂(Fg,b))+ ∧ �Ck(R∞)+ ∧ Σk ((EGL+ 2 (R))+ ∧ GL+ 2 (R) Z)∧k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 10 LUCIANA BASUALDO BONATTO Comparing the spectral sequences associated to these filtrations, it is enough to show that the induced map on these subquotients is a homology isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Consider the map of cofibrations: EDiff∂(Fg,b) × Diff∂(Fg,b) Ck(Fg,b) EDiff∂(Fg,b) × Diff∂(Fg,b) ( �Ck(Fg,b) × Σk Z∧k) Xk/Xk−1 BDiff∂(Fg,b) × Ck(R∞, BGL+ 2 (R)) BDiff∂(Fg,b) × ( �Ck(R∞) × Σk ((EGL+ 2 (R))+ ∧ GL+ 2 (R) Z)∧k) Yk/Yk−1 τk×εk τk×εk By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3 with X = ∗ the left-hand map induces a homology isomorphism in degrees ≤ 2 3g, and by the same result with X = Z∧n, so does the middle map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then by the five lemma, the right-hand map also induces a homology isomorphism in degrees ≤ 2 3g, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Homological Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Fg+1,b be a surface of genus g+1 and b ≥ 1 boundary components, which is obtained from Fg,b by a boundary connected sum with F1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then extending diffeomorphisms by the identity on F1,1 gives a map of topological groups s : Diff∂(Fg,b) �→ Diff∂(Fg+1,b) which we refer to as the stabilisation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, the inclusion Fg,1 �→ Fg+1,b induces a continuous map of labelled configuration spaces C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) → C(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) which is s-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Together this implies we have an induced map on the Borel constructions: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For b ≥ 1, the stabilisation map on the Borel constructions s∗ : C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → C(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+1,b) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The above result is a corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, however care has to be taken with respect to the model we have used for the Borel constructions and classifying spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Namely, it is not clear how to define a stabilisation map on the level of embedding spaces Emb(Fg,b, R∞) → Emb(Fg+1,b, R∞) which induces the desired map on Borel constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This can be remedied by taking as model for EDiff∂(Fg,b) a weakly contractible subspace of Emb(Fg,b, R∞) that still has a free and has proper action of Diff∂(Fg,b) and in which the stabilisation is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Fix a boundary component of Fg,b and an embedding S1 �→ {0}×R∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote by Emb∂(Fg,b, (−∞, 0]×R∞) the space of all extensions to an embedding of Fg,b which are standard on a collar neighbourhood of the marked boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By the same arguments as above, Emb∂(Fg,b, (−∞, 0] × R∞) is a model for EDiff∂(Fg,b), and the inclusion map Emb∂(Fg,b, (−∞, 0] × R∞) �→ Emb(Fg,b, R∞) is a Diff∂(Fg,b)-equivariant weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Fixing an embedding e : F1,2 �→ R∞ which restricts to the chosen embedding on a collar of the boundary, we get an inclusion Emb∂(Fg,b, (−∞, 0] × R∞) → Emb∂(Fg+1,b, (−∞, 0] × R∞) given by extending any embedding of Fg,b by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is clearly compatible with the stabilisation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using as model for EDiff∂(Fg,b) the space Emb∂(Fg,b, (−∞, 0]× R∞) described above, we get a commutative diagram DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 11 C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) C(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+1,b) BDiff∂(Fg,b) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧ GL+ 2 (R) Z) BDiff∂(Fg+1,b) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (EGL+ 2 (R))+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' s τ×ε τ×ε s×id By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 the vertical maps induce homology isomorphisms in degrees ≤ 2 3g, and by Harer’ Stability Theorem [Har85, Iva87, Iva89, Iva93, Bol12, RW16] and Kun- neth’s Theorem, so does the bottom map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Therefore the top map must also induce homology isomorphisms in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ The Decoupling Theorem also allows us to identify what the homology stabilises to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let C(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F∞) and BDiff∂(F∞) be respectively C(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F∞) := colim(C(F1,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F1,1) s−→ C(F2,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F2,1) s−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' ) BDiff∂(F∞) := colim(BDiff∂(F1,1 s−→ BDiff∂(F2,1) s−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For Z a pointed connected GL+ 2 (R)-space, the decoupling map induces τ × ε : C(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F∞) → Ω∞MTSO(2) × Ω∞Σ∞ � EGL+ 2 (R)+ ∧ GL+ 2 (R) Z � which is a homology isomorphism in all degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4 imply that the decoupling map on the colimits τ × ε : C(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞, EGL+ 2 (R)+ ∧ GL+ 2 (R) Z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5) induces a homology isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By [GTMW09, MW07], BDiff∂(F∞) admits a map to Ω∞MTSO(2) which is a homology isomorphism, and by [Seg73], the right-most config- uration space in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5) is homotopy equivalent to Ω∞Σ∞ � EGL+ 2 (R)+ ∧GL+ 2 (R) Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Monoid of Moduli of Labelled Configuration Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Gluing two surfaces Fg,1 and Fh,1 along part of their boundary defines a map of topological groups Diff∂(Fg,1) × Diff∂(Fh,1) → Diff∂(Fg+h,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This can be made into an associative operation if we fix once and for all oriented surfaces Fg,b of genus g and one boundary component, and compatible with the stabilisation (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using these choices for our surfaces, we can see that the above map gives an associative operation in the collection of diffeomorphism groups of all Fg,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' An example of such surfaces and multiplication is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' µ , = Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Example of the map µ : MC(C)2 × MC(C)3 → MC(C)5 where C is the space of colours, with white as the basepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Up to homotopy, the above gluing process is equivalent to gluing the boundary circles of surfaces Fg,1 and Fh,1, to two of the three boundary circles of F0,3, what is called the pair of pants multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We choose to think of this multiplication in terms of the first description because in this way the product is strictly associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This operation also induces a multiplication on the classifying spaces BDiff∂(Fg,1), which can be made associative by picking a convenient model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1, we will use as EDiff∂(Fg,1) a certain subspace of Emb(Fg,1, R∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Namely, we can fix 12 LUCIANA BASUALDO BONATTO embeddings δg : S1 �→ [0, g] × R∞ such that δg(S1) ∩ ({0} × R∞) = {0} × (−1, 1) × {0} and δg(S1) ∩ ({g} × R∞) = {g} × (−1, 1) × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote by Embδg(Fg,1, [0, g] × R∞) the space of all extensions to an embedding of Fg,1 which are standard on a collar neighbourhood of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By the same arguments as above, Embδg(Fg,1, [0, g] × R∞) is a model for EDiff∂(Fg,1), and the inclusion map Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,1, R∞) is a Diff∂(Fg,1)-equivariant weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By picking embeddings δg which are compatible with the stabilisation map, we can define a multiplication Embδg(Fg,1, [0, g] × R∞) × Embδh(Fh,1, [0, h] × R∞) → Embδg+h(Fg+h,1, [0, g + h] × R∞) by extending an embedding of Fg,1 by the translation of the embedding of Fh,1 in the first coordinate by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is clearly associative and it is compatible with the associative multiplication on the diffeomorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Taking as model for EDiff∂(Fg,1) the spaces Embδg(Fg,1, [0, g] × R∞) we obtain an associative multiplication on classifying spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This structure equips the space � g≥0BDiff∂(Fg,1) with the structure of a topological monoid, which we refer to as the surface monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is equivalent to the one studied by Tillmann in [Til00], which was essential in the proof of the Madsen-Weiss Theorem [MW07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The operation on diffeomorphism groups and spaces EDiff∂(Fg,1) described above, together with the fixed identifications Fg+h,1 = Fg,1#∂Fh,1, induce an associative mul- tiplication also on the Borel constructions (see Figure 1) µ : C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1) × C(Fh,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fh,1) → C(Fg+h,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+h,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote by MC(Z)g := C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The monoid of moduli of configurations labelled by Z is the topological monoid given by the disjoint union MC(Z) := � g≥0MC(Z)g together with the operation µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any pointed GL+ 2 (R)+-space Z, the decoupling map induces a weak equivalences on group completions ΩB �� g C(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,1) � ≃ ΩB �� g BDiff∂(Fg,1) � ×ΩB C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' EGL+ 2 (R)+ ∧ GL+ 2 (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Segal showed in [Seg73] that the space Ω∞Σ∞(X) is the group completion of the con- figuration space C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' X) seen as a topological monoid where the operation is roughly given by transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using the inclusion Embδg(Fg,1, [0, g] × R∞) �→ Emb(Fg,b, R∞), we see that the decoupling map of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 induces a map between the spaces using the current model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof of the above theorem will consist of showing that the decoupling induces a monoidal map between MC(Z) and the monoids Tillmann and Segal, and to show that this induces a homotopy equivalence on group completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The maps τ and ε defined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1 are compatible with these monoidal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For the map τ, this result follows directly from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For ε, some care has to be taken into making the configuration spaces into actual topological monoids (see [Seg73]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Namely, instead of C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' X), we use the homotopy equivalent space C′(R∞, X) = {(c, t) ∈ C(R∞, X) × R : t ≥ 0, c ⊂ (0, t) × R∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The monoidal structure is given by juxtaposition, ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (c, t), (c′, t′) �→ (c ∪ Tt(c′), t + t′) where Tt(−) is the map that translates a configuration by t on the first direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map τ of the decoupling theorem is then equivalent to the monoidal map MC(Z) → C′(R∞, X) taking an element of [e, c] ∈ MC(Z)g to (τ([e, c]), g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 13 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is enough to show that the map of monoids given by the decou- pling induces a homotopy equivalence on group completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As the group completions are loop spaces, they are in particular simple and, by the Whitehead theorem for simple spaces, it suffices to show that it induces a homology equivalence on the group comple- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Both monoids are homotopy commutative, hence the group completion theorem [MS76] can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Therefore it is enough to prove that the induced map on the limit spaces (defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1) τ × ε : C(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(F∞) → BDiff∂(F∞) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' EGL+ 2 (R)+ ∧ GL+ 2 (R) Z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6) is a homology equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This holds by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Decoupling Configuration Spaces with Partially Summable Labels In this section we prove the main result of this paper, which is a decoupling result for configuration spaces with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this case, the labelling space is equipped with a partial multiplication and the particles are allowed to collide whenever their labels can be multiplied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) of configurations in M with par- tially summable labels in P has been defined in [Sal01] and its definition requires more sophisticated tools such as the Fulton-MacPherson configuration spaces and operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1 we recall these definitions and the concept of a partial d-monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To prove the decoupling theorem for CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P), we develop a semi-simplicial resolu- tion for this space in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, denoted |DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='13 we show that indeed this space is weakly equivalent to CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In the decoupling context, we naturally encounter another space of discs with configurations which is constructed in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' With these semi-simplicial spaces, we prove Theorem F, combining Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We then use this to deduce Theorem E and Corollary G, which are, respectively Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='19 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Fulton-MacPherson configuration space and partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this section we recall the concept of a configuration space with summable labels in a partial monoid as described in [Sal01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A partial d-monoid P is, in essence, a space with a continuous operation that is not defined on all collections of elements, but only on a subset of composable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The data defining such a partial monoid consists of the subset of composable elements, together with an operation on this subset which is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We are interested in partial monoids that moreover have the structure of an Ed-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Essential to this construction is Fd, the Fulton-MacPherson operad in dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is a cofibrant replacement for the little d-discs operad and therefore will be used to make precise the notion of a Ed-partial monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Crucially for us, the Fulton-MacPherson operad is defined in terms of configuration spaces of points, in which the particles in the configuration are allowed to collide, but keeping track of the relative position of the particles before they collided and the order in which collided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We follow the definition of the Fulton-MacPherson configuration space Ck(M) of a manifold M as described in [Sin04].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To make it precise, we consider M as embedded in some Rm and we denote by k the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We record the directions of particles in a collision, by defining for (i, j) ∈ �C2(k), a map πi,j : �Ck(Rm) → Sm−1 sending a configuration (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , xk) to the unit vector in the direction xi − xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The order of collision is recorded by defining for (i, j, ℓ) ∈ �C3(k) the map si,j,ℓ : �Ck(Rm) → [0, ∞] sending (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , xk) to |xi − xj|/|xi − xℓ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 14 LUCIANA BASUALDO BONATTO Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1 ([Sin04], Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The Fulton-MacPherson configuration space Ck(M) of the manifold M is the closure of the image of the map i × (πi,j| � Ck(M)) × (si,j,k| � Ck(M)) : �Ck(M) −→ M k × (Sm−1)k(k−1) × [0, ∞]k(k−1)(k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space Ck(M) is homotopy equivalent to the ordered configuration space �Ck(M) [Sin04, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5] and whenever M is compact, Ck(M) is a compactification of �Ck(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, this construction is functorial with respect to embeddings [Sin04, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' any embedding f : M �→ N induces an embedding f∗ : Ck(M) → Ck(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The following result gives a convenient way to represent elements of Ck(M), which will be used throughout the chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 ([Sin04], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Each element in Ck(M) is uniquely deter- mined by: (1) A configuration of points P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , Pl in the interior of M, with 1 ≤ l ≤ n (we refer to these as the infinitesimal configuration), (2) For each 1 ≤ i ≤ l, a tree Ti with fi leaves (twigs), no bivalent vertices, so that �l i=1 fi = n, and for each vertex in Ti of valence m an element in Cm(TPiM)/G(d), where G(d) is the group of affine transformations of Rd gen- erated by translations and positive dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (3) A global ordering of the k leaves of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This interpretation of the elements of Ck(M) also provides a way of describing the map f∗ : Ck(M) → Ck(N) induced by an embedding f : M �→ N into a d′-manifold N: it takes a point with infinitesimal configurations P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , Pl ∈ M to one with infinitesimal configurations f(P1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , f(Pl) ∈ N, it preserves the trees and ordering of the leaves, but changes the labels of the vertices of the trees by taking a label ξ ∈ Cm(TPiM)/G(d) to the label DPif(ξ) ∈ Cm(Tf(Pi)N)/G(d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The Fulton-MacPherson operad Fd is built out of subspaces of these configurations in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Intuitively, for each k ≥ 0 the space Fd(k) is the subspace of the ordered configurations of k points in Rd, in which the points have collided at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The Fulton-MacPherson operad in dimension d, denoted Fd, is defined by taking Fd(k) to be the pullback Fd(k) Ck(Rd) 0 (Rd)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' ⌟ In other words, it is the subspace of Ck(Rd) with infinitesimal configuration given by a single point at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' With the description of this space as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, the composition of this operad is given by grafting of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Pictorially, we will often represent elements of Fd(k) as trees of configurations such as in the rightmost picture of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As shown in [Sal01], there exists a model structure on the category of topological op- erads in which Fd is a cofibrant replacement of the little d-discs operad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' An algebra over Fd, called by Salvatore a d-monoid, consists of a space A together with Σk-equivariant maps Fd(k) × Ak → A which commute with the structure maps of Fd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 15 10 9 8 5 1 7 6 4 3 2 11 12 10 9 8 5 1 7 6 4 3 2 11 12 8 5 1 3 9 7 2 12 10 6 4 11 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Representation of an element in the space F2(12) in terms of a tree of infinitesimal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4 ([Sal01], Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A partial d-monoid is a space P to- gether with monomorphisms of Σk-spaces i : Compk �→ Fd(k) ×Σk P k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' and composition maps ρ : Compk → P such that (1) The unit η × P : P → Fd(1) × P factors uniquely through �η : P → Comp1 and the composition ρ ◦ �η is the identity idP ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (2) For f ∈ Fd(k) and [fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' pj] ⊂ Compmj, for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , k, the element [f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' ρ(f1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' p1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ρ(fk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' pk)] ∈ Fd(k) ×Σk P k belongs to Compk if and only if [µ(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , fk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , pk] belongs to Compm1+···+mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, if that is the case, then their image by ρ coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A pointed space (P, 0) is a partial d-monoid with unit if in addition it satisfies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For every k there is an inclusion u : Fd(k) ×Σk � k P �→ Compk such that the composition with i◦u is the subspace inclusion, and ρ◦i : Fd(k)×Σk � k P → P is the projection onto the P coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The definition above is better understood when in comparison to Ed-algebras, which Salvatore calls d-monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Given a d-monoid M and a infinitesimal configuration f ∈ Fd(k), we can always compose any k elements of M via the composition rule described by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In a partial d-monoid P, that is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We instead are given a subset of the k- tuples of P which are composable via the operation described by a given f ∈ Fd(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space Compk should be then thought of as the pairs of possible composition rules in Fd together with the tuples of elements of P which can be composed with this composition rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map ρ is computes the results of compositions, whenever they are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (a) Any space X admits the structure of a trivial partial d-monoid, by defining Comp1 = {[1, x] : x ∈ X} and Compk = ∅, for all k ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this case, the only composition rule that can be performed is the identity and no other collection of points in X is composable in any way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (b) When X is a space equipped with a basepoint ∗, we can define a unital partial d-monoid by setting Compk = Fd(k) × ∨kX and ρ(f, x) = xi, where xi is the unique non-basepoint coordinate, or ∗ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this case, the basepoint acts as a unit and compositions are only defined when done with the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 16 LUCIANA BASUALDO BONATTO (c) Every d-monoid is trivially a partial d-monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In particular, every Ωd-space is a partial d-monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (d) The canonical inclusion id,n : Rd �→ Rd+n also allows us to construct a partial (d + n)-monoid P from a partial d-monoid by adding n trivial composition directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We call this the naive upgrade of a partial d-monoid P and denote it Td,nP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The underlying space of Td,nP is P, and we define CompTd,nP k to be the image of CompP k �→ Fd(k) ×Σk P k (in,k)∗ �−−−−→ Fd+n(k) ×Σk P k = Fd+n(k) ×Σk (Td,nP)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In [Sal01], it was always assumed that the partial d-monoids were good, in the sense described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A partial d-monoid P is good if for every k the inclusion CompP k �→ Fd(k) ×Σk P k is a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' From now on, we always assume partial d-monoids to be good and to have a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For future applications, we are further interested in partial d-monoids with compatible actions of GLd(R), so we introduce this concept here and perform our constructions in this more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Recall from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 that an element of Fd(k) is described by a tree with k ordered leaves and a decoration of the vertices by elements of �C|v|(Rd)/G(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The action of GLd(R) on Rd induces an action on �C|v|(Rd)/G(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This gives an action of this group on Fd(k) for every k, and it is simple to check that the operad maps µ are all GLd(R) equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7 ([Sal01], Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The framed Fulton-MacPherson operad de- noted fFd, is the operad defined by fFd(k) = Fd(k) × GLd(R)k, with structure map �µ((x, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , gk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='(x1, g1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , gm1 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , (xk, g1 k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , gmk k )) = (µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' g1x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , gkxk), g1g1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , gkgmk k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The construction above is an instance of the construction A ⋊ G, the semi-direct product of an operad A and group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This construction, and the proof that the above indeed defines an operad can be found in [SW03, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8 ([Sal01], Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' A framed partial d-monoid with unit is a pointed GLd(R)-space P together with monomorphisms of Σk ≀ GLd(R)-spaces i : fCompk �→ fFd(k) ×Σk P k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' and GLd(R)-equivariant composition maps ρ : fCompk → P satisfying properties 1-3 of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The GLd(R)-bundle of frames on M induces a (GLd(R))k-bundle fCk(M) on Ck(M), acted on by Σk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This is called the framed configuration space of k points in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Using the description of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, an element of the space fCk(M) can be uniquely determined by an infinitesimal configuration in M with labelled trees, together with additional k frames of the tangent planes associated to the k leaves of the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the space of framed configurations fC(M) = � k fCk(M) is a right fFd-module, with GLd(R)-equivariant multiplication maps m : fCk(M) ×Σk (fFd(n1) × · · · × fFd(nk)) → fCn1+···+nk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' defined by grafting the element of fFd(ni) on the i-th leaf of the element of fCk(M), for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , k and using the frame on the leaves to identify Cm(Rd)/G(d) with a configuration on the tangent space of M ([Sal01, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to verify that any co-dimension zero embedding e : M �→ N induces a right fFd-homomorphism e∗ : fC(M) �→ fC(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 17 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='9 ([Sal01], Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let P be a framed partial d-monoid, and let M be a manifold of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the space of configurations in M with partially summable labels in P, denoted CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is defined as the co-equalizer of the following � k � fCk(M) ×Σk � � π∈Map(n,k) k� i=1 fCompP π−1(i) �� � k fCk(M) ×Σk P k (m×id)◦(id ×i) id ×ρk An element of CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is then an equivalence class of elements in � k fCk(M) ×Σk P k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' From the description of fCk(M) above, we can see that an element in CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) can be represented by an infinitesimal configuration w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , wl of ℓ < k points in M together with trees Ti, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ℓ, where the vertices of Ti are labelled by elements in xi j ∈ fFd, and the leaves of the trees are labelled by elements of pi k ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The equivalence relation defining CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) implies that if some leaves labelled by p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , pk are departing from a vertex labelled by x ∈ fFd(k) and the composition ρ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , pk) is defined, then we identify this configuration with the one in which such leaves are removed and their vertex is replaces by a leaf labelled by ρ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For a framed partial d-monoid P and a co-dimension zero embedding f : M �→ N, we get an induced map f∗ : � k fCk(M) ×Σk P k → � k fCk(N) ×Σk P k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' With the description above, a point in the domain with infinitesimal configurations w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , wℓ in M, with trees Ti, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ℓ, where the vertices of Ti are labelled by elements xi j ∈ fFd, and the leaves of the trees are labelled by elements of pi k ∈ P, is taken to the point with infinitesimal configuration φ(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , φ(xℓ), trees Ti, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ℓ, and corresponding labels dxiφ · xi j and dxiφ · pi k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Here we are using the standard actions of GLd(R) on fFd and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By the equivariance condition in the definition of a framed partial monoid, the map preserves the equivalence classes described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Therefore any such co-dimension zero embedding induces a map f∗ : CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1) Seeing the group Diff∂(M) as a subspace of Emb(M, M), the above construction shows that CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) admits an action of Diff∂(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We will be interested in a decou- pling theorem for the space CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Disc models for configuration spaces with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let M be a smooth compact d-manifold, possibly with boundary, and P a framed partial d- monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The group Σk ≀GLd(R) has a canonical inclusion into Diff∂( � k Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We consider Emb( � k Rd, M) to be a Σk ≀ GLd(R)-space with action given by pre-composition by the inverse, and CΣ( � k Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) to be a Σk ≀ GLd(R)-space with action induced by the action of Diff∂( � k Rd), as described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='10 ([MT14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Z be a pointed GLd(R)-space, The space of tubular configurations in M with labels in Z, denoted D(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z), is the quotient � �� k≥0 Emb( � k Rd, M) × Σk≀GLd(R) Zk � � � ∼ where (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) ∼ (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ek−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk−1) whenever zk is the basepoint of Z, for ei : Rd �→ M and zi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 18 LUCIANA BASUALDO BONATTO The space D(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an embedding e : � k Rd �→ M, and z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) ∈ Zk, we define φ · [e, z] = [φ ◦ e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' De(01)φ · z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , De(0k)φ · zk] where 0i denotes the origin of the i-th component of � k Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='11 ([MT14], Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The inclusion of the origin i : ∗ �→ Rd induces a Diff∂(M)-equivariant weak equivalence i∗ : D(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) ≃ −→ C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' One can think of D(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) as disc models for configuration spaces with labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To construct a disc model for summable labels, we need much more structure: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space of surrounded configurations in M with summable labels in P, denoted DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P), is the quotient � �� k≥0 Emb( � k Rd, M) × Σk≀GLd(R) CΣ( � k Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) � � � ∼ where (e : � mRd → M, ξ) ∼ (e′ : � n Rd → M, ξ′) if there are injections k �→ m and k �→ n such that the induced inclusions i1 : � k Rd → � mRd and i2 : � k Rd → � n Rd satisfy ξ ⊂ Im i1, ξ′ ⊂ Im i2 e ◦ i1 = e′ ◦ i2, and e∗(ξ) = e′ ∗(ξ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Here e∗ denotes the map of configurations with summable labels induced by a codimen- sion zero embedding, as described in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Since the Fulton-MacPherson configuration spaces are functorial for co-dimension zero embeddings, we get a map p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2) taking a class (e, ξ) to the configuration e∗(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By definition, this map does not depend on the choice of representative (e, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) admits a partial ordering by declaring (e, ξ) < (e′, ξ′) if, for some representative of the classes, e(ξ) = e′(ξ′) and Im e ⊃ Im e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote by DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• the semi-simplicial nerve of the poset DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By the definition of the partial order, the map p induces an augmentation DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In particular, DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• is an augmented topological flag complex (Defini- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2), since the space of n-simplices is indeed an open subspace of the (n + 1)-tuples of vertices with the same image under the augmentation map, and the condition for a tuple to form an n-simplex is just given by checking the pairwise order relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It might be helpful to keep in mind the following visualisation for elements in the space |DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|: any point can be expressed by a tuple ((e0, ξ0) < · · · < (ek, ξk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' t0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , tk) t0 · · · + tk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The poset construction implies that for such a tuple e0(ξ0) = · · · = ek(ξk) and Im e0 ⊃ · · ⊃ Im ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We can visualise it as a collection of descending embedded discs around the configuration ei(ξi) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Each collection of embedded discs has weights adding up to 1 and when the weight associated to a collection of embeddings goes to zero, that collection disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The equivalence relation guarantees that we can always get a representative for the collection of discs such that each component has at least one particle of the configuration inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) induces a weak homotopy equivalence |DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 19 t0 t2 t1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Element in a 2-simplex of |DΣ(F3)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof of the above will be a direct application of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3 [GRW14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As discussed before, the augmented semi-simplicial space p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is an augmented topological flag complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We need to verify that the hy- potheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3 are satisfied: (i) To see that p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) has local sections, take f : Dn → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) and a point (e, ξ) ∈ p−1(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the image of e is an open subset of M containing f(x), and therefore the subspace V of configurations contained in Im e is an open subset of CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) containing f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let U = f −1(V ), which is an open neighbourhood of x in Dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the map F : U → DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) taking y to (e, e−1(f(y))) satisfies p ◦ F = f|U and F(x) = (e, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (ii) p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is surjective, since any configuration ξ admits a tubular neighbourhood e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then (e, e−1(ξ)) is an element of DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) in the pre-image of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (iii) For a configuration ξ ∈ CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) and a non-empty finite subset {(e1, ξ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , (ek, ξk)} in its pre-image, we can always find e an embedding of Rd’s containing ei(ξi) and contained in all ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For instance by taking e to small open discs around the points in the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3, the map |DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is a weak homotopy equiva- lence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ The space DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is equipped with an action of Diff∂(M): for ψ ∈ Diff∂(M), an embedding e : � k Rd �→ M, and ξ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , pk) a configuration of points in � k Rd with labels in P, we define φ · [e, ξ] = [φ ◦ e, ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to verify this action is well-defined and preserves the partial order in DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It follows directly from the definition that the augmentation map p : DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is Diff∂(M)- equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Since the partial order in DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) is compatible with the Diff∂(M)-action, it in- duces a fibrewise partial order on DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) ×Diff∂(M) Emb(M, R∞) over BDiff∂(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then the semi-simplicial nerve of the poset DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) ×Diff∂(M) Emb(M, R∞) is simply is the fibrewise semi-simplicial nerve DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• ×Diff∂(M) Emb(M, R∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) → CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) induces a weak homotopy equiva- lence |DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• × Diff∂(M) Emb(M, R∞)| −→ CΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The above follows directly from the definition of the definition of the partial order on DΣ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) ×Diff∂(M) Emb(M, R∞) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14 allows us to use a disc model for the space of configurations with partially summable labels when proving the decoupling in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We finish this section by introducing another space of discs and configurations which will be used in the decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Z be a pointed GLd(R)-space and let M be a smooth compact manifold of dimension n > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space of d-tubular configurations in M with labels in 20 LUCIANA BASUALDO BONATTO Z, denoted Dd(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z), is the quotient � �� k≥0 Emb( � k Rd, M) × Σk≀GL+ d (R) Zk � � � ∼ where (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk) ∼ (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ek−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , zk−1) whenever zk is the basepoint of Z, for ei : Rd �→ M and zi ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The difference between the spaces Dd(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) and D(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='10) is that on the former we look at embedded discs of a lower dimension than the ambient manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let n > d, and denote by Ed,n denote the total space of the canonical GL+ d (R)-bundle over the oriented Grassmanian Gr+(d, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The inclusion of the origin i : ∗ �→ Rd induces a weak equivalence i∗ : Dd(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z) ≃ −→ C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (Ed,n)+ ∧ GL+ d (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This result should be seen as an analogue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='11 in the setting of d-tubular configurations in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Recall that Emb( � k Rd, Rn) ≃ �Ck(Rn) × (Ed,n)k, where the map to �Ck(Rn) is induced by the inclusion of the origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then we get weak equivalences Emb( � k Rd, M) × (Σk≀GL+ d (R)) Zk ≃ −→ ( �Ck(Rn) × (Ed,n)k) × (Σk≀GL+ d (R)) Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' These respect the equivalence relations and therefore induce a map i∗ : Dd Σ(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) ≃ −→ C(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (Ed,n)+ ∧ GL+ d (R) Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof then follows from the same arguments of [MT14, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ We also need an analogue of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='12 for the case of d-tubular configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let M be a smooth compact manifold of dimension n > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space of d-surrounded configurations in M with summable labels in P, denoted Dd Σ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P), is the quotient � �� k≥0 Emb( � k Rd, M) × Σk≀GL+ d (R) CΣ( � k Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) � � � ∼ where (e : � mRd → M, ξ) ∼ (e′ : � n Rd → M, ξ′) if there are injections k �→ m and k �→ n such that the induced inclusions i1 : � k Rd → � mRd and i2 : � k Rd → � n Rd satisfy ξ ⊂ Im i1, ξ′ ⊂ Im i2 and e ◦ i1 = e′ ◦ i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We equip this space with a partial ordering by declaring (e, ξ) < (e′, ξ′) if, for some representative of the classes, e(ξ) = e′(ξ′) and Im e ⊃ Im e′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We denote the semi- simplicial nerve of this poset by Dd Σ(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The decoupling theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this section we prove the main theorem of this paper, which is a decoupling result for CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As in Section 3, we take as a model for EDiff∂(Fg,b) the space Emb(Fg,b, R∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For every g and b ≥ 0 we have τΣ × εΣ : DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) × Emb(Fg,b, R∞) −→ Emb(Fg,b, R∞) × D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) ((e : � k R2 �→ Fg,b, ξ), f : Fg,b �→ R∞) �−→ (f, (f ◦ e, ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 21 This map is Diff∂(Fg,b)-equivariant with respect to the diagonal action on the domain and the action on Emb(Fg,b, R∞) on the target, and it preserves the poset structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Hence it induces a map τΣ × εΣ fitting into the following diagram |DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• × Diff∂(Fg,b) Emb(Fg,b, R∞)| |BDiff∂(Fg,b) × D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) × Diff∂(Fg,b) Emb(Fg,b, R∞) BDiff∂(Fg,b) × |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| ≃ τΣ×εΣ ≃ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map τΣ × εΣ : |DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• × Diff∂(Fg,b) Emb(Fg,b, R∞)| → |BDiff∂(Fg,b) × D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14 imply Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof of the above, will consist on showing that τΣ × εΣ is a level-wise homology isomorphism of semi-simplicial spaces in degrees ≤ 2 3g and then show that this implies that the same holds on the geometric realisations, as done in [GRW17, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' To do this, we will use the spectral sequence recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='11, and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, the decoupling result for the space of non-colliding configurations with labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Throughout the proof, it will be helpful to keep in mind Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Correspondence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3) between an element of DΣ(F3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)2 (top) and D(F3, Z2) (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The dotted regions represent the em- bedded R2’s and the arrows indicate their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We start by showing that DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• × Diff∂(Fg,b) Emb(Fg,b, R∞) → BDiff∂(Fg,b) × D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• induces level-wise homology isomorphisms in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let Zp be the subspace of DΣ(R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p consisting of those tuples e = ((e0, ξ0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , (ep, ξp)) with e0 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This space is pointed by the unique class on the empty configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We % %22 LUCIANA BASUALDO BONATTO show there is a Diff∂(Fg,b)-equivariant homeomorphism DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p ∼= D(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3) where D(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) is the space of tubular configurations with labels in Zp (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Let [(e0, ξ0) < · · · < (ep, ξp)] denote the an element in DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then, by definition e0(ξ0) = · · · = ep(ξp) and Im e0 ⊃ · · · ⊃ Im ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Denote by ij : R2 �→ � k R2 the map induced by the inclusion {j} �→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , k}, for 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The map DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p → DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) takes a sequence ((e0, ξ0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , (ep, ξp)), to the class represented by the embedding e0 and and the label associated to the jth component R2 ⊂ � k R2 given by ((id, ξ0), ((e0 ◦ ij)−1 ◦ e1, ξ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' , ((e0 ◦ ij)−1 ◦ ep, ξp)) ∈ Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For intuition behind this homeomorphism, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It is simple to explicitly con- struct an inverse for this map and check this is a Diff∂(Fg,b)-equivariant homeomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='11, the inclusion of the origins defines Diff∂(Fg,b)-equivariant map D(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) → C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) which is a weak-equivalence of Diff∂(Fg,b)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Together with the homeomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='3) this implies that DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p × Diff∂(Fg,b) Emb(Fg,b, R∞) ≃ −→ C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) × Diff∂(Fg,b) Emb(Fg,b, R∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4) Similarly, we now show that D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p ≃ −→ C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (ESO(2))+ ∧ SO(2) Zp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5) By the same argument as above, we get a homeomorphism D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p ∼= D2(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) where D2(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) is the space of d-tubular configurations with labels in Zp as in Def- inition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='16, the inclusion of the origins induces a weak homotopy equivalence D2(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) ≃ C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (ESO(2))+ ∧SO(2) Zp) which implies the homotopy equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Then we have a commutative diagram DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p × Diff∂(Fg,b) Emb(Fg,b, R∞) C(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Zp) × Diff∂(Fg,b) Emb(Fg,b, R∞) BDiff∂(Fg,b) × D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)p BDiff∂(Fg,b) × C(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' ESO(2)+ ∧ SO(2) Zp) ≃ (τΣ×εΣ)p τ×ε ≃ where the top and bottom maps are weak equivalences by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The right-hand map induces homology isomorphisms in degrees ≤ 2 3g by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2 with Z = Zp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Therefore so does the map (τΣ × εΣ)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This implies that the map between the spectral sequences associated to the semi- simplicial spaces X• = DΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•×Diff∂(Fg,b)Emb(Fg,b, R∞), and Y• = BDiff∂(Fg,b)× D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• E1 p,q = Hq(Xp) Hp+q(|X•|) E′1 p,q = Hq(Yp) Hp+q(|Y•|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' τΣ×εΣ induces an isomorphism on the E1-pages for all q ≤ 2 3g, and therefore the right-hand map is also an isomorphism in such degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ DECOUPLING GENERALISED CONFIGURATION SPACES ON SURFACES 23 As in the case for labelled configuration spaces, the Decoupling Theorem for Sum- mable Labels, allows us to deduce homological stability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For b ≥ 1, the map induced by gluing F1,1 along the boundary CΣ(Fg,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg,b) → CΣ(Fg+1,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Z)//Diff∂(Fg+1,b) induces a homology isomorphism in degrees ≤ 2 3g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof follows from the same arguments as in the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Monoids of configurations on surfaces with partially summable labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' In this section, we show that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18 implies a splitting for the group completion of the monoid of configuration on surfaces with partially summable labels, analogous to Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2, gluing two surfaces Fg,1 and Fh,1 along part of their boundary defines an associative multiplication BDiff∂(Fg,1) × BDiff∂(Fh,1) → BDiff∂(Fg+h,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For (P, 0) a framed partial 2-monoid with unit, the operation on diffeomorphism groups and spaces EDiff∂(Fg,1) described above, together with the fixed identifications Fg+h,1 = Fg,1#∂Fh,1, induce an associative multiplication also on the Borel construc- tions µ : CΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1)×CΣ(Fh,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fh,1) → CΣ(Fg+h,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg+h,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Analogous to the construction of Chapter 3, we denote the associated Borel construction by MCΣ(P)g = CΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' This multiplication makes MCΣ(P) = � g≥0MCΣ(P)g into a topological monoid, which we refer to as the monoid of configurations with summable labels in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' On the other hand, the poset D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) can be made into a partially ordered topo- logical monoid, using the same strategy Segal used to define a topological monoid equiv- alent to C(R∞, X), as we recalled in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' For any path-connected framed partial 2-monoid with unit P, there is a homotopy equivalence ΩB(MCΣ(P)) ≃ ΩB �� g BDiff∂(Fg,1) � × ΩB |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The proof of the above result consists of constructing a zig-zag of monoids (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='6) � g≥0 |DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(Fg,1)| � g≥0 BDiff∂(Fg,1) × |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| MCΣ(P) ≃ τΣ×εΣ The vertical arrow is induced by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14 while the horizontal arrow is induced by the decoupling map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' We describe the monoidal structures on the top spaces using a general construction for posets recalled in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The space � g≥0 |DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(Fg,1)| is homotopy equiva- lent to the geometric realisation of the semi-simplicial nerve of the poset � g≥0 DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1) 24 LUCIANA BASUALDO BONATTO with partial order defined component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As before, gluing surfaces along part of their boundary makes � g≥0DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(Fg,1) into a partially ordered topological monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Moreover, one can verify that the augmentations DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)• → CΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P) induce a map of monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='1, this gives a map of monoids � g≥0 |DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(Fg,1)| → MCΣ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7) On the other hand, the decoupling map gives a map of topological monoids � g≥0 |DΣ(Fg,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(Fg,1)| → � g≥0 BDiff∂(Fg,1) × |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8) just as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' All that remains is to verify that the maps (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='8) induce homotopy equiva- lences on group completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' As the group completions are loop spaces, they are simple and, by Whitehead’s theorem, it suffices to show the maps induce homology equivalences on the group completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' All monoids are homotopy commutative, hence the group completion theorem [MS76] can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' It implies that it is enough to show that the induced maps on limit spaces |DΣ(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(F∞)| → CΣ(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)//Diff∂(F∞) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='9) |DΣ(F∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•//Diff∂(F∞)| → BDiff∂(F∞) × |D2 Σ(R∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' P)•| (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='10) are homology equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' The first map is a weak equivalence by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='14, while the second map is a homology equivalence by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='18 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' □ References [AF15] David Ayala and John Francis, Factorization homology of topological manifolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNAyT4oBgHgl3EQfSvfA/content/2301.00093v1.pdf'} +page_content=' 8 (2015), no.' 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b/JdAyT4oBgHgl3EQfsPlo/content/tmp_files/2301.00573v1.pdf.txt @@ -0,0 +1,929 @@ +Lagrangian Relaxation for Mixed-Integer Linear +Programming: Importance, Challenges, Recent +Advancements, and Opportunities +Mikhail A. Bragin +Department of Electrical and Computer Engineering, +University of Connecticut, Storrs, USA +Abstract +Operations in areas of importance to society are frequently modeled as Mixed- +Integer Linear Programming (MILP) problems. +While MILP problems suffer from +combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve +the associated difficulties through decomposition. Due to the non-smooth nature of +Lagrangian dual functions, the coordination aspect of the method has posed serious +challenges. This paper presents several significant historical milestones (beginning with +Polyak’s pioneering work in 1967) toward improving Lagrangian Relaxation coordi- +nation through improved optimization of non-smooth functionals. Finally, this paper +presents the most recent developments in Lagrangian Relaxation for fast resolution of +MILP problems. The paper also briefly discusses the opportunities that Lagrangian +Relaxation can provide at this point in time. +Keywords: Combinatorial Optimization; Decomposition and Coordination; Lagrangian Re- +laxation; Discrete Optimization; Duality; Mixed-Integer Linear Programming +1 +Introduction +The aim of this paper is to review Lagrangian-Relaxation-based methods for separable +Mixed-Integer Linear Programming (MILP) problems, which are formally defined as follows: +min +(x,y):={xi,yi}I +i=1 +� +I +� +i=1 +� +(cx +i )T xi + (cy +i )T yi +� +� +, +(1) +whereby I subsystems are coupled through the following constraints +s.t. +I +� +i=1 +Ax +i xi + +I +� +i=1 +Ay +i yi − b = 0, {xi, yi} ∈ Fi, i = 1, . . . , I. +(2) +1 +arXiv:2301.00573v1 [math.OC] 2 Jan 2023 + +The primal problem (1)-(2) is assumed to be feasible and the feasible region F ≡ �I +i=1 Fi +with Fi ⊂ Znx +i × Rny +i is assumed to be bounded and finite. +Because of integer variables, Lagrangian Relaxation leads to non-smooth optimization +in the dual space. Accordingly, key non-smooth optimization methods will also be reviewed. +1.1 +Importance and Difficulties of MILP Problems +MILP has multiple applications in problems of importance to society: ambulance relocation +(Lee et al., 2022), balanced item placement (Gasse et al., 2022), cost-sharing for ride-sharing +(Hu et al., 2021), drop box location (Schmidt and Albert, 2022), efficient failure detection +in large-scale distributed systems (Er-Rahmadi and Ma, 2022), home healthcare routing +(Dastgoshade et al., 2020), home service routing and appointment scheduling (Tsang and +Shehadeh, 2022), inventory control under demand and lead time uncertainty (Thorsen and +Yao, 2017), job-shop scheduling (Liu et al., 2021), facility location (Basciftci et al., 2021), +flow-shop scheduling (Hong et al., 2019; Balogh et al., 2022; ¨Oztop et al., 2022), freight +transportation (Archetti et al., 2021), location and inventory prepositioning of disaster re- +lief supplies (Shehadeh and Tucker, 2022), machine scheduling with sequence-dependent +setup times (Yalaoui and Nguyen, 2021), maritime inventory routing (Gasse et al., 2022), +multi-agent path finding with conflict-based search (Huang et al., 2021), multi-depot electric +bus scheduling (Gkiotsalitis et al., 2023), multi-echelon/multi-facility green reverse logis- +tics network design (Reddy et al., 2022), optimal physician staffing (Prabhu et al., 2021), +optimal search path with visibility (Morin et al., 2023), oral cholera vaccine distribution +(Smalley et al., 2015), outpatient colonoscopy scheduling (Shehadeh et al., 2020), pharma- +ceutical distribution (Zhu and Ursavas, 2018), plant factory crop scheduling (Huang et al., +2020), post-disaster blood supply (Hamdan and Diabat, 2020; Kamyabniya et al., 2021), real +assembly line balancing with human-robot collaboration (Nourmohammadi et al., 2022), +reducing vulnerability to human trafficking (Kaya et al., 2022), restoration planning and +crew routing (Morshedlou et al., 2021), ridepooling (Gaul et al., 2022), scheduling of un- +conventional oil field development (Soni et al., 2021), security-constrained optimal power +flow (Velloso et al., 2021), semiconductor manufacturing (Chang and Dong, 2017), surgery +scheduling (Kayvanfar et al., 2021), unit commitment (Kim et al., 2018; Chen et al., 2019; +Li and Zhai, 2019; Chen et al., 2020; Li et al., 2020; van Ackooij et al., 2021), vehicle shar- +2 + +ing and task allocation (Arias-Melia et al., 2022), workload apportionment (Gasse et al., +2022), and many others. Because of integer variables x, MILP problems are NP-hard and +instances of practical sizes are generally difficult to solve to optimality due to the combina- +torial complexity: the computational effort increases super-linearly (i.e., exponentially) as +the problem size increases. For a number of practical problems, the computational effort +may be significant to obtain even a feasible solution. Additionally, many important prob- +lems require short solving times (ranging from 20 minutes down to a few seconds), as well +as high-quality solutions. +By the same token, with decreasing problem size, NP-hardness leads to the super-linear +reduction of complexity. +The dual decomposition and coordination Lagrangian Relax- +ation method is promising to exploit this reduction of complexity; the method essentially +“reverses” combinatorial complexity upon decomposition, thereby drastically reducing the +effort required to solve subproblems (each subproblem i corresponds to a subsystem i). +Lagrangian Relaxation is also deeply rooted in economic theory, whereby the solutions +obtained are rested upon the economical principle of “supply and demand.” When the +“demand” exceeds the “supply,” Lagrangian multipliers (which can be viewed as “shadow +prices”) increase (and vice versa) thereby discouraging subsystems from making less “eco- +nomically viable” decisions. Notwithstanding the advantage of the decomposition aspect, +the “price-based” coordination of the method (to appropriately coordinate the subprob- +lems), however, has been the subject of intensive research for many decades because of the +fundamental difficulties of the underlying non-smooth optimization of the associated dual +functions as explained ahead. +The purpose of this paper is to present a brief overview of the key milestones in the +development of the Lagrangian Relaxation method for MILP problems as well as in the +optimization of convex non-smooth functions. The rest of the paper is structured as follows: +1. At the beginning of Section 2, the Lagrangian dual problem is presented and the +difficulties of Lagrangian Relaxation on a pathway to solving MILP problems are +explained. In subsequent subsections, the difficulties are resolved one by one; +2. In subsection 2.1, early research on non-smooth optimization (Polyak, 1967, 1969) is +presented to lay the foundation for further developments; specifically, the Polyak’s +formula (Polyak, 1969) depending on the optimal dual value q(λ∗) to ensure geomet- +3 + +ric/linear convergence rate is presented; +3. In subsection 2.2, the subgradient-level method (Goffin and Kiwiel, 1999) is presented +to ensure convergence without the need to know q(λ∗); +4. In subsection 2.3, the fundamental difficulties associated with subgradient methods +(high computational effort and zigzagging of multipliers) are explained; +5. In subsections 2.4 and 2.5, two separate research thrusts (surrogate (Kaskavelis and +Caramanis, 1998; Zhao et al., 1999) and incremental (Nedic and Bertsekas, 2001)) to +reduce computational effort as well to alleviate zigzagging of multipliers are reviewed; +the former thrust still requires q(λ∗) for convergence; the latter thrust avoids the need +to know q(λ∗) following the “subgradient-level” ideas presented in subsection 2.2; +6. In subsection 2.6, Surrogate Lagrangian Relaxation (SLR) (Bragin et al., 2015) that +proved convergence without q(λ∗) by exploiting “contraction mapping” while inher- +iting convergence properties of the surrogate method of subsection 2.4 is reviewed; +7. In subsection 2.7, further methodological advancements for SLR are presented; to ac- +celerate the reduction of constraint violations while enabling the use of MILP solvers, +“absolute-value” penalties have been introduced (Bragin et al., 2018); to efficiently +coordinate distributed entities while avoiding the synchronization overhead, compu- +tationally distributed version of SLR has been developed to efficiently coordinate +distributed subsystems in an asynchronous way (Bragin et al., 2020); +8. In subsection 2.8, Surrogate “Level-Based” Lagrangian Relaxation (Bragin and Tucker, +2022) is reviewed; this first-of-the-kind method exploits the linear-rate convergence +potential intrinsic to the Polyak’s formula presented in subsection 2.1 but without +the knowledge q(λ∗) and without heuristic adjustments of its estimates presented in +subsection 2.2. Rather, an estimate of q(λ∗) has been innovatively determined purely +through a simple constraint satisfaction problem; the surrogate concept (Zhao et al., +1999; Bragin et al., 2015) ensures low computational requirements as well as the alle- +viated zigzagging; accelerated reduction of constraint violations is achieved through +“absolute-value” penalties (Bragin et al., 2018) enabling the use of MILP solvers; +9. In Section 3, a brief conclusion is provided with future directions delineated. +4 + +2 +Lagrangian Duality for Discrete Programs and Non- +Smooth Optimization +The Lagrangian dual problem that corresponds to the original MILP problem (1)-(2) is the +following non-smooth optimization problem: +max +λ {q(λ) : λ ∈ Ω ⊂ Rm}, +(3) +where the convex dual function is defined as follows: +q(λ) = min +(x,y) +� +L(x, y, λ), {xi, yi} ∈ Fi, i = 1, . . . , I +� +. +(4) +Here L(x, y, λ) ≡ �I +i=1 +� +(cx +i )T xi + (cy +i )T yi +� ++ λT · +� �I +i=1 Ax +i xi + �I +i=1 Ay +i yi − b +� +is the +Lagrangian function obtained by relaxing coupling constraints (2) by using Lagrangian +multipliers λ. The minimization of L(x, y, λ) within (4) with respect to {x, y} is referred +to as the relaxed problem, which is separable into subproblems due to the additivity of +L(x, y, λ). This feature will be exploited starting from subsection 2.4. +Even though the original primal problem (1) is non-convex, q(λ) is always continuous +and concave with the feasible set Ω being always convex. Due to integer variables x in +the primal space, q(λ) is non-smooth with facets (each representing a particular solution +to (4)) intersecting at ridges where derivatives of q(λ) exhibit discontinuities; in particular, +q(λ) is non-differentiable at λ∗. As a result, subgradients (referred to in early literature as +“generalized gradients” or “subdifferentials”) are generally non-ascending: when λ are up- +dated along subgradients, dual values may decrease. Moreover, subgradients may almost be +perpendicular to the directions toward optimal multipliers λ∗ thereby leading to zigzagging +of λ across ridges of the dual function (see Figure 1 for illustrations). +While Lagrangian multipliers λ are generally fixed parameters within (4), λ are “dual” +decision variables with respect to the dual problem (3). Traditionally, (3) is maximized by +iteratively updating λ by making a series of steps sk along subgradients g(xk, yk) as: +λk+1 = λk + sk · g(xk, yk), +(5) +where {xk, yk} is a concise way to denote an optimal solution {x∗(λk), y∗(λk)} to the re- +laxed problem (4) with multipliers equal to λk. Within Lagrangian Relaxation, subgradients +5 + + +A +B +* +Figure 1: An example of a dual function that illustrates the difficulties associated with +subgradient methods. +Solid lines denote the level curves, dash-dotted lines denote the +ridges of the dual function whereby the gradients are not defined (possible subgradient +directions at points A and B are shown by solid arrows), and dashed lines denote the +subgradient direction from point B toward optimal multipliers. This Figure is taken from +(Bragin and Tucker, 2022) with permission. +are defined as levels of constraint violations g(xk, yk) ≡ +��I +i=1 Ax +i xk +i + �I +i=1 Ay +i yk +i − b +� +; +for compactness, g(xk, yk) will be denoted simply as gk as appropriate. +If inequality +constraints �I +i=1 Ax +i xi + �I +i=1 Ay +i yi ≤ b are present, they are generally converted into +equality constraints by introducing non-negative real-valued slack variables z such that +�I +i=1 Ax +i xi + �I +i=1 Ay +i yi + z = b. Multipliers are then updated per (5) with subsequent +projection onto the positive orthant - a set delineated by constraints λ ≥ 0. +Because the dual problem results from the relaxation of coupling constraints, dual values +are generally less than primal values q(λk) < f(xk, yk),1 i.e., there is a duality gap - the +relative difference between q(λk) and f(xk, yk).2 +Because of the discrete nature of the +primal problem (1)-(2), even at optimality, the duality gap is non-zero, that is q(λ∗) < +f(x∗, y∗). Consequently, maximization of the dual function does not lead to an optimal +primal solution (x∗, y∗) or even a feasible solution. To obtain solutions feasible with respect +1In this particular case, f(xk, yk) ≡ �I +i=1 +� +(cx +i )T xk +i + (cy +i )T yk +i +� +such that {xk, yk} satisfy constraints (2). +2Dual values can be used to quality the quality of the solution {xk, yk}. +6 + +to the original problem (1)-(2), solutions to the relaxed problem {xk +i , yk +i } are typically +perturbed heuristically.3 Generally, the closer the multipliers are to their optimal values λ∗, +the smaller the levels of constraint violations (owing to the concavity of the dual function), +and, therefore, the easier the search for feasible solutions. +In short summary, the roadblocks on the way of Lagrangian Relaxation to efficiently +solve MILP problems are the following: +1. Non-differentiability of the dual function; +(a) Subgradient directions are non-ascending; +(b) Necessary and sufficient conditions for extrema are inapplicable; +2. High computational effort is required to compute subgradient directions if the number +of subsystems is large; +3. Zigzagging of multipliers across ridges of the dual function leading to many iterations +required for convergence; this difficulty follows from the non-differentiability of the +dual function, but this difficulty deserves a separate resolution; +4. Solutions {xk +i , yk +i } to the relaxed problem, when put together, do not satisfy con- +straints (2). Moreover, +5. Even at λ∗, levels of constraint violations may be large and the heuristic effort to +“repair” the relaxed problem solution {xk +i , yk +i } may still be significant. +In order to resolve difficulty 1(b), stepsizes need to approach zero (yet, this condition +alone is not sufficient), as will be discussed in the subsection that follows. This requirement +puts a restriction on the methods that will be reviewed. +Scope. By examining the above-mentioned difficulties (which will also be referred to +as D1(a), D1(b), D2, D3, D4, and D5), several stages in the development of Lagrangian +Relaxation and its applications to optimizing non-smooth dual functions and solving MILP +problems will be chronologically reviewed, along with specific features of the methods that +3To solve MILP problems, Lagrangian Relaxation is often regarded as a heuristic. However, in dual +space, the Lagrangian relaxation method is exact; the method is also capable of helping to improve solutions +through the multipliers update, unlike many other heuristic methods. +7 + +address the above difficulties. In view of the above difficulties such as D1(b) and D2, several +research directions, though having their own merit, will be excluded, for example: +1. The Method of Multipliers. +The Alternate Direction Method of Multipliers +(ADMM), which is derived from Augmented Lagrangian Relaxation (ALR) (the +“Method of Multipliers”), introduces quadratic penalties to penalize violations of +relaxed constraints, improving the convergence of Lagrangian Relaxation. The two +methods (ADMM and ALR), however, only converge when solving continuous primal +problems. Without stepsizes approaching zero, neither method converges [in the dual +space] when solving discrete primal problems and does not resolve D1(b). Neverthe- +less, the penalization idea underlying ALR led to the development of other LR-based +methods with improved convergence as described in subsection 2.7. +2. The Bundle Method. The Bundle Method’s idea is to obtain the so-called ε−ascent +direction to update multipliers (Zhao and Luh, 2002). +Considering that the non- +differentiability of dual functions may generally result in non-ascending subgradient +directions, the Bundle method resolves D1(a). Since the relaxed problems need to be +solved several times (Zhao and Luh, 2002), the effort required to obtain multiplier- +updating directions exceeds that required in subgradient methods, thus the method +does not resolve D2. +2.1 +Minimization of “Unsmooth Functionals” +Optimization of non-smooth convex functions, a direction that stems from the seminal work +of Polyak (1967), is a broader subject than the optimization of q(λ) within Lagrangian +Relaxation. To present the underlying principles that support Lagrangian Relaxation to +efficiently solve MILP problems, the work of Polyak (1967) is discussed next. +Subgradient Method with “Non-Summable” Stepsize. While subgradients are gen- +erally non-descending (non-ascending) for minimization (maximization) problems (Polyak, +1967, p.33), convergence to the optimal solution optimizing a non-smooth function (e.g., to +λ∗ maximizing q(λ)) was proven under the following (frequently dubbed as non-summable) +8 + +stepsizing formula satisfying the following conditions: +sk > 0, +lim +k→∞ sk = 0, +∞ +� +k=1 +sk = ∞. +(6) +Subgradient Method with Polyak’s Stepsize. +As Polyak noted in his later work +(Polyak, 1969, p.15), non-summable stepsizes lead to very slow convergence. Intending to +achieve geometric (also referred to as linear4) rate of convergence so that ∥λk − λ∗∥ is +monotonically decreasing, Polyak developed the stepsizing formula, which, for the problem +under consideration, is presented in the following way: +0 < sk < γ · q(λ∗) − q(λk) +��g(xk, yk) +��2 , γ < 2. +(7) +In the simplest form, a rendition of the Polyak’s result can be presented as follows. First, +consider a binomial expansion of ∥λ∗ − λk+1∥2 as +∥λ∗ − λk+1∥2 = ∥λ∗ − λk∥2 − 2 · sk · (gk)T · (λ∗ − λk) + (sk)2 · ∥gk∥2. +(8) +Owing to the concavity of the dual function, +q(λ∗) − q(λk) ≤ (gk) · (λ∗ − λk). +(9) +Therefore, (8) becomes: +∥λ∗ − λk+1∥2 ≤ ∥λ∗ − λk∥2 − 2 · sk · (q(λ∗) − q(λk)) + (sk)2 · ∥gk∥2. +(10) +From (7), it follows that +sk · ∥gk∥2 < 2 · (q(λ∗) − q(λk)). +(11) +Therefore, (10) simplifies to +∥λ∗ − λk+1∥2 < ∥λ∗ − λk∥2. +(12) +Within (7)-(12) and thereafter in the paper, the standard Euclidean norm will be used +(unless specified otherwise). +4Superlinear convergence is also possible, however, 1. A reformulation of the dual problem (Charisopoulos +and Davis, 2022) is required; 2. Within the Lagrangian Relaxation framework, a dual function is generally +unavailable as argued in subsection 2.3. +9 + +The Polyak’s stepsizing (7) can be regarded as a creative workaround of D1(a) in the +sense that a more computationally difficult problem of obtaining ascending directions at +every iteration (as in the Bundle method) to ensure convergence is replaced with a provably +easier problem of reducing ∥λ∗ − λk∥ at every step to guarantee convergence. +Through the decades following 1969, two distinct research directions, along the lines +of “non-summable” (6) and “Polyak” (7) stepsizes continued with various research groups +adhering to either one or the other. +Both directions guarantee convergence to λ∗ that +maximizes the dual function q(λ) (thereby resolving D1(b)), although, up to this stage in +the discussion, convergence by using Polyak’s stepsizing is purely theoretical, since optimal +dual value q(λ∗) required within (7) is unknown. +The Subgradient-Level Method was +developed to achieve convergence in practice. +2.2 +The Subgradient-Level Method +The Subgradient-Level Method (Goffin and Kiwiel, 1999) overcame difficulties associated +with the unavailability of optimal [dual] value, which is needed to compute Polyak’s step- +size (7) through adaptively adjusting a level estimate based on the detection of sufficient +descent and oscillations of the [dual] solutions. +In terms of the problem (3), the procedure of the method is explained as follows: the +“level” estimate qk +lev = qkj +rec + δj is used in place of the optimal dual value q(λ∗), where qk +rec +is the best (highest) dual value (“record objective value”) obtained up to an iteration k, +and δj is an adjustable parameter with j denoting the jth update of qk +lev. The main premise +behind this is when δj is “too large,” then multipliers will exhibit oscillations while traveling +significant (predefined) distance R without improving the “record” value. In this case, the +parameter δj is updated as δj+1 = β · δj with β = 1 +2. On the other hand, if δj is such that +the dual value is sufficiently increased: q(λk) ≥ qk +lev +τ ·δj, with τ = 1 +2, then the parameter +δj is unchanged and the distance traveled by multipliers is reset to 0 to avoid premature +reduction of δj by β in future iterations. +Followed by an examination of resolutions of D2 and D3, further discussions of the +implementation of Polyak’s stepsizing to resolve D1 will be deferred to future subsections. +10 + +2.3 +Fundamental Difficulties of Sub-gradient Methods +High Computational Effort (D2). In the methods reviewed thus far, non-smooth func- +tions are assumed to be given. However, a dual function q(λ) cannot be obtained computa- +tionally efficiently. In fact, even for a given value of multipliers λk, minimization within (4) +to obtain a dual value q(λk) and the corresponding subgradient is time-consuming. Even +then, only one possible value of the subgradient can generally be obtained; a complete +description of the subgradient is generally non-attainable (Goffin, 1977). +Zigzagging of Multipliers (D3). As hypothesized by (Goffin, 1977), the slow conver- +gence of subgradient methods is due to ill-conditioning. The condition number µ is formally +defined as (Goffin, 1977): µ = inf{µ(λ) : λ ∈ Rm/P}, where P = {λ ∈ Rm : q(λ) = q(λ∗)} +(the set of optimal solutions) and µ(λ) = minu +uT ·(λ∗−λ) +∥uT ∥·∥λ∗−λ∥ (the cosine of the angle that +subgradient form with directions toward the optimal multipliers).5 It was then confirmed +experimentally when solving, for example, scheduling (Czerwinski and Luh, 1994, Fig. 3(b), +p. 104) as well as power systems problems (Guan et al., 1995, Fig. 4, p. 774) that the +ill-conditioning leads to the zigzagging of multipliers across the ridges of the dual function. +To address these two difficulties, the notions of “surrogate,” “interleaved” and “incre- +mental” subgradients, which do not require relaxed problems to be fully optimized to speed +up convergence, emerged in the late 1990s, and early 2000s as reviewed next. +2.4 +The Interleaved Subgradient and the Surrogate Subgradient +Methods +Within the Interleaved Subgradient method proposed by (Kaskavelis and Caramanis, 1998), +multipliers are updated after solving one subproblem at a time +min +(xi,yi) +� +(cx +i )T xi+(cy +i )T yi+λT · +� +Ax +i xi+Ay +i yi +� +, {xi, yi}∈Fi +� +, +(13) +rather than solving all the subproblems as in subgradient methods. +This significantly +reduces computational effort, especially for problems with a large number of subsystems. +The more general Surrogate Sub-gradient method with proven convergence was then +developed by Zhao et al. (1999) whereby the exact optimality of the relaxed problem (or +5The condition number for the dual function illustrated in Figure 1 is 0 since the subgradient emanating +from point B forms a right angle with the direction toward optimal multipliers. +11 + +even subproblems) is not required. As long as the following “surrogate optimality condition” +L(˜xk, ˜yk, λk) < L(˜xk−1, ˜yk−1, λk) +(14) +is satisfied, the multipliers are updated as +λk+1 = λk + sk · g(˜xk, ˜yk), +(15) +by using the following formula +0 < sk < γ · q(λ∗) − L(˜xk, ˜yk, λk) +∥g(˜xk, ˜yk)∥2 +, γ < 1. +(16) +The convergence to λ∗ is guaranteed (Zhao et al., 1999). Unlike that in Polyak’s formula, +parameter γ is less than 1 to guarantee that q(λ∗) > L(˜xk, ˜yk, λk) so that the step-sizing +formula (16) is well-defined in the first place, as proven in (Zhao et al., 1999, Proposition +3.1, p. 703). Here, “tilde” indicates that the corresponding solutions do not need to be +necessarily subproblem-optimal. Solutions {˜xk, ˜yk} form a set S(˜xk−1, ˜yk−1, λk) ≡ {(x, y) : +L(x, y, λk) < L(˜xk−1, ˜yk−1, λk)}. Once a member {˜xk, ˜yk} ∈ S(˜xk−1, ˜yk−1, λk) is found, +i.e., the surrogate optimality condition (14) is satisfied, the optimization of the relaxed +problem stops and multipliers are updated per (15). A case when S(˜xk−1, ˜yk−1, λk) = ∅ +indicates that for given λk no solution better than {˜xk−1, ˜yk−1} can be found indicating +that (˜xk−1, ˜yk−1) = (x∗(λk), y∗(λk)) is subproblem-optimal for given value λk, and the +multipliers are updated by using a subgradient direction. +The convergence proof is quite similar to that presented in (8)-(12). The only caveat is +that L(˜xk, ˜yk, λk) is not a function; unlike the dual function q(λk), L(˜xk, ˜yk, λk) can take +multiple values for given λk. Therefore, the analogue of (9) cannot follow from concavity +of L(˜xk, ˜yk, λk). It follows from the fact that the surrogate dual value is obtained without +solving all the subproblems, hence +q(λ∗) ≤ L(˜xk, ˜yk, λ∗). +(17) +Adding and subtracting g(˜xk, ˜yk)T · λk from the previous inequality leads to +q(λ∗) − L(˜xk, ˜yk, λk) ≤ g(˜xk, ˜yk)T · (λ∗ − λk). +(18) +The procedure described in (10)-(12) follows analogously. +12 + +In addition to the reduction of computational effort, a concomitant reduction of multi- +plier zigzagging has been also observed. Indeed, with an exception of the aforementioned +situation whereby S(˜xk−1, ˜yk−1, λk) = ∅, a solution to one subproblem (13) is sufficient to +satisfy (14). In this case, only one term within each summation of surrogate subgradient +�I +i=1 Ax +i xi+�I +i=1 Ay +i yi−b will be updated thereby preventing surrogate sub-gradients from +changing drastically, and from zigzagging of multipliers as the result. +2.5 +Incremental Subgradient Methods +In the incremental subgradient method, a subproblem i is solved before multipliers are up- +dated (similar to the interleaved method). However, as opposed to updating all multipliers +at once, the incremental subgradient method updates multipliers incrementally. After the +ith subgradient component is calculated, multipliers are updated as +ψk +i = ψk +i−1 + sk · +� +Ax +i xk +i + Ay +i yk +i − βi +� +. +(19) +Here βi are the vectors such that �I +i=1 βi = b, for example, βi = +b +I . Only after all i +subproblems are solved, are the multipliers “fully” updated as +λk+1 = ψk +I . +(20) +Convergence results of the subgradient-level method (Goffin and Kiwiel, 1999) have been +extended for the subgradient method. Variations of the method were proposed with β and +τ belonging to an interval [0, 1] rather than being equal to 1 +2. Moreover, to improve conver- +gence, rather than using constant R, a sequence of Rl was proposed such that �∞ +l=1 Rl = ∞. +2.6 +The Surrogate Lagrangian Relaxation Method +Based on contraction mapping, Surrogate Lagrangian Relaxation (SLR) (Bragin et al., +2015) overcomes the difficulty associated with the lack of knowledge about the optimal +dual value. At consecutive iterations, the distance between multipliers must decrease, i.e., +���λk+1 − λk��� ≤ αk · +���λk − λk−1��� , +0 ≤ αk ≤ 1. +(21) +Based on (15) and (21), the step-sizing formula has been derived: +sk = αk · sk−1 ��˜g(xk−1) +�� +∥˜g(xk)∥ +. +(22) +13 + +Moreover, a specific formula to set αk has been developed to guarantee convergence: +αk = 1 − +1 +M · k1− 1 +kr , M ≥ 1, 0 ≤ r ≤ 1. +(23) +Since αk → 1, stepsizes within SLR are “non-summable.” Linear convergence can only be +guaranteed outside of a neighborhood of λ∗ (Bragin et al., 2015, Proposition 2.5, p. 187). +When multipliers are close to their optimal values6 and subproblems are “sufficiently +coordinated,” solutions to the relaxed problem are close to feasible solutions. As a result, +only a few subproblems cause infeasibility. This leads to the resolution of the Difficulty +D4 (“solutions to the relaxed problem, when put together may not constitute a feasible +solution to the original problem”). +An “automatic” procedure to identify and “repair” +a few subproblem solutions that cause the infeasibility of the original problem has been +developed by Bragin et al. (2018). +2.7 +Further Methodological Advancements +Surrogate Absolute-Value Lagrangian Relaxation (Bragin et al., 2018). The Sur- +rogate Absolute-Value Lagrangian Relaxation (SAVLR) method is designed to guarantee +convergence and to speed up the reduction of constraint violations while avoiding nonlin- +earity and nonconvexity that would have occurred if traditional quadratic terms had been +used. In the SAVLR method, the following dual problem is considered: +max +λ {qρ(λ) : λ ∈ Ω ⊂ Rm}, +(24) +where +qρ(λ) = min +(x,y) +� +I +� +i=1 +� +(cx +i )T xi + (cy +i )T yi +� ++ λT · +� I +� +i=1 +Ax +i xi + +I +� +i=1 +Ay +i yi − b +� ++ +ρ · +����� +I +� +i=1 +Ax +i xi + +I +� +i=1 +Ay +i yi − b +����� +1 +, {xi, yi} ∈ Fi, i = 1, . . . , I +� +. +(25) +The above minimization involves the exactly-linearizable piece-wise linear penalties, which +penalize constraint violations thereby ultimately reducing the number of subproblems that +cause infeasibility mentioned in 2.6 and consequently reducing the effort required by heuris- +tics to find primal solutions. This resolves Difficulty D5. +6A quality measure to quantify the quality of multipliers (i.e., how close the multipliers are to their +optimal values) will be discussed in subsection 2.8 +14 + +Distributed and Asynchronous Surrogate Lagrangian Relaxation (DA-SLR) +(Bragin et al., 2020). With the emergence of technologies supporting distributed com- +putational capabilities of multiple distributed entities as well as communication enabled by +the Internet of Things, computational tasks can be accomplished much more efficiently by +using distributed computing resources than by using a single computer. With the assump- +tion of a single coordinator, the DA-SLR methodology has been developed to efficiently +coordinate distributed subsystems in an asynchronous manner while avoiding the overhead +of synchronization. Compared to the sequential Surrogate Lagrangian Relaxation (Bragin +et al., 2015), numerical testing shows a faster convergence (12 times speed-up to achieve a +gap of 0.03% for one instance of the generalized assignment problem). +A short summary is in order here. While in theory, Polyak’s formula offers a geometric +rate of convergence, the convergence rate of the Subgradient-Level Method, however, is +not discussed either in the original paper by Goffin and Kiwiel (1999), or in subsequent +applications of the subgradient “level-based” ideas (e.g., by Nedic and Bertsekas (2001)). +Likely, because of the requirements that the multipliers need to travel, either explicitly or +implicitly, an infinite distance, the geometric/linear rate of convergence cannot be achieved. +While SLR-based methods avoid estimating the optimal dual value, the requirement (23) +to avoid premature termination results in the [stepsize] non-summability. Ideally, the goal +is to avoid multiplier zigzagging, to reduce the computational effort required to obtain +multiplier-updating directions, and to achieve linear convergence. The first step in this +direction is explained in the following subsection. +2.8 +Surrogate Level-Based Lagrangian Relaxation +To exploit the linear convergence potential inherent to Polyak’s stepsizing formula, the Sur- +rogate “Level-Based” Lagrangian Relaxation (SLBLR) method has been recently developed +(Bragin and Tucker, 2022). It was proven that once the following feasibility problem +∥λ − λkj+1∥ ≤ ∥λ − λkj∥, +∥λ − λkj+2∥ ≤ ∥λ − λkj+1∥, +(26) +. . . +∥λ − λkj+nj∥ ≤ ∥λ − λkj+nj−1∥, +15 + +admits no feasible solution with respect to λ (which are the decision variables in the problem +above) for some kj and nj, then the “level value” equals to +qj = +max +κ∈[kj,kj+nj] qκ,j > q(λ∗), +(27) +where +qκ,j = 1 +γ · sκ · ∥g(˜xκ, ˜yκ)∥2 + L(˜xκ, ˜yκ, λκ). +(28) +In subsequent iterations, Polyak’s stepsizing formula is used +sk = ζ · γ · qj − L(˜xk, ˜yk, λk) +∥g(˜xk, ˜yk)∥2 +, ζ < 1, k = kj+1, . . . , kj+1 + nj+1 − 1. +(29) +In essence, the above formula is used until the above feasibility problem admits no solution +again, at which point the level value is reset from qj to qj+1 and the multiplier-updating +process continues. It is worth noting that the optimization problem (3) is the maximization +problem and the quality of its solutions (Lagrangian multipliers) can be quantified through +the upper bound provided by {qj}. +The assumption here is that the above feasibility problem (26) becomes infeasible “peri- +odically” and “infinitely often” thereby triggering recalculations of “level” values qj, which +will decrease and approach q(λ∗) from above. The assumption is realistic since the only +sure-fire way to ensure that (26) is always feasible is to know the optimal dual value. +Given the above and with the addition of “absolute-value” penalties to facilitate the fea- +sible solution search, the SLBLR method addresses all the difficulties D1-D5. The method +inherits features from Polyak’s formula (7), reduction of computational effort as well as the +alleviation of zigzagging from surrogate methods (Zhao et al., 1999; Bragin et al., 2015) +and the acceleration of reduction of the constraint violation from (Bragin et al., 2018). The +decisive advantage of SLBLR is provided by the practically operationalizable and efficient +decision-based procedure described above to determine “level” values without the need for +estimation or heuristic adjustment of estimates of the optimal dual value. Results reported +in (Bragin and Tucker, 2022) indicate that the SLBLR method solves a wider range of gen- +eralized assignment problems (GAPs) to optimality as compared to other methods. With +other things being equal, the “level-based” stepsizing of the SLBLR method (Bragin and +Tucker, 2022) is more advantageous as compared to the “non-summable” stepsizing of the +16 + +SAVLR method (Bragin et al., 2018). Additionally, SLBLR successfully solves other prob- +lems such as stochastic job-shop scheduling and pharmaceutical scheduling outperforming +commercial solvers by at least two orders of magnitude. +The SLBLR method (Bragin and Tucker, 2022) is not restricted to MILP problems since +linearity is not required for the above-mentioned determination of level values. The method +is modular and has the potential to support plug-and-play capabilities. For example, while +applications to pharmaceutical scheduling have been tested by using fixed data (Bragin and +Tucker, 2022), the method is also suitable to handle urgent requests to manufacture new +pharmaceutical products since such products can be introduced into the relaxed problem +“on the fly” and the corresponding subproblems can keep being coordinated through La- +grangian multipliers without the major intervention of the scheduler and without disrupting +the overall production schedule. +3 +Conclusions and Future Directions +This paper intends to summarize the key difficulties encountered on a path to efficiently +solve MILP problems as well as to provide a brief summary of important milestones of a +more than half-a-century-long research journey to address these difficulties by using La- +grangian Relaxation. +Moreover, the most recent SLBLR method is 1. +general having +the potential to handle general MIP problems since linearity is not required to exploit +the geometric rate of convergence; 2. flexible and modular having the potential to support +plug-and-play capabilities with real-time response to unpredictable and/or disruptive events +such as natural hazards, operational faults, and cyber-physical events, including generator +outages in power systems, receiving an urgent order in manufacturing or encountering a +traffic jam in transportation. With communication and distributed computing capabilities, +these events can be handled by a continuous update of Lagrangian multipliers, improving +system resilience; the method is thus also suitable for fast re-optimization; 3. compatible +with other optimization methods such as quantum-computing as well as machine-learning +methods, which can potentially be used to further improve subproblem solving thereby con- +tributing to drastically reducing the CPU time supported by the fast coordination aspect +of the method. +17 + +Acknowledgements +This work was supported in part by the U.S. National Science Foundation under Grant +ECCS-1810108. +References +Archetti, C., L. Peirano, and M. G. Speranza (2021). Optimization in multimodal freight +transportation problems: A survey. European Journal of Operational Research 299(1), +1–20. +Arias-Melia, P., J. Liu, and R. Mandania (2022). The vehicle sharing and task allocation +problem: Milp formulation and a heuristic solution approach. 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Transportation Research Part E: Logistics and +Transportation Review 116, 190–207. +22 + diff --git a/JdAyT4oBgHgl3EQfsPlo/content/tmp_files/load_file.txt b/JdAyT4oBgHgl3EQfsPlo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a3802002ef9e1c46aa8e712d8bc3e135dca6042 --- /dev/null +++ b/JdAyT4oBgHgl3EQfsPlo/content/tmp_files/load_file.txt @@ -0,0 +1,791 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf,len=790 +page_content='Lagrangian Relaxation for Mixed-Integer Linear Programming: Importance, Challenges, Recent Advancements, and Opportunities Mikhail A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Bragin Department of Electrical and Computer Engineering, University of Connecticut, Storrs, USA Abstract Operations in areas of importance to society are frequently modeled as Mixed- Integer Linear Programming (MILP) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' While MILP problems suffer from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope to resolve the associated difficulties through decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Due to the non-smooth nature of Lagrangian dual functions, the coordination aspect of the method has posed serious challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This paper presents several significant historical milestones (beginning with Polyak’s pioneering work in 1967) toward improving Lagrangian Relaxation coordi- nation through improved optimization of non-smooth functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Finally, this paper presents the most recent developments in Lagrangian Relaxation for fast resolution of MILP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The paper also briefly discusses the opportunities that Lagrangian Relaxation can provide at this point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Keywords: Combinatorial Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Decomposition and Coordination;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Lagrangian Re- laxation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Discrete Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Duality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Mixed-Integer Linear Programming 1 Introduction The aim of this paper is to review Lagrangian-Relaxation-based methods for separable Mixed-Integer Linear Programming (MILP) problems, which are formally defined as follows: min (x,y):={xi,yi}I i=1 � I � i=1 � (cx i )T xi + (cy i )T yi � � , (1) whereby I subsystems are coupled through the following constraints s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' I � i=1 Ax i xi + I � i=1 Ay i yi − b = 0, {xi, yi} ∈ Fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' , I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (2) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='00573v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='OC] 2 Jan 2023 The primal problem (1)-(2) is assumed to be feasible and the feasible region F ≡ �I i=1 Fi with Fi ⊂ Znx i × Rny i is assumed to be bounded and finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Because of integer variables, Lagrangian Relaxation leads to non-smooth optimization in the dual space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Accordingly, key non-smooth optimization methods will also be reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='1 Importance and Difficulties of MILP Problems MILP has multiple applications in problems of importance to society: ambulance relocation (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), balanced item placement (Gasse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), cost-sharing for ride-sharing (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), drop box location (Schmidt and Albert, 2022), efficient failure detection in large-scale distributed systems (Er-Rahmadi and Ma, 2022), home healthcare routing (Dastgoshade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020), home service routing and appointment scheduling (Tsang and Shehadeh, 2022), inventory control under demand and lead time uncertainty (Thorsen and Yao, 2017), job-shop scheduling (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), facility location (Basciftci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), flow-shop scheduling (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Balogh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' ¨Oztop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), freight transportation (Archetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), location and inventory prepositioning of disaster re- lief supplies (Shehadeh and Tucker, 2022), machine scheduling with sequence-dependent setup times (Yalaoui and Nguyen, 2021), maritime inventory routing (Gasse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), multi-agent path finding with conflict-based search (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), multi-depot electric bus scheduling (Gkiotsalitis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2023), multi-echelon/multi-facility green reverse logis- tics network design (Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), optimal physician staffing (Prabhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), optimal search path with visibility (Morin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2023), oral cholera vaccine distribution (Smalley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015), outpatient colonoscopy scheduling (Shehadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020), pharma- ceutical distribution (Zhu and Ursavas, 2018), plant factory crop scheduling (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020), post-disaster blood supply (Hamdan and Diabat, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Kamyabniya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), real assembly line balancing with human-robot collaboration (Nourmohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), reducing vulnerability to human trafficking (Kaya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), restoration planning and crew routing (Morshedlou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), ridepooling (Gaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), scheduling of un- conventional oil field development (Soni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), security-constrained optimal power flow (Velloso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), semiconductor manufacturing (Chang and Dong, 2017), surgery scheduling (Kayvanfar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), unit commitment (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Li and Zhai, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' van Ackooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2021), vehicle shar- 2 ing and task allocation (Arias-Melia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), workload apportionment (Gasse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2022), and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Because of integer variables x, MILP problems are NP-hard and instances of practical sizes are generally difficult to solve to optimality due to the combina- torial complexity: the computational effort increases super-linearly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', exponentially) as the problem size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' For a number of practical problems, the computational effort may be significant to obtain even a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Additionally, many important prob- lems require short solving times (ranging from 20 minutes down to a few seconds), as well as high-quality solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' By the same token, with decreasing problem size, NP-hardness leads to the super-linear reduction of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The dual decomposition and coordination Lagrangian Relax- ation method is promising to exploit this reduction of complexity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the method essentially “reverses” combinatorial complexity upon decomposition, thereby drastically reducing the effort required to solve subproblems (each subproblem i corresponds to a subsystem i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Lagrangian Relaxation is also deeply rooted in economic theory, whereby the solutions obtained are rested upon the economical principle of “supply and demand.” When the “demand” exceeds the “supply,” Lagrangian multipliers (which can be viewed as “shadow prices”) increase (and vice versa) thereby discouraging subsystems from making less “eco- nomically viable” decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Notwithstanding the advantage of the decomposition aspect, the “price-based” coordination of the method (to appropriately coordinate the subprob- lems), however, has been the subject of intensive research for many decades because of the fundamental difficulties of the underlying non-smooth optimization of the associated dual functions as explained ahead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The purpose of this paper is to present a brief overview of the key milestones in the development of the Lagrangian Relaxation method for MILP problems as well as in the optimization of convex non-smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The rest of the paper is structured as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' At the beginning of Section 2, the Lagrangian dual problem is presented and the difficulties of Lagrangian Relaxation on a pathway to solving MILP problems are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsequent subsections, the difficulties are resolved one by one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='1, early research on non-smooth optimization (Polyak, 1967, 1969) is presented to lay the foundation for further developments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' specifically, the Polyak’s formula (Polyak, 1969) depending on the optimal dual value q(λ∗) to ensure geomet- 3 ric/linear convergence rate is presented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='2, the subgradient-level method (Goffin and Kiwiel, 1999) is presented to ensure convergence without the need to know q(λ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='3, the fundamental difficulties associated with subgradient methods (high computational effort and zigzagging of multipliers) are explained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='5, two separate research thrusts (surrogate (Kaskavelis and Caramanis, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1999) and incremental (Nedic and Bertsekas, 2001)) to reduce computational effort as well to alleviate zigzagging of multipliers are reviewed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the former thrust still requires q(λ∗) for convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the latter thrust avoids the need to know q(λ∗) following the “subgradient-level” ideas presented in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='6, Surrogate Lagrangian Relaxation (SLR) (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015) that proved convergence without q(λ∗) by exploiting “contraction mapping” while inher- iting convergence properties of the surrogate method of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='4 is reviewed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='7, further methodological advancements for SLR are presented;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' to ac- celerate the reduction of constraint violations while enabling the use of MILP solvers, “absolute-value” penalties have been introduced (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' to efficiently coordinate distributed entities while avoiding the synchronization overhead, compu- tationally distributed version of SLR has been developed to efficiently coordinate distributed subsystems in an asynchronous way (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='8, Surrogate “Level-Based” Lagrangian Relaxation (Bragin and Tucker, 2022) is reviewed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' this first-of-the-kind method exploits the linear-rate convergence potential intrinsic to the Polyak’s formula presented in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='1 but without the knowledge q(λ∗) and without heuristic adjustments of its estimates presented in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Rather, an estimate of q(λ∗) has been innovatively determined purely through a simple constraint satisfaction problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the surrogate concept (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015) ensures low computational requirements as well as the alle- viated zigzagging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' accelerated reduction of constraint violations is achieved through “absolute-value” penalties (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018) enabling the use of MILP solvers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In Section 3, a brief conclusion is provided with future directions delineated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 4 2 Lagrangian Duality for Discrete Programs and Non- Smooth Optimization The Lagrangian dual problem that corresponds to the original MILP problem (1)-(2) is the following non-smooth optimization problem: max λ {q(λ) : λ ∈ Ω ⊂ Rm}, (3) where the convex dual function is defined as follows: q(λ) = min (x,y) � L(x, y, λ), {xi, yi} ∈ Fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' , I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (4) Here L(x, y, λ) ≡ �I i=1 � (cx i )T xi + (cy i )T yi � + λT · � �I i=1 Ax i xi + �I i=1 Ay i yi − b � is the Lagrangian function obtained by relaxing coupling constraints (2) by using Lagrangian multipliers λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The minimization of L(x, y, λ) within (4) with respect to {x, y} is referred to as the relaxed problem, which is separable into subproblems due to the additivity of L(x, y, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This feature will be exploited starting from subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Even though the original primal problem (1) is non-convex, q(λ) is always continuous and concave with the feasible set Ω being always convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Due to integer variables x in the primal space, q(λ) is non-smooth with facets (each representing a particular solution to (4)) intersecting at ridges where derivatives of q(λ) exhibit discontinuities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' in particular, q(λ) is non-differentiable at λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' As a result, subgradients (referred to in early literature as “generalized gradients” or “subdifferentials”) are generally non-ascending: when λ are up- dated along subgradients, dual values may decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Moreover, subgradients may almost be perpendicular to the directions toward optimal multipliers λ∗ thereby leading to zigzagging of λ across ridges of the dual function (see Figure 1 for illustrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' While Lagrangian multipliers λ are generally fixed parameters within (4), λ are “dual” decision variables with respect to the dual problem (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Traditionally, (3) is maximized by iteratively updating λ by making a series of steps sk along subgradients g(xk, yk) as: λk+1 = λk + sk · g(xk, yk), (5) where {xk, yk} is a concise way to denote an optimal solution {x∗(λk), y∗(λk)} to the re- laxed problem (4) with multipliers equal to λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Within Lagrangian Relaxation, subgradients 5 A B \uf06c* Figure 1: An example of a dual function that illustrates the difficulties associated with subgradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Solid lines denote the level curves, dash-dotted lines denote the ridges of the dual function whereby the gradients are not defined (possible subgradient directions at points A and B are shown by solid arrows), and dashed lines denote the subgradient direction from point B toward optimal multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This Figure is taken from (Bragin and Tucker, 2022) with permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' are defined as levels of constraint violations g(xk, yk) ≡ ��I i=1 Ax i xk i + �I i=1 Ay i yk i − b � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' for compactness, g(xk, yk) will be denoted simply as gk as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' If inequality constraints �I i=1 Ax i xi + �I i=1 Ay i yi ≤ b are present, they are generally converted into equality constraints by introducing non-negative real-valued slack variables z such that �I i=1 Ax i xi + �I i=1 Ay i yi + z = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Multipliers are then updated per (5) with subsequent projection onto the positive orthant - a set delineated by constraints λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Because the dual problem results from the relaxation of coupling constraints, dual values are generally less than primal values q(λk) < f(xk, yk),1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', there is a duality gap - the relative difference between q(λk) and f(xk, yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='2 Because of the discrete nature of the primal problem (1)-(2), even at optimality, the duality gap is non-zero, that is q(λ∗) < f(x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Consequently, maximization of the dual function does not lead to an optimal primal solution (x∗, y∗) or even a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' To obtain solutions feasible with respect 1In this particular case, f(xk, yk) ≡ �I i=1 � (cx i )T xk i + (cy i )T yk i � such that {xk, yk} satisfy constraints (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2Dual values can be used to quality the quality of the solution {xk, yk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 6 to the original problem (1)-(2), solutions to the relaxed problem {xk i , yk i } are typically perturbed heuristically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='3 Generally, the closer the multipliers are to their optimal values λ∗, the smaller the levels of constraint violations (owing to the concavity of the dual function), and, therefore, the easier the search for feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In short summary, the roadblocks on the way of Lagrangian Relaxation to efficiently solve MILP problems are the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Non-differentiability of the dual function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (a) Subgradient directions are non-ascending;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (b) Necessary and sufficient conditions for extrema are inapplicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' High computational effort is required to compute subgradient directions if the number of subsystems is large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Zigzagging of multipliers across ridges of the dual function leading to many iterations required for convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' this difficulty follows from the non-differentiability of the dual function, but this difficulty deserves a separate resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Solutions {xk i , yk i } to the relaxed problem, when put together, do not satisfy con- straints (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Moreover, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Even at λ∗, levels of constraint violations may be large and the heuristic effort to “repair” the relaxed problem solution {xk i , yk i } may still be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In order to resolve difficulty 1(b), stepsizes need to approach zero (yet, this condition alone is not sufficient), as will be discussed in the subsection that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This requirement puts a restriction on the methods that will be reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' By examining the above-mentioned difficulties (which will also be referred to as D1(a), D1(b), D2, D3, D4, and D5), several stages in the development of Lagrangian Relaxation and its applications to optimizing non-smooth dual functions and solving MILP problems will be chronologically reviewed, along with specific features of the methods that 3To solve MILP problems, Lagrangian Relaxation is often regarded as a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' However, in dual space, the Lagrangian relaxation method is exact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the method is also capable of helping to improve solutions through the multipliers update, unlike many other heuristic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 7 address the above difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In view of the above difficulties such as D1(b) and D2, several research directions, though having their own merit, will be excluded, for example: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Method of Multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Alternate Direction Method of Multipliers (ADMM), which is derived from Augmented Lagrangian Relaxation (ALR) (the “Method of Multipliers”), introduces quadratic penalties to penalize violations of relaxed constraints, improving the convergence of Lagrangian Relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The two methods (ADMM and ALR), however, only converge when solving continuous primal problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Without stepsizes approaching zero, neither method converges [in the dual space] when solving discrete primal problems and does not resolve D1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Neverthe- less, the penalization idea underlying ALR led to the development of other LR-based methods with improved convergence as described in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Bundle Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Bundle Method’s idea is to obtain the so-called ε−ascent direction to update multipliers (Zhao and Luh, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Considering that the non- differentiability of dual functions may generally result in non-ascending subgradient directions, the Bundle method resolves D1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Since the relaxed problems need to be solved several times (Zhao and Luh, 2002), the effort required to obtain multiplier- updating directions exceeds that required in subgradient methods, thus the method does not resolve D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='1 Minimization of “Unsmooth Functionals” Optimization of non-smooth convex functions, a direction that stems from the seminal work of Polyak (1967), is a broader subject than the optimization of q(λ) within Lagrangian Relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' To present the underlying principles that support Lagrangian Relaxation to efficiently solve MILP problems, the work of Polyak (1967) is discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Subgradient Method with “Non-Summable” Stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' While subgradients are gen- erally non-descending (non-ascending) for minimization (maximization) problems (Polyak, 1967, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='33), convergence to the optimal solution optimizing a non-smooth function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', to λ∗ maximizing q(λ)) was proven under the following (frequently dubbed as non-summable) 8 stepsizing formula satisfying the following conditions: sk > 0, lim k→∞ sk = 0, ∞ � k=1 sk = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (6) Subgradient Method with Polyak’s Stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' As Polyak noted in his later work (Polyak, 1969, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='15), non-summable stepsizes lead to very slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Intending to achieve geometric (also referred to as linear4) rate of convergence so that ∥λk − λ∗∥ is monotonically decreasing, Polyak developed the stepsizing formula, which, for the problem under consideration, is presented in the following way: 0 < sk < γ · q(λ∗) − q(λk) ��g(xk, yk) ��2 , γ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (7) In the simplest form, a rendition of the Polyak’s result can be presented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' First, consider a binomial expansion of ∥λ∗ − λk+1∥2 as ∥λ∗ − λk+1∥2 = ∥λ∗ − λk∥2 − 2 · sk · (gk)T · (λ∗ − λk) + (sk)2 · ∥gk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (8) Owing to the concavity of the dual function, q(λ∗) − q(λk) ≤ (gk) · (λ∗ − λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (9) Therefore, (8) becomes: ∥λ∗ − λk+1∥2 ≤ ∥λ∗ − λk∥2 − 2 · sk · (q(λ∗) − q(λk)) + (sk)2 · ∥gk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (10) From (7), it follows that sk · ∥gk∥2 < 2 · (q(λ∗) − q(λk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (11) Therefore, (10) simplifies to ∥λ∗ − λk+1∥2 < ∥λ∗ − λk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (12) Within (7)-(12) and thereafter in the paper, the standard Euclidean norm will be used (unless specified otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 4Superlinear convergence is also possible, however, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' A reformulation of the dual problem (Charisopoulos and Davis, 2022) is required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Within the Lagrangian Relaxation framework, a dual function is generally unavailable as argued in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 9 The Polyak’s stepsizing (7) can be regarded as a creative workaround of D1(a) in the sense that a more computationally difficult problem of obtaining ascending directions at every iteration (as in the Bundle method) to ensure convergence is replaced with a provably easier problem of reducing ∥λ∗ − λk∥ at every step to guarantee convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Through the decades following 1969, two distinct research directions, along the lines of “non-summable” (6) and “Polyak” (7) stepsizes continued with various research groups adhering to either one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Both directions guarantee convergence to λ∗ that maximizes the dual function q(λ) (thereby resolving D1(b)), although, up to this stage in the discussion, convergence by using Polyak’s stepsizing is purely theoretical, since optimal dual value q(λ∗) required within (7) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Subgradient-Level Method was developed to achieve convergence in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='2 The Subgradient-Level Method The Subgradient-Level Method (Goffin and Kiwiel, 1999) overcame difficulties associated with the unavailability of optimal [dual] value, which is needed to compute Polyak’s step- size (7) through adaptively adjusting a level estimate based on the detection of sufficient descent and oscillations of the [dual] solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In terms of the problem (3), the procedure of the method is explained as follows: the “level” estimate qk lev = qkj rec + δj is used in place of the optimal dual value q(λ∗), where qk rec is the best (highest) dual value (“record objective value”) obtained up to an iteration k, and δj is an adjustable parameter with j denoting the jth update of qk lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The main premise behind this is when δj is “too large,” then multipliers will exhibit oscillations while traveling significant (predefined) distance R without improving the “record” value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In this case, the parameter δj is updated as δj+1 = β · δj with β = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' On the other hand, if δj is such that the dual value is sufficiently increased: q(λk) ≥ qk lev +τ ·δj, with τ = 1 2, then the parameter δj is unchanged and the distance traveled by multipliers is reset to 0 to avoid premature reduction of δj by β in future iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Followed by an examination of resolutions of D2 and D3, further discussions of the implementation of Polyak’s stepsizing to resolve D1 will be deferred to future subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='3 Fundamental Difficulties of Sub-gradient Methods High Computational Effort (D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In the methods reviewed thus far, non-smooth func- tions are assumed to be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' However, a dual function q(λ) cannot be obtained computa- tionally efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In fact, even for a given value of multipliers λk, minimization within (4) to obtain a dual value q(λk) and the corresponding subgradient is time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Even then, only one possible value of the subgradient can generally be obtained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' a complete description of the subgradient is generally non-attainable (Goffin, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Zigzagging of Multipliers (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' As hypothesized by (Goffin, 1977), the slow conver- gence of subgradient methods is due to ill-conditioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The condition number µ is formally defined as (Goffin, 1977): µ = inf{µ(λ) : λ ∈ Rm/P}, where P = {λ ∈ Rm : q(λ) = q(λ∗)} (the set of optimal solutions) and µ(λ) = minu uT ·(λ∗−λ) ∥uT ∥·∥λ∗−λ∥ (the cosine of the angle that subgradient form with directions toward the optimal multipliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='5 It was then confirmed experimentally when solving, for example, scheduling (Czerwinski and Luh, 1994, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 3(b), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 104) as well as power systems problems (Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1995, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 774) that the ill-conditioning leads to the zigzagging of multipliers across the ridges of the dual function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' To address these two difficulties, the notions of “surrogate,” “interleaved” and “incre- mental” subgradients, which do not require relaxed problems to be fully optimized to speed up convergence, emerged in the late 1990s, and early 2000s as reviewed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='4 The Interleaved Subgradient and the Surrogate Subgradient Methods Within the Interleaved Subgradient method proposed by (Kaskavelis and Caramanis, 1998), multipliers are updated after solving one subproblem at a time min (xi,yi) � (cx i )T xi+(cy i )T yi+λT · � Ax i xi+Ay i yi � , {xi, yi}∈Fi � , (13) rather than solving all the subproblems as in subgradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This significantly reduces computational effort, especially for problems with a large number of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The more general Surrogate Sub-gradient method with proven convergence was then developed by Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (1999) whereby the exact optimality of the relaxed problem (or 5The condition number for the dual function illustrated in Figure 1 is 0 since the subgradient emanating from point B forms a right angle with the direction toward optimal multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 11 even subproblems) is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' As long as the following “surrogate optimality condition” L(˜xk, ˜yk, λk) < L(˜xk−1, ˜yk−1, λk) (14) is satisfied, the multipliers are updated as λk+1 = λk + sk · g(˜xk, ˜yk), (15) by using the following formula 0 < sk < γ · q(λ∗) − L(˜xk, ˜yk, λk) ∥g(˜xk, ˜yk)∥2 , γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (16) The convergence to λ∗ is guaranteed (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Unlike that in Polyak’s formula, parameter γ is less than 1 to guarantee that q(λ∗) > L(˜xk, ˜yk, λk) so that the step-sizing formula (16) is well-defined in the first place, as proven in (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1999, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 703).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Here, “tilde” indicates that the corresponding solutions do not need to be necessarily subproblem-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Solutions {˜xk, ˜yk} form a set S(˜xk−1, ˜yk−1, λk) ≡ {(x, y) : L(x, y, λk) < L(˜xk−1, ˜yk−1, λk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Once a member {˜xk, ˜yk} ∈ S(˜xk−1, ˜yk−1, λk) is found, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', the surrogate optimality condition (14) is satisfied, the optimization of the relaxed problem stops and multipliers are updated per (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' A case when S(˜xk−1, ˜yk−1, λk) = ∅ indicates that for given λk no solution better than {˜xk−1, ˜yk−1} can be found indicating that (˜xk−1, ˜yk−1) = (x∗(λk), y∗(λk)) is subproblem-optimal for given value λk, and the multipliers are updated by using a subgradient direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The convergence proof is quite similar to that presented in (8)-(12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The only caveat is that L(˜xk, ˜yk, λk) is not a function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' unlike the dual function q(λk), L(˜xk, ˜yk, λk) can take multiple values for given λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Therefore, the analogue of (9) cannot follow from concavity of L(˜xk, ˜yk, λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' It follows from the fact that the surrogate dual value is obtained without solving all the subproblems, hence q(λ∗) ≤ L(˜xk, ˜yk, λ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (17) Adding and subtracting g(˜xk, ˜yk)T · λk from the previous inequality leads to q(λ∗) − L(˜xk, ˜yk, λk) ≤ g(˜xk, ˜yk)T · (λ∗ − λk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (18) The procedure described in (10)-(12) follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 12 In addition to the reduction of computational effort, a concomitant reduction of multi- plier zigzagging has been also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Indeed, with an exception of the aforementioned situation whereby S(˜xk−1, ˜yk−1, λk) = ∅, a solution to one subproblem (13) is sufficient to satisfy (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In this case, only one term within each summation of surrogate subgradient �I i=1 Ax i xi+�I i=1 Ay i yi−b will be updated thereby preventing surrogate sub-gradients from changing drastically, and from zigzagging of multipliers as the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='5 Incremental Subgradient Methods In the incremental subgradient method, a subproblem i is solved before multipliers are up- dated (similar to the interleaved method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' However, as opposed to updating all multipliers at once, the incremental subgradient method updates multipliers incrementally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' After the ith subgradient component is calculated, multipliers are updated as ψk i = ψk i−1 + sk · � Ax i xk i + Ay i yk i − βi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (19) Here βi are the vectors such that �I i=1 βi = b, for example, βi = b I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Only after all i subproblems are solved, are the multipliers “fully” updated as λk+1 = ψk I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (20) Convergence results of the subgradient-level method (Goffin and Kiwiel, 1999) have been extended for the subgradient method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Variations of the method were proposed with β and τ belonging to an interval [0, 1] rather than being equal to 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Moreover, to improve conver- gence, rather than using constant R, a sequence of Rl was proposed such that �∞ l=1 Rl = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='6 The Surrogate Lagrangian Relaxation Method Based on contraction mapping, Surrogate Lagrangian Relaxation (SLR) (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015) overcomes the difficulty associated with the lack of knowledge about the optimal dual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' At consecutive iterations, the distance between multipliers must decrease, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', ���λk+1 − λk��� ≤ αk · ���λk − λk−1��� , 0 ≤ αk ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (21) Based on (15) and (21), the step-sizing formula has been derived: sk = αk · sk−1 ��˜g(xk−1) �� ∥˜g(xk)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (22) 13 Moreover, a specific formula to set αk has been developed to guarantee convergence: αk = 1 − 1 M · k1− 1 kr , M ≥ 1, 0 ≤ r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (23) Since αk → 1, stepsizes within SLR are “non-summable.” Linear convergence can only be guaranteed outside of a neighborhood of λ∗ (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 187).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' When multipliers are close to their optimal values6 and subproblems are “sufficiently coordinated,” solutions to the relaxed problem are close to feasible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' As a result, only a few subproblems cause infeasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This leads to the resolution of the Difficulty D4 (“solutions to the relaxed problem, when put together may not constitute a feasible solution to the original problem”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' An “automatic” procedure to identify and “repair” a few subproblem solutions that cause the infeasibility of the original problem has been developed by Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='7 Further Methodological Advancements Surrogate Absolute-Value Lagrangian Relaxation (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The Sur- rogate Absolute-Value Lagrangian Relaxation (SAVLR) method is designed to guarantee convergence and to speed up the reduction of constraint violations while avoiding nonlin- earity and nonconvexity that would have occurred if traditional quadratic terms had been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' In the SAVLR method, the following dual problem is considered: max λ {qρ(λ) : λ ∈ Ω ⊂ Rm}, (24) where qρ(λ) = min (x,y) � I � i=1 � (cx i )T xi + (cy i )T yi � + λT · � I � i=1 Ax i xi + I � i=1 Ay i yi − b � + ρ · ����� I � i=1 Ax i xi + I � i=1 Ay i yi − b ����� 1 , {xi, yi} ∈ Fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' , I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (25) The above minimization involves the exactly-linearizable piece-wise linear penalties, which penalize constraint violations thereby ultimately reducing the number of subproblems that cause infeasibility mentioned in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='6 and consequently reducing the effort required by heuris- tics to find primal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' This resolves Difficulty D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 6A quality measure to quantify the quality of multipliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', how close the multipliers are to their optimal values) will be discussed in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='8 14 Distributed and Asynchronous Surrogate Lagrangian Relaxation (DA-SLR) (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' With the emergence of technologies supporting distributed com- putational capabilities of multiple distributed entities as well as communication enabled by the Internet of Things, computational tasks can be accomplished much more efficiently by using distributed computing resources than by using a single computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' With the assump- tion of a single coordinator, the DA-SLR methodology has been developed to efficiently coordinate distributed subsystems in an asynchronous manner while avoiding the overhead of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Compared to the sequential Surrogate Lagrangian Relaxation (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015), numerical testing shows a faster convergence (12 times speed-up to achieve a gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='03% for one instance of the generalized assignment problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' A short summary is in order here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' While in theory, Polyak’s formula offers a geometric rate of convergence, the convergence rate of the Subgradient-Level Method, however, is not discussed either in the original paper by Goffin and Kiwiel (1999), or in subsequent applications of the subgradient “level-based” ideas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', by Nedic and Bertsekas (2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Likely, because of the requirements that the multipliers need to travel, either explicitly or implicitly, an infinite distance, the geometric/linear rate of convergence cannot be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' While SLR-based methods avoid estimating the optimal dual value, the requirement (23) to avoid premature termination results in the [stepsize] non-summability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Ideally, the goal is to avoid multiplier zigzagging, to reduce the computational effort required to obtain multiplier-updating directions, and to achieve linear convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The first step in this direction is explained in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='8 Surrogate Level-Based Lagrangian Relaxation To exploit the linear convergence potential inherent to Polyak’s stepsizing formula, the Sur- rogate “Level-Based” Lagrangian Relaxation (SLBLR) method has been recently developed (Bragin and Tucker, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' It was proven that once the following feasibility problem ∥λ − λkj+1∥ ≤ ∥λ − λkj∥, ∥λ − λkj+2∥ ≤ ∥λ − λkj+1∥, (26) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' ∥λ − λkj+nj∥ ≤ ∥λ − λkj+nj−1∥, 15 admits no feasible solution with respect to λ (which are the decision variables in the problem above) for some kj and nj, then the “level value” equals to qj = max κ∈[kj,kj+nj] qκ,j > q(λ∗), (27) where qκ,j = 1 γ · sκ · ∥g(˜xκ, ˜yκ)∥2 + L(˜xκ, ˜yκ, λκ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (28) In subsequent iterations, Polyak’s stepsizing formula is used sk = ζ · γ · qj − L(˜xk, ˜yk, λk) ∥g(˜xk, ˜yk)∥2 , ζ < 1, k = kj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' , kj+1 + nj+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' (29) In essence, the above formula is used until the above feasibility problem admits no solution again, at which point the level value is reset from qj to qj+1 and the multiplier-updating process continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' It is worth noting that the optimization problem (3) is the maximization problem and the quality of its solutions (Lagrangian multipliers) can be quantified through the upper bound provided by {qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The assumption here is that the above feasibility problem (26) becomes infeasible “peri- odically” and “infinitely often” thereby triggering recalculations of “level” values qj, which will decrease and approach q(λ∗) from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The assumption is realistic since the only sure-fire way to ensure that (26) is always feasible is to know the optimal dual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Given the above and with the addition of “absolute-value” penalties to facilitate the fea- sible solution search, the SLBLR method addresses all the difficulties D1-D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The method inherits features from Polyak’s formula (7), reduction of computational effort as well as the alleviation of zigzagging from surrogate methods (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2015) and the acceleration of reduction of the constraint violation from (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The decisive advantage of SLBLR is provided by the practically operationalizable and efficient decision-based procedure described above to determine “level” values without the need for estimation or heuristic adjustment of estimates of the optimal dual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Results reported in (Bragin and Tucker, 2022) indicate that the SLBLR method solves a wider range of gen- eralized assignment problems (GAPs) to optimality as compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' With other things being equal, the “level-based” stepsizing of the SLBLR method (Bragin and Tucker, 2022) is more advantageous as compared to the “non-summable” stepsizing of the 16 SAVLR method (Bragin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Additionally, SLBLR successfully solves other prob- lems such as stochastic job-shop scheduling and pharmaceutical scheduling outperforming commercial solvers by at least two orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The SLBLR method (Bragin and Tucker, 2022) is not restricted to MILP problems since linearity is not required for the above-mentioned determination of level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' The method is modular and has the potential to support plug-and-play capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' For example, while applications to pharmaceutical scheduling have been tested by using fixed data (Bragin and Tucker, 2022), the method is also suitable to handle urgent requests to manufacture new pharmaceutical products since such products can be introduced into the relaxed problem “on the fly” and the corresponding subproblems can keep being coordinated through La- grangian multipliers without the major intervention of the scheduler and without disrupting the overall production schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 3 Conclusions and Future Directions This paper intends to summarize the key difficulties encountered on a path to efficiently solve MILP problems as well as to provide a brief summary of important milestones of a more than half-a-century-long research journey to address these difficulties by using La- grangian Relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Moreover, the most recent SLBLR method is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' general having the potential to handle general MIP problems since linearity is not required to exploit the geometric rate of convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' flexible and modular having the potential to support plug-and-play capabilities with real-time response to unpredictable and/or disruptive events such as natural hazards, operational faults, and cyber-physical events, including generator outages in power systems, receiving an urgent order in manufacturing or encountering a traffic jam in transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' With communication and distributed computing capabilities, these events can be handled by a continuous update of Lagrangian multipliers, improving system resilience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' the method is thus also suitable for fast re-optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' compatible with other optimization methods such as quantum-computing as well as machine-learning methods, which can potentially be used to further improve subproblem solving thereby con- tributing to drastically reducing the CPU time supported by the fast coordination aspect of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 17 Acknowledgements This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' National Science Foundation under Grant ECCS-1810108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' References Archetti, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' Transportation Research Part E: Logistics and Transportation Review 116, 190–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAyT4oBgHgl3EQfsPlo/content/2301.00573v1.pdf'} diff --git a/JtAyT4oBgHgl3EQf5_rl/content/2301.00816v1.pdf b/JtAyT4oBgHgl3EQf5_rl/content/2301.00816v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..aafdb9bd67a683210bc911e1e358a306229857fd --- /dev/null +++ b/JtAyT4oBgHgl3EQf5_rl/content/2301.00816v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:222f509df37db5be8ed442c979a1d26952df04a3732474b24c25579806c68167 +size 794150 diff --git a/JtAyT4oBgHgl3EQf5_rl/vector_store/index.faiss b/JtAyT4oBgHgl3EQf5_rl/vector_store/index.faiss new file 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Pace,2 Jung Hoon Han,3 Yizhi You,4, ∗ and Hyun-Yong Lee1, 5, 6, † +1Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea +2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA +3Department of Physics, Sungkyunkwan University, Suwon 16419, Korea +4Department of Physics, Northeastern University, 360 Huntington Ave, Boston, MA 02115, USA +5Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea +6Interdisciplinary Program in E·ICT-Culture-Sports Convergence, Korea University, Sejong 30019, Korea +(Dated: January 13, 2023) +Rank-2 toric code (R2TC), a prototypical archetype of the discrete rank-2 symmetric gauge theory, has prop- +erties that differ from those of the standard toric code. Specifically, it features a blending of UV and IR in its +ground state, restricted mobility of its quasiparticles, and variations in the braiding statistics of its quasiparticles +based on their position. In this paper, we investigate various aspects of ZN rank-2 gauge theory in (2 + 1)- +dimensional spacetime. Firstly, we demonstrate that U(1) rank-2 gauge theory can arise from U(1) × U(1) +rank-1 gauge theory after condensing the gauge charges in a specific way. This construction scheme of U(1) +rank-2 gauge theory carries over to the ZN case simply by Higgsing U(1) to ZN, after which the resulting +rank-2 gauge theory can be tuned to the R2TC. The holonomy operators of R2TC are readily identified using +this scheme and are given clear physical interpretation as the pair creation/annihilation of various monopoles +and dipoles. Explicit tensor network construction of the ground states of R2TC are given as two copies of the +ground states of Kitaev’s toric code that are ‘sewn together’ according to the condensation scheme. In addi- +tion, through a similar anyon condensation protocol, we present a double semion version of rank-2 toric code +whose flux excitations exhibit restricted mobility and semionic statistics. Finally, we identify the generalized +discrete symmetries of the R2TC, which are much more complex than typical 1-form symmetries. They in- +clude conventional and unconventional 1-form symmetries, such as framed 1-form symmetries and what we call +sublattice 1-form symmetries. Using these, we interpret the R2TC’s unique properties (UV/IR mixing, position- +dependent braiding, etc.) from the modern perspective of generalized spontaneous symmetry breaking and ’t +Hooft anomalies. +CONTENTS +I. Introduction +1 +II. Condensation Scheme +3 +A. Condensation of rank-2 U(1) lattice gauge fields +3 +B. Condensation of stabilizers and holonomies +4 +C. Higher-order instanton and confinement +in the rank-2 U(1) gauge theory +6 +D. Conservation laws +6 +III. Applications of the Condensation Scheme +7 +A. Tensor network representation of +ZN R2TC wavefunctions +7 +B. Twisted rank-2 gauge theory from anyon +condensation +8 +IV. Holonomy Construction +10 +A. Pre-projection holonomies +10 +B. Post-projection holonomies +11 +C. Physical interpretation of the holonomies +13 +D. Field-theoretic derivation of the holonomies +13 +E. Understanding the position-dependent braiding +15 +V. Generalized Symmetries +16 +∗ Electronic address: y.you@northeastern.com +† Electronic address: hyunyong@korea.ac.kr +A. Reviewing the 1-form symmetries of the R1TC +17 +B. Symmetries of the R2TC +18 +1. Construction of symmetry operators +18 +2. Analysis and discussion of R2TC symmetries +21 +VI. Summary and Outlook +23 +Acknowledgments +23 +A. Review of discrete differential geometry for +d-dimensional cubic lattices +23 +References +25 +I. +INTRODUCTION +Long-range entangled phases of quantum matter are com- +monly described by fractionalized quasiparticles and emer- +gent gauge fields which provide an effective description cap- +turing the phase’s universal properties [1]. Indeed, canoni- +cal examples include fractional quantum Hall liquids [2] and +quantum spin liquids [3]. Unsurprisingly, long-range entan- +gled quantum matter with increasingly exotic properties is de- +scribed by increasingly rich generalizations of conventional +gauge theory. A particular example is abelian gauge theories +whose gauge fields are symmetric tensor fields instead of vec- +arXiv:2301.04706v1 [cond-mat.str-el] 11 Jan 2023 + +2 +tor fields. These higher-rank gauge theories1 have attracted +substantial interest recently in the study of fracton phases [4– +8] and topological order [7–12]. +One of the simplest archetype of discrete gauge theories can +be obtained from Higgsing the U(1) theory into Z2 by con- +densing charge-2 gauge charges. Following this protocol, we +can obtain the rank-2 Z2 gauge theory starting from a rank- +2 U(1) gauge theory and Higgsing U(1) down to Z2 [7, 8]. +In the zero-correlation length limit, the resultant gauge the- +ory can be interpreted as an exactly solvable Hamiltonian, so- +called rank-2 toric code (R2TC) [9–11, 13]. +The R2TC features several interesting properties. One of +them is the sensitive dependence of the ground state degener- +acy (GSD) on the system size Lx × Ly against N, the Hilbert +space dimension of the local spin state |s⟩ (s = 0, · · · , N −1). +The GSD varying from N 3 to N 6 was first discovered in Ref. +[9] and was soon clarified as a rigorous formula [10] +GSD = N 3gcd(Lx, N)gcd(Ly, N)gcd(Lx, Ly, N), (1.1) +where gcd stands for the greatest common divisor among +the two or three integers. The fact that the GSD, a macro- +scopic property, depends sensitively on the number of unit +cells, a microscopic property, is a manifestation of what’s +known as UV/IR mixing [10, 12, 14, 15]. The braiding pro- +cess between a pair of quasiparticles showed interesting po- +sition dependence that is not seen in Kitaev’s toric code, and +requires a new form of field theory called the dipolar BF the- +ory (dBF) [11] for comprehensive understanding. A differ- +ent interpretation of the dipolar braiding in terms of multi- +component mutual Chern-Simons theory was given in Ref. +[10]. Lastly, quasiparticle excitations in the model showed +restricted mobility such as the ability to hop only in multiples +of N lattice sites in certain directions [7, 8]. We note that +many of these features were already apparent in the plaquette +model of Wen [16], and more recently several models sharing +similar features were proposed [12, 17, 18]. +The restricted mobility exhibited by quasiparticle excita- +tions in R2TC is clearly shared in a more rigorous way in +fracton models such as the X-cube model [19]. A standard +way of constructing these fracton models is to use the net- +work construction scheme first proposed in Refs. [20, 21]. In +it, one starts with layers of 2D toric codes with fully mobile +quasiparticle excitations, and produces a fractonic model with +immobile excitations by imposing constraints among the lay- +ers. It is a natural question then if a similar scheme does exist +to construct R2TC -a model based on rank-2 gauge theory - +from the R1TC which is rooted in conventional rank-1 gauge +theory. This paper answers this question in the affirmative. +In accomplish this, we get to exploit the idea of coupling +two gauge theories together through constraint, in a process +often called the anyon condensation. Several past works have +1 We will generally denote abelian symmetric tensor gauge theory simply +as higher-rank gauge theory. This is not to be confused with higher-form +gauge theories whose gauge fields are differential forms—antisymmetric +tensors. +exploited the condensation idea to produce the X-cube model +from layers of 2D toric codes [20, 21] (hereafter referred to +as rank-1 toric code, or R1TC for short), or to produce rank-2 +gauge theories from rank-1 theories [22, 23]. Our condensa- +tion scheme share the similar spirit as these works, but dif- +fers greatly in details of how we implement the constraint. +In particular, we make a clear comparison between the con- +densation scheme of Ref. [23] and our own in Sec. II B in +an effort to emphasize the consequences of various conden- +sation schemes. In the past, R2TC was obtained by Higgsing +the symmetric rank-2 gauge field [7, 9] but the origin of this +higher-rank gauge field was left obscure. We show here that it +emerges naturally in the course of constraining the two copies +of rank-1 gauge fields in a certain way. +Furthermore, the GSD of Eq. (1.1) is closely related to +the existence of six independent Wilson line operators in the +model, which have been identified previously in the spin oper- +ator [9] and the field theory language [10], respectively. Still +lacking was a clear physical picture accompanying these Wil- +son operators, such as the creation/annihilation of electric and +magnetic quasiparticle pairs in the case of R1TC. It turns out +that the condensation scheme provides a helpful guide in con- +structing the full set of holonomies 2 needed to fully account +for the GSD, which are also amenable to physically appealing +interpretations. In addition to the braiding of charges, braid- +ing of dipoles play an important role in accounting for the +degeneracy of the R2TC ground states. +As an added benefit of the condensation picture, we find +some useful applications in explicitly constructing the first- +quantized ground state wave function of R2TC in tensor net- +work (TN) form, as two copies of R1TC ground states sewn +together through some constraining tensor that directly re- +flects the constraint. As another example we construct the +rank-2 generalization of a model with semionic flux statis- +tics [24] by coupling two copies of the pristine double semion +model through anyon condensation. +As a final topic of the paper, we explore the generalized +symmetries of the R2TC. Modern generalizations of sym- +metry [25–31] have opened up an exciting frontier for the +discovery of new phases of quantum matter [32–38] and in +the conceptual organization of both known and new quan- +tum phases [29]. +For instance, these generalizations have +allowed topological order to be understood in a symmetry +framework [29, 34, 39–41]. It is therefore natural to wonder +if the interesting properties of the R2TC can be understood +in this unifying, modern point of view of topological quan- +tum matter. Here, we construct all of the symmetry operators +for the R2TC for general N. In the ground state sub-Hilbert +space, the symmetries we identify are all 1-form symmetries. +However, they are not all conventional 1-form symmetries: +some rely on a framing structure of the lattice (framed 1-form +symmetries) and others on a sublattice structure of the lattice +(sublattice 1-form symmetries). Furthermore, these symme- +tries have a rich mixed ’t Hooft anomaly structure. We show +2 In this paper, we use the words holonomies interchangeably with Wilson +line operators. + +3 +FIG. 1. Illustration of two possible condensation processes leading +to the appearance of new (a) magnetic and (b) electric flux opera- +tors given by Eq. (2.3) and Eq. (2.8), respectively. There are two +inter-penetrating sublattices Λ1, Λ2 shown by dashed and solid lines, +respectively. The dark squares (circles) represent Vvh (Vhv) sites. +The coordinate ⃗vi refers to the Vvh sites. The pink and blue arrows +represent rank-1 gauge field patterns from the sublattice Λ1 and Λ2, +respectively, that combine to give new flux patterns in the rank-2 +gauge theory. +that the R2TC ground state spontaneously breaks all of these +1-form symmetries. This allows us to interpret the unconven- +tional properties of the R2TC (position-dependent braiding, +UV/IR mixing, etc.) all in terms of these symmetries. +Organization of the paper is as follows. In Sec. II we out- +line the condensation scheme that leads one from two copies +of rank-1 lattice gauge theory to the rank-2 lattice gauge the- +ory and ultimately to the R2TC. In Sec. III we discuss two +applications of the condensation idea in the construction of +the ground state of rank-2 toric code out of those of the rank- +1 toric code, and the construction of ‘twisted’ rank-2 gauge +theory resulting in a new model with semionic flux statistics. +In Sec. IV we carefully go through the procedure by which +all the holonomies in the R2TC can be derived. Physical in- +terpretation of the holonomies thus constructed is given. In +Sec. V, after first reviewing the generalized symmetries of the +R1TC, we discuss the R2TC from the point of view of general- +ized symmetries. Additional themes such as instanton effects +in rank-1 gauge theories in 2+1D (Sec. II C), field-theoretic +understanding of the holonomy and the position-dependent +braiding (Secs. IV D and IV E) are discussed. The summary +and outlook follows in Sec. VI. +II. +CONDENSATION SCHEME +We show how rank-2 gauge theoires can emerge from two +copies of rank-1 gauge theory through the condensation of +certain components of the gauge fields. We outline this proce- +dure first from the perspective of U(1) gauge theory, followed +by that of ZN gauge fields. Discussion of instanton suppres- +sion in the rank-2 gauge theory is given as well. +A. +Condensation of rank-2 U(1) lattice gauge fields +Consider two interpenetrating square lattices denoted Λ1 +(dashed lines) and Λ2 (solid lines) as in Fig. 1. Each square +lattice has gauge degrees of freedom (Aµ +a, Eµ +a ) residing at the +µ = x, y-oriented links of the respective square sublattice +labeled by a = 1, 2, satisfying the canonical commutation +[Aµ +a, Eµ′ +a′ ] = iδaa′δµµ′. The two square lattices are super- +posed in such a way that horizontal bonds in Λ1 and verti- +cal bonds in Λ2 intersect at one set of sites belonging to Vhv, +while vertical bonds of Λ1 and horizontal bonds of Λ2 cross +at sites belonging to Vvh. In Fig. 1, sites belonging to Vvh and +Vhv are designated by dark squares and circles, respectively. +The coordinate ⃗vi = xiˆx + yiˆy, or sometimes just i, is used to +label the Vvh sites. (Note that what we call sites are the links in +the individual sublattice.) To reduce the notational clutter, we +will also use i to label the vertex position ⃗r1,i = ⃗vi−ˆy/2 of the +Λ1 lattice and the vertex position ⃗r2,i = ⃗vi−ˆx/2 of the Λ2 lat- +tice as well. According to this notation scheme, the y-oriented +fields (Ay +1,i, Ey +1,i) and the x-oriented fields (Ax +2,i, Ex +2,i) both +reside on the same site ⃗vi = xiˆx + yiˆy. The sites in Vhv +and the fields defined on them are then assigned appropriate +coordinates in reference to those given to Vvh sites. +Each square lattice Λ1, Λ2 hosts its own gauge-invariant +quantities (a = 1, 2), +Ga(⃗ra,i) = (∇ · Ea)i, +Ba(⃗ra,i) = (∇ × Aa)i, +(2.1) +which are the lattice divergence and lattice curl. Suppose that +we impose the constraint Ex +1,i+ˆy = Ey +2,i+ˆx at half the links, +e.g. on the Vhv sites (dark circles in Fig. 1). In other words, +we write the combined Hilbert spaces of the two lattice gauge +theory’s as |Ψ⟩ = |ψ1⟩ ⊗ |ψ2⟩ where |ψa⟩ belongs to the +Hilbert space of Λa, and insist that only the subset of Hilbert +spaces obeying the following constraint survives: +(Ex +1,i+ˆy − Ey +2,i+ˆx)|ψ⟩ = 0. +(2.2) +To be clear, i refers to all the Vhv sites. Such constraint nec- +essarily precludes operators that do not commute with it, such +as (∇ × Aa)i, while (∇ · Ea)i is still allowed. In their place, +a new operator that commutes with the constraint can be con- +structed by noting that [Ex +1,i+ˆy − Ey +2,i+ˆx, Ax +1,i+ˆy + Ay +2,i+ˆx] = +0. We find +Bi = ∆x(∇ × A1)i − ∆y(∇ × A2)i +(2.3) +indeed involves only the combination Ax +1 + Ay +2 at all the Vhv +sites where the constraint Eq. (2.3) is imposed, and therefore +commutes with it 3. The meaning of discrete derivatives ∆x +and ∆y is clear from Fig. 1. +Due to the constraint, one must identify Ex +1,i+ˆy = Ey +2,i+ˆx +as one gauge field and accordingly introduce a new label +(Ex +2,i, Ey +1,i, Ex +1,i+ˆy = Ey +2,i+ˆx) → (Exx +i , Eyy +i , Exy +i ). +(2.4) +3 It is easy to convince that no simpler operator exists that commutes with +the constraint. + +(b) +a4 +A similar re-labeling +(Ax +2,i, Ay +1,i, Ax +1,i+ˆy + Ay +2,i+ˆx) → (Axx +i , Ayy +i , Axy +i ) +(2.5) +yields symmetric rank-2 gauge fields (Aa +i , Ea +i ) (a += +xx, xy, yy) obeying the canonical relation [Aa +i , Eb +j] += +iδijδab4. +There are two electric charges (ex +i , ey +i ) and one vector +charge mi in the projected Hilbert space obeying Eq. (2.2) +given by +ex +i =Exx +i+ˆx − Exx +i ++ Exy +i +− Exy +i−ˆy, +ey +i =Exy +i +− Exy +i−ˆx + Eyy +i+ˆy − Eyy +i , +mi =Axx +i+ˆy + Axx +i−ˆy − 2Axx +i ++ Ayy +i+ˆx + Ayy +i−ˆx − 2Ayy +i +− Axy +i ++ Axy +i−ˆx + Axy +i−ˆy − Axy +i−ˆx−ˆy. +(2.6) +Here mi is simply the re-writing of Bi in Eq. (2.3). The sym- +metric rank-2 gauge fields as well as the new mutually com- +muting generators formed by them emerge naturally from the +condensation process just outlined. Upon Higgsing, the three +charge operators in Eq. (2.6) become the three commuting +spin operators of R2TC [9]. Among the tensor gauge fields, +xx and yy components reside at the Vvh sites where no con- +densation has taken place, and the xy component resides at the +Vhv sites where condensation reduces the degrees of freedom +from two to one. +The constraint expressed in Eq. (2.2) is by no means the +unique one. Instead of condensing E, one can condense the A +fields through the constraint +(Ax +1,i+ˆy − Ay +2,i+ˆx)|Ψ⟩ = 0, +(2.7) +at the Vhv sites. In this case, (∇·Ea)i is no longer an allowed +operator but a new quantity +Gi = ∆x(∇ · E2)i + ∆y(∇ · E1)i +(2.8) +emerges as a viable operator in the constrained Hilbert space +- see Fig. 1(b). After re-labeling +(Ax +2,i, Ay +1,i, Ax +1,i+ˆy = Ay +2,i+ˆx) → (Axx +i , Ayy +i , Axy +i ) +(Ex +2,i, Ey +1,i, Ex +1,i+ˆy + Ey +2,i+ˆx) → (Exx +i , Eyy +i , Exy +i ), +(2.9) +one arrives at two magnetic charges (mx +i , my +i ) and one electric +charge ei defined by +mx +i =Ayy +i+ˆx − Ayy +i +− Axy +i ++ Axy +i−ˆy, +my +i =Axy +i +− Axy +i−ˆx − Axx +i+ˆy + Axx +i , +ei =Eyy +i+ˆy + Eyy +i−ˆy − 2Eyy +i ++ Exx +i+ˆx + Exx +i−ˆx − 2Exx +i ++ Exy +i +− Exy +i−ˆx − Exy +i−ˆy + Exy +i−ˆx−ˆy, +(2.10) +where ei is a mere re-writing of Gi in Eq. (2.8). Higging them +leads to R2TC with a scalar electric charge and vector mag- +netic charges, which is dual to the theory with vector electric +and scalar magnetic charge [9, 10]. In the ensuing discussion, +we will adopt this version of R2TC that has scalar-electric and +vector-magnetic charges unless otherwise specified. +4 This follows from the original gauge fields obeying the canonical relation +[Aµ +a,i, Eν +b,j] = iδabδijδµν. +B. +Condensation of stabilizers and holonomies +The previous subsection showed which operators survive +under the projection (condensation) of two rank-1 lattice +gauge theory’s to the constrained Hilbert space. The oper- +ators that become the stabilizers in the R2TC emerged nat- +urally. In this subsection we elaborate how the condensation +idea plays out for the various spin operators and stabilizers. To +be specific, we first construct the stabilizers and holonomies +in the pre-projected Hilbert space consisting of two copies of +R1TCs. Then we examine which of these operators survive, +or become modified, under the projection. Stabilizers of the +R2TC are recovered once again in this way. Although at first +sight this discussion seems redundant in light of the aforemen- +tioned projection scheme outlined in the context of U(1) gauge +fields, there is a nice benefit to the present discussion in that it +paves the way for the efficient identification and construction +of holonomy operators of R2TC in Sec. IV. The insight gained +in this subsection will also be pivotal in the construction of TN +wave functions in Sec. III A. +As before we consider two interpenetrating square lattices +Λ1 and Λ2, and place ZN spins on the links. There are gener- +alized Pauli operators satisfying ZX = ωXZ (ω = e2πi/N) +at the links of each sublattice, which follow from the Higgsing +formula [9]: +X = e2πiA, Z = e2πiE/N. +(2.11) +We place R1TC on each of the sublattices Λ1 and Λ2, with the +star (aa(⃗ra,i)) and the plaquette (ba(⃗ra,i)) operators defined, +respectively, by (a = 1, 2) +aa,i =Za,x(⃗ra,i)Za,x(⃗ra,i−ˆx)−1Za,y(⃗ra,i)Za,y(⃗ra,i−ˆy)−1, +ba,i =Xa,x(⃗ra,i)Xa,x(⃗ra,i+ˆy)−1Xa,y(⃗ra,i)−1Xa,y(⃗ra,i+ˆx). +(2.12) +Here, the subscript i indicate the vertex of square lattice ⃗ra,i, +and the extra subscripts x, y in the X, Z operators indicates +the direction of the bond on which the operators are defined. +States in each R1TC are denoted as |ψ⟩1 and |ψ⟩2, respec- +tively. The eigenstates of X operator are X|n⟩ = ωn|n⟩. The +constraint, Eq. (2.7), implies Ax +1,i+ˆy|ψ⟩ = Ay +2,i+ˆx|ψ⟩ or, after +Higgsing, +X1,x(⃗r1,i + ˆy)|ψ⟩ = X2,y(⃗r2,i + ˆx)|ψ⟩. +(2.13) +In other words, only the following product of states in the pre- +projection Hilbert space survives the projection, +|n⟩1 ⊗ |n⟩2 +P +−→ |n⟩. +(2.14) +Besides, Eq. (2.9) states that Ex +1,i+ˆy + Ey +2,i+ˆx must be identi- +fied with Exy +i +as well, which in the ZN language means +Z1,x(⃗r1,i + ˆy)Z2,y(⃗r2,i + ˆx)|ψ⟩ = Z(⃗vi)|ψ⟩. +This constraint can be expressed in the Z-basis Z|m⟩ = +ωm|m⟩ as the projection +|m1⟩1 ⊗ |m2⟩2 +P +−→ |m1 + m2⟩. +(2.15) + +5 +In both Eqs. (2.14) and (2.15) the mapping acts only at the +Vhv sites where the gauge field constraint has been imposed. +One can think of the operator projection as follows: +P [O|Ψ⟩] = O′ [P|Ψ⟩] = O′|ψ⟩. +(2.16) +Here |Ψ⟩ and O refer to the pre-projected state and the oper- +ator, respectively, while |ψ⟩ and O′ are their post-projection +counterparts. Based on the above consideration, one can iden- +tify the operator mapping +X1,x(⃗r1,i + ˆy) +P +−→ X0(⃗vi), +X2,y(⃗r2,i + ˆx) +P +−→ X0(⃗vi), +Z1,x(⃗r1,i + ˆy)Z2,y(⃗r2,i + ˆx) +P +−→ Z0(⃗vi), +Z1,x(⃗r1,i + ˆy) +P +−→ 0, +Z2,y(⃗r2,i + ˆx) +P +−→ 0, +(2.17) +where the new subscript 0 indicates the condensed sites Vhv. +Operators at the Vvh sites are not affected by the projection +and are simply re-labeled as +X1,y(⃗r1,i) +P +−→ X2(⃗vi), +X2,x(⃗r2,i) +P +−→ X1(⃗vi), +Z1,y(⃗r1,i) +P +−→ Z2(⃗vi), +Z2,x(⃗r2,i) +P +−→ Z1(⃗vi). +(2.18) +The post-projected X, Z operators are defined with respect to +the site ⃗ri, and carry three internal indices 0,1,2. The pre- +projection plaquette operators b1(⃗r1,i) and b2(⃗r2,i), with sup- +ports on Λ1 and Λ2 respectively, survive the projection P and +become, after some re-labeling, +b1,i +P +−→ bx +i =X2(⃗vi)−1X2(⃗vi + ˆx)X0(⃗vi)−1X0(⃗vi − ˆy), +b2,i +P +−→ by +i =X1(⃗vi)X1(⃗vi + ˆy)−1X0(⃗vi)X0(⃗vi − ˆx)−1. +(2.19) +Despite the re-labeling, they are the same stabilizers from the +two underlying R1TCs. +On the other hand, the pre-projection star operators a1,i and +a2,i from Λ1 and Λ2 become zero under the projection as they +contain only Z1,x or Z2,y, but not both. To survive the projec- +tion, Z1,x(⃗r1,i + ˆy) and Z2,y(⃗r2,i + ˆx) must appear simulta- +neously, as in the following operator +ai =a1,ia−1 +1,i−ˆxa2,ia−1 +2,i−ˆy, +(2.20) +which becomes, under the projection ai +P +−→ ai, +ai =Z0(⃗vi)Z0(⃗vi−ˆx)−1Z0(⃗vi−ˆy)−1Z0(⃗vi−ˆx−ˆy) +⊗ Z2(⃗vi−ˆy)Z2(⃗vi)−2Z2(⃗vi+ˆy) +⊗ Z1(⃗vi−ˆx)Z1(⃗vi)−2Z1(⃗vi+ˆx). +(2.21) +The three post-projection stabilizers ai, bx +i , and by +i are mutu- +ally commuting, and are none other than the stabilizers of the +R2TC Hamiltonian. +So far the discussion seems limited to the recovery of +stabilizers that make up the R2TC. Importantly, though, +FIG. 2. +The superoperator � +Xx(⃗ri) (left panel) is defined at x-bond +(blue oval) with respect to ⃗ri, and � +Xy(⃗ri) (right panel) at the y-bond. +there is an additional stabilizer one can identify in the pre- +projected Hilbert space that is not given as a mere product of +ai, b1,i, b2,i. It is given by +b3,i = � +Xx(⃗vi) � +Xy(⃗vi + ˆx) � +Xx(⃗vi + ˆy)−1 � +Xy(⃗vi)−1. +(2.22) +The super-operators ˜Xx, ˜Xy are defined by +� +Xx(⃗vi) = +� +X1,x(⃗r1,i + ˆy) +�yi−y0−1� +X2,x(⃗r2,i + ˆx) +�xi−x0 +� +Xy(⃗vi) = +� +X1,y(⃗r1,i + ˆy) +�yi−y0� +X2,y(⃗r2,i + ˆx) +�xi−x0−1, +(2.23) +and illustrated in Fig. 2. The arbitrary constants x0 and y0 +are kept here to simplify certain algebraic relations among the +holonomies, and do not serve other purpose. Other stabiliz- +ers b1,i, b2,i, ai have the matching lattice gauge theory expres- +sions given in Eq. (2.10). As for b3,i, the corresponding gauge +field expression is the lattice curl +m′ +i = (A′)x +i − (A′)x +i+ˆy − (A′)y +i + (A′)y +i+ˆx, +(2.24) +where +(A′)x +i =(xi − x0)Axx +i+ˆx + (yi − y0 − 1)Axy +i , +(A′)y +i =(yi − y0)Ayy +i+ˆy + (xi − x0 − 1)Axy +i . +(2.25) +Despite the apparent complexity of the definition of b3,i, the +virtue of this choice is that it allows us to express the product +of b3,i as a product of boundary operators and thereby leads +naturally to the new holonomies, as discussed thoroughly in +Sec. +IV. In fact, there is another choice, namely b3,i = +X1,x(⃗r1,i + ˆy)X2,y(⃗r2,i + ˆx)−1, which is composed of op- +erators from both sublattices and commutes with ai, b1,i, b2,i. +Such a choice amounts to the condensation scheme adopted in +Ref. [23]. This choice, however, does not allow the transfor- +mation of the bulk product to the boundary product, hence no +new holonomy operators can be generated. +The new field m′ +i commutes with mx +i , my +i , ei and may +seem to constitute the fourth charge in the rank-2 theory, but +one can show that, after projection, b3,i becomes b3,i +P +−→ +� +bx +i+ˆy +�yi−y0 � +by +i+ˆx +�xi−x0 - a composite of existing stabiliz- +ers. The main use of identifying the stabilizer b3,i is that, +through it, we come to identify the two super- � +X operators as +given in Eq. (2.23). Naively, two copies of R1TC will gen- +erate only four holonomies, made of products of X1,i or X2,i + +6 +along horizontal and vertical directions of the torus. The exis- +tence of super-operators allows the construction of two addi- +tional holonomies, as products of � +Xx along the x- and of � +Xy +along the y-direction of the torus, and in total account for the +six holonomies generating the GSD of R2TC, Eq. (1.1). +C. +Higher-order instanton and confinement +in the rank-2 U(1) gauge theory +While this paper focuses mainly on the ZN gauge theory +on a lattice, it is instructive to touch upon the physics of U(1) +rank-2 compact gauge theory in the continuum for compari- +son. The Maxwell theory for the gauge fields of Eq. (2.6) is +given by the effective Lagrangian, +L = +� +(Exx)2 + (Eyy)2 + 2(Exy)2� +− 1 +2g B2, +(2.26) +with a quadratic dispersion ω ∼ k2 due to the fact that B is +given by second spatial derivatives, B = ∂2 +yAxx + ∂2 +xAxx − +∂x∂yAxy. For a compact gauge theory with gapless fluctua- +tions, the key question is whether the theory becomes confined +due to the proliferation of instantons. To delineate the instan- +ton event, we consider the pure gauge theory in the charge- +neutral sector ex = ey = 0 in Eq. (2.6) that allows the solu- +tion +Exx = ∂2 +yh, Eyy = ∂2 +xh, Exy = −∂x∂yh. +(2.27) +The h field can be viewed as the height operator that is +canonically conjugate with the flux [B(⃗r), h(⃗r′)] = iδ(⃗r −⃗r′) +so that the instanton operator ei2πh creates a 2π flux [42]. +Such an instanton event, once proliferated, can potentially +lead to a confined phase. The low energy effective theory of +the height field can be obtained by integrating out the gaussian +fluctuation of B, +Lh = −g(∂th)2 + (∇2h)2. +(2.28) +The quantum theory of h is defined in 2+1D space-time with +a quadratic dispersion reminiscent of the Rokhsar-Kivelson +point in 2D compact gauge theory, suggesting that the instan- +ton operator has a power-law decay correlation whose opera- +tor dimension depends on g. The relevance of 2π flux tunnel- +ing event and the proliferation of topological defects depends +on the parameters of the theory. +On the other hand, there exists another kind of higher-order +instanton events that are more relevant. For instance the in- +stanton operator ei∂xh creating a flux-dipole - a pair of 2π and +−2π fluxes spatially separated along the x-link - has the cor- +relator 5 +e−(∂xh(0)∂xh(⃗r)) +r→∞ +−−−→ Const. +(2.29) +5 The importance of flux-dipole tunneling events in governing the phase of +matter was pointed out in the context of one-dimensional dipolar boson +Hubbard model recently [43, 44]. +These higher-order instanton terms creating flux-dipole tun- +neling events display long-range order and thus can prolif- +erate. As a result, the theory would be confined due to the +proliferation of instanton-dipoles. This unique feature is due +to the fact that the dipole flux is conserved in our higher-rank +gauge theory and thus the 2π flux tunneling event must appear +in a quadrupolar process, i.e., creating a pair of opposite flux- +dipoles from the vacuum and separating them apart. The cor- +relation function in Eq. (2.29) implies the interaction between +flux-dipoles are short-ranged so they will proliferate and gap +out the low-energy modes. +D. +Conservation laws +Before the explicit construction of holonomies, it is useful +to identity the full content of conserved charges in the theory. +Physically, it is the braiding of one of these conserved charges +around the non-contractible loop of the torus that defines the +holonomy. The discussion is most conveniently carried out in +the continuum language. +The three expressions in Eq. (2.10) can be cast in the con- +tinuum as +mx = ∂xAyy − ∂yAxy, +my = ∂xAxy − ∂yAxx, +e = ∂2 +xExx + ∂2 +yEyy + ∂x∂yExy. +(2.30) +The three charge densities obey the continuity equations as +derived recently [11], +∂tmx + ∂xJxx +m + ∂yJxy +m = 0, +∂tmy + ∂xJxy +m − ∂yJyy +m = 0, +∂te + ∂2 +xJxx +e ++ ∂x∂yJxy +e ++ ∂2 +yJyy +e += 0, +(2.31) +where Jab +m and Jab +e (a, b = x, y) are symmetric rank-2 current +densities for the magnetic and electric charges, respectively 6. +By assuming vanishing currents at the boundary, one can show +that all three monopole charges are conserved: +∂t +� +edV = ∂t +� +mxdV = ∂t +� +mydV = 0. +(2.32) +In addition, we have three dipole conservation laws +∂t +� +xedV =− +� +d2r x +� +∂2 +xJxx +e ++ ∂x∂yJxy +e ++ ∂2 +yJyy +e +� += 0, +∂t +� +yedV =− +� +d2r y +� +∂2 +xJxx +e ++ ∂x∂yJxy +e ++ ∂2 +yJyy +e +� += 0, +∂t +� +(xmy + ymx)dV = − +� +d2r [x∂xJxy +m − y∂yJxy +m ] = 0. +(2.33) +6 The continuity equations derived in [11] were for the vector electric charge. +Here we are dealing with the dual theory with vector magnetic charge. The +two theories are dual to each other [9], and the continuity equations for the +quasiparticles have the same structure. + +7 +Altogether we have the conservation of three monopoles +and three dipoles. We will now construct the six magnetic and +six electric holonomies associated with the x- and y-winding +around the torus of the six conserved quantities. +III. +APPLICATIONS OF THE CONDENSATION SCHEME +Two useful applications of the condensation idea are con- +sidered. One is the explicit construction of the tensor network +wave function for the ground state of R2TC. The second is +the construction of the rank-2 version of the double-semion +model. +A. +Tensor network representation of +ZN R2TC wavefunctions +In this section, we show that ZN R2TC wave function can +be obtained by stacking two copies of ZN R1TC wave func- +tion followed by a certain isometric operation that reflects the +gauge-field constraint of the previous section. To this end, +we begin with the tensor network (TN) representation of the +R1TC ground state wave function that is composed of two +types of tensors g and T as below: +, +(3.1) +where +gm +ij = δi,jδj,m, +Tlurd = δr+u,l+d. +(3.2) +The delta function in the second line is implemented mod N. +The physical index m represents the qudit state in the Z-basis, +i.e., Z|m⟩ = ωm|m⟩, and all subscripts denote the virtual +indices of dimension N. One can easily show that g and T +tensors satisfy the following relations: +[Zn]mm′gm′ +ij = [Zn′]ii′[Zn−n′]jj′gm +i′j′, +(3.3) +[Xn]mm′gm′ +ij = [X−n]ii′[X−n]jj′gm +i′j′, +(3.4) +[Zn]ll′[Z−n]uu′[Z−n]rr′[Zn]dd′Tl′u′r′d′ = Tlurd, +(3.5) +and +[Xnl]ll′[Xnu]uu′[Xnr]rr′[Xnd]dd′Tl′u′r′d′ = Tlurd, +(3.6) +if (nr−nl+nu−nd) mod N = 0. Graphical representations +of the above equations are the following: +(3.7) +Note that the T-tensor generates the string-net configura- +tions corresponding to the domain wall configurations of the +N-state Potts model on the square lattice. For example, Z2 +R1TC wave function is depicted as a superposition of closed- +loop configurations, i.e., the domain wall of the Ising model. +Using the above relations, one can easily verify that the TN +wave function |ψ⟩, obtained by contracting all the virtual in- +dices, is the ground state of the ZN R1TC Hamiltonian, i.e., +ai|ψ⟩ = |ψ⟩ and bi|ψ⟩ = |ψ⟩, or graphically as below, +. +(3.8) +Now, we consider the square lattice (Λ1) and its dual (Λ2) +together, and accommodate the ZN R1TC wave function on +each lattice, i.e., |R1TC⟩Λ1 ⊗ |R1TC⟩Λ2. Then, there are +two types of vertices in the system: Vhv (vh) at which horizon- +tal (vertical) bonds in Λ1 and vertical (horizontal) bonds in Λ2 +cross each other. Generally, two unentangled qudits live on +the vertex Vhv ⊕ Vvh. Now we impose the following isome- +try on the two qudits labeled by quantum numbers (m1, m2) +residing on the Vvh vertices: +P m +m1m2 = δm,m1+m2, +(3.9) +where the delta function is implemented mod N, and +P m +m1m2P m′ +m1m2 = δmm′. The two-qudit state is mapped to +a single-qudit state through isometry and, furthermore, the re- +sulting TN exactly represents the ground state of ZN R2TC. +The TN wave function thus constructed is written in the Z- +basis, Z|m⟩ = ωm|m⟩, and the constraint Eq. +(2.15) is +faithfully reflected through the isometry tensor P m +m1m2 = +δm,m1+m2. +The TN representation for the R2TC ground state is illus- + +m +m +u +r=T7n +7n-n +-n +-n +n +Yni +-n +n8 +trated below: +, (3.10) +where the square lattice (dual lattice) in solid (dotted) line de- +notes Λ1 (2), and the gray square stands for the T-tensor given +in Eq. (3.2). The isometry P satisfies the relations, +[Zn]mm′P m′ +m1m2 = [Zn]m1m′ +1[Zn]m2m′ +2P m +m′ +1m′ +2, +[Xn]mm′P m′ +m1m2 = [X−n′]m1m′ +1[Xn′−n]m2m′ +2P m +m′ +1m′ +2, +(3.11) +or graphically +. +(3.12) +Using Eqs. (3.7) and (3.12), it is straightforward to derive the +following relation: +[Xn]mm′P m′ +m1m2gm1 +ud gm2 +lr += P m +m1m′ +2gm1 +ud [X−n]ll′[X−n]rr′gm2 +l′r′, +[Zn]mm′P m′ +m1m2gm1 +ud gm2 +lr += P m +m1m2gm1 +u′dgm2 +l′r [Zn]uu′[Zn]ll′ += P m +m1m2gm1 +u′dgm2 +lr′ [Zn]uu′[Zn]rr′ += P m +m1m2gm1 +ud′gm2 +lr′ [Zn]dd′[Zn]rr′ += P m +m1m2gm1 +ud′gm2 +l′r [Zn]dd′[Zn]ll′, +(3.13) +or graphically +. (3.14) +Now, using Eqs. (3.7) and (3.14), we show that the above +TN wave function, |ψ⟩, is the ground state of the ZN R2TC +Hamiltonian, i.e., bx +i |ψ⟩ = |ψ⟩, by +i |ψ⟩ = |ψ⟩ as below +, (3.15) +and ai|ψ⟩ = |ψ⟩ in the following way, +. +(3.16) +This completes the proof that the TN ground state wave +function of R2TC is given as two copies of those of R1TC +with an additional isometry. To summarize, the ground state +wave function of the R1TC is constructed using the well- +known tensors given in Eq. (3.2). Two copies of such TN +wave functions are introduced one for each of the two in- +terpenetrating square lattices. Then the isometry operation +P m +m1m2 given in Eq. (3.9) acts on half of the overlapping sites +(the Vvh sites) to reduce the two qudits (m1, m2) to a single +qudit m = m1 + m2. +There are a large number of ground states given by the GSD +formula, Eq. (1.1), and our TN construction captures only one +of them. The rest of the states can be generated by applying +holonomy operators, to be derived in the next section, to the +existing TN wave function. +B. +Twisted rank-2 gauge theory from anyon condensation +In this section, we utilize the coupled layer construction +protocol to build a twisted rank-2 gauge theory in 2D with +dipole conservation, whose gauge flux turns out to have +semionic statistics. The strategy is to combine two intersect- +ing Z2 twisted gauge theories from string-net models [24] and +implement anyon condensation to impose restricted mobility +for quasiparticle excitations. To avoid technical complexities, +the discussion in this section is limited to N = 2. +To construct the “semionic” version of Z2 gauge theory +from commuting projectors, we will need to start from triva- +lent 2D lattices such as the Fisher lattice shown in Fig. 3. A +small diamond shape is added at each vertex of the square lat- +tice so that every vertex is connected to three links with Z2 +qubits living on them. + +7u +m. +m +u +m +om +m +m. +0 +mmo +aXn +Zn +Zn +Vn'-nh +n +Zn +Th79 +FIG. 3. (a) The double semion model on the Fisher lattice. The flux +operator � +i∈Y Zi is defined on the vertex with three Z operators +on the adjoining links. The charge operators � X are defined on +the diamond and the octagon. (b) The string operator in the double +semion model (see text for details). (c) Intersecting bilayers of the +Fisher lattice, illustrated as one solid and one dashed lines. The cir- +cles are the intersection between x-link from the first layer and the +y-link from the second layer where we put a strong coupling term +−JzZ1Z2. (d) The charge operator after perturbation contains the +product of four octagon operators. +Now we begin with the conventional double semion model +that manifests a twisted Z2 gauge theory [24] on the Fisher +lattice with the Hamiltonian, +H = − +� +Y +� +i∈Y +Zi − +� +O +F0 +� +i∈O +Xi − +� +D +F1 +� +i∈D +Xi +F0 = +� +i∈VO +Si, F1 = +� +i∈VD +Si, +Si = +� +1 0 +0 i +� +(3.17) +Here Y is any set of three co-planar links entering a vertex, +O is the octagon, and D is the diamond on the Fisher lattice. +VO (VD) refers to the eight (four) links pointing outward from +an octagon (diamond), and are indicated by S in Fig. 3(a). +In the Z basis, the first term in H imposes a condition that +the parity of the gauge flux entering any vertex be even. The +remaining two terms in H provide dynamics to the gauge field +while preserving this parity at each vertex. Specifically, they +effectively bind the charge, given by � X, to the gauge flux +as measured by F0 and F1. +This charge-flux binding has important consequences for +the braiding statistics, as creating a flux excitation would si- +multaneously generate half Z2 charge. The string operator L +that creates a pair of flux is +Lm = +� +a∈red +Xa +� +a∈blue +Sa +� +a∈green +(−1)Ga, +Ga = 1 +4(1 − Za), +(3.18) +where the X operator along the red links in Fig. 3(b) flips +the spins along the string, analogous to the flux operator in +the toric code. The red string has a product of X operators +while the operators on blue (green) lines living on either side +of the string embellish it with an additional sign structure that +endows the semionic statistics between the flux, as well as +ensuring that Lm commutes with the Hamiltonian except near +the endpoints of the line. Two Lm operators anti-commute +with each other when they intersect. The extra sign structure +embellished a half-charge with the flux, and, as a result, the +flux excitation carries a half gauge charge and thus displays +semion statistics. +Now we take two intersecting layers of the Fisher lattice +with the x-link from the first layer intersecting the y-link from +the second layer and vice versa, as shown in Fig. 3(c). Links +that form the diamonds do not overlap between the layers. We +then strongly couple the qubits from distinct layers on the cir- +cled intersections in Fg. 3 through the interaction, +− JzZ1 +i Z2 +i +(3.19) +with Jz ≫ 1. In this strong coupling limit, the vertex oper- +ators on Y -junctions and the charge operators � X on dia- +monds are unaffected. However, charge operators around oc- +tagons do not commute with the Jz term, and instead, a prod- +uct of four such terms appearing at the third order in pertur- +bation theory does. In Fig. 3(d), we illustrated the new charge +operator after perturbation, composed of the product of four +octagon operators from Jz = 0. The resulting Hamiltonian +takes a form very similar to the R2TC, except that the dia- +mond plaquette terms and the quadrupolar octagon terms are +supplemented with a product of S operators over outward- +pointing edges. It can be checked that all the terms in the new +Hamiltonian commute with one another. +What is the excitation structure of this semionic version of +R2TC? The charge excitations share a similar character as the +scalar charge theory in Eq. (2.10). The novelty comes from +the vector flux excitation. For example, consider a 1D parti- +cle moving in the x-direction which creates a flux mx. Sup- +pose that the line on which this 1D excitation move intersects +that of another flux mx moving in the y- direction, which can +only hop on the even-numbered sites. The string operators +associated with the two 1D particles anti-commute. This anti- +commutation of string operators is related to the fact that two +such flux excitations can undergo a full braiding, so their mu- +tual statistics is well defined. In this case, the two types of +flux have mutual statistics θ = π, which contrasts with trivial +mutual statistics θ = 0 in the original R2TC model. + +(b) +(a) +S +S +S +X +X +S +(c) +(d) +X +S +C10 +IV. +HOLONOMY CONSTRUCTION +A. +Pre-projection holonomies +There are six pre-projection holonomies consisting of the +product of X operators. The first four holonomies are taken +directly from those of two independent R1TC’s, +W pp +1 (yi) = +Lx +� +xi=1 +X1,x(⃗r1,i), +W pp +2 (xi) = +Ly +� +yi=1 +X1,y(⃗r1,i), +W pp +3 (yi) = +Lx +� +xi=1 +X2,x(⃗r2,i), +W pp +4 (xi) = +Ly +� +yi=1 +X2,y(⃗r2,i) +(4.1) +for a torus of size Lx × Ly. Here, ⃗r1,i = ⃗vi − ˆy/2, and ⃗r2,i = +⃗vi − ˆx/2, where ⃗vi = (xi, yi). All of them commute with +the pre-projection stabilizers ai, b1,i, b2,i and b3,i introduced +in Sec. II B. The two additional holonomies are constructed +from the super-operators � +Xx and � +Xy: +W pp +5 (yi) = +lcm(Lx,N) +� +xi=1 +� +Xx(⃗vi), +W pp +6 (xi) = +lcm(Ly,N) +� +yi=1 +� +Xy(⃗ri). +(4.2) +Note that in these two cases the product over xi (yi) goes +around the torus multiple times, i.e. by +cx =lcm(Lx, N)/Lx = N/gcd(Lx, N), +cy =lcm(Ly, N)/Ly = N/gcd(Ly, N), +(4.3) +to ensure that the holonomy action on a ground state returns +another ground state with no residual excitations [10]. As a +consequence we have +[W pp +5 ]gcd(Lx,N) = [W pp +6 ]gcd(Ly,N) = 1 +while for other holonomies it is (W)N += 1. +The six +holonomies are seemingly coordinate-dependent, but this de- +pendence goes away when their actions on the ground state +are examined. We omit the proof, which is purely technical, +since the result is well-anticipated. +The holonomies we constructed can be motivated in a dif- +ferent way. One can show that the product of b1,i or b2,i over +all sites inside a rectangle S = [x1, x2] × [y1, y2] is equal +to the product of X’s or X−1’s along its four boundaries as +all terms in the interior cancel out. These boundary operators +precisely take the form of W pp +1 +through W pp +4 . In a similar +fashion, product of b3,i over a closed area leads to the cancel- +lation of all terms in the interior, leaving only the product of +super-X operators along the boundary. These boundary op- +erators motivate the W pp +5 , W pp +6 +holonomies. Collectively we +refer to the six logical operators in Eqs. (4.1) and (4.2) as +X-holonomies. +The six Z-holonomies are constructed by following a sim- +ilar reasoning. One can show that the product of ai in Eq. +(2.20) inside a closed region reduces to the boundary product, +which motivates the two Z-holonomies: +� +W pp +5 (xi) = +Ly +� +yi=1 +Z2,x(⃗r2,i + ˆx)Z2,x(⃗r2,i)−1, +� +W pp +6 (yi) = +Lx +� +xi=1 +Z1,y(⃗r1,i + ˆy)Z1,y(⃗r1,i)−1. +(4.4) +Hereafter we drop the explicit coordinate dependence from +the holonomy operators. Taking the product of (ai)yi−y′ +0 on +a closed region and extracting the boundary terms gives two +other Z-holonomies: +� +W pp +1 += +lcm(Ly,N) +� +yi=1 +�Z1,x(⃗vi), +� +W pp +2 += +Lx +� +xi=1 +�Z1,y(⃗vi). +(4.5) +Finally, the product of (ai)xi−x′ +0 gives +� +W pp +3 += +Ly +� +yi=1 +�Z2,x(⃗vi), +� +W pp +4 += +lcm(Lx,N) +� +xi=1 +�Z2,y(⃗vi), +(4.6) +Various super-Z operators appearing in the holonomies are +�Z1,x(⃗vi) =Z1,x(⃗r1,i)Z2,y(⃗r2,i +ˆx − ˆy) +⊗ +� +Z2,x(⃗r2,i)Z2,x(⃗r2,i + ˆx)−1�yi−y′ +0−2 , +�Z1,y(⃗vi) =Z1,y(⃗r1,i) +� +Z1,y(⃗r1,i)Z1,y(⃗r1,i + ˆy)−1�yi−y′ +0−1 , +�Z2,x(⃗vi) =Z2,x(⃗r2,i) +� +Z2,x(⃗r2,i)Z2,x(⃗r2,i + ˆx)−1�xi−x′ +0−1 , +�Z2,y(⃗vi) =Z2,y(⃗r2,i)Z1,x(⃗r1,i − ˆx + ˆy) +⊗ +� +Z1,y(⃗r1,i)Z1,y(⃗r1,i + ˆy)−1�xi−x′ +0−2 . (4.7) +Note that � +W1 and � +W4 involve the product of super-operators +over the circumference of the torus cy and cx times, respec- +tively. Arbitrary constants x′ +0, y′ +0 are introduced for generality +and for simplifying certain aspect of the holonomy algebra. +Invoking ZX += ωXZ, one can verify the following +Heisenberg algebra among the X- and Z-holonomies: +� +� +W pp +1 , W pp +1 +� += ωcy, +� +� +W pp +2 , W pp +2 +� += ω, +� +� +W pp +3 , W pp +3 +� += ω, +� +� +W pp +4 , W pp +4 +� += ωcx, +� +� +W pp +5 , W pp +5 +� += ωcx, +� +� +W pp +6 , W pp +6 +� += ωcy. +(4.8) +The commutator here means [A, B] = ABA−1B−1. In addi- +tion, the following set of holonomies show nontrivial commu- +tation:� +� +W pp +1 , W pp +5 +� +=ωcxcy[y′ +0−y0− 1 +2 (N−gcd(Ly,N))] +� +� +W pp +2 , W pp +6 +� +=ωcy[y′ +0−y0+ 1 +2 (N−gcd(Ly,N))] +� +� +W pp +3 , W pp +5 +� +=ωcx[x′ +0−x0+ 1 +2 (N−gcd(Lx,N))] +� +� +W pp +4 , W pp +6 +� +=ωcxcy[x′ +0−x0− 1 +2 (N−gcd(Lx,N))]. +(4.9) + +11 +However, the following choice removes the non-trivial phase +factors among them 7, +y′ +0 =y0 + 1 +2 +� +N − gcd(Ly, N) +� +, +x′ +0 =x0 + 1 +2 +� +N − gcd(Lx, N) +� +, +(4.10) +and the nontrivial Heisenberg algebra is spanned entirely by +Eq. (4.8). The holonomies of the R2TC are then obtained by +projection of the pre-projection holonomies constructed here. +B. +Post-projection holonomies +As one can see from the projection schemes, Eqs. (2.17)- +(2.18), the X-operators remain intact through the projection +except some re-labeling. The pre-projection X-holonomies +of Eqs. (4.1) and (4.2) become, after re-labeling, +W1 = +Lx +� +xi=1 +X0(⃗vi), +W2 = +Ly +� +yi=1 +X2(⃗vi) +W3 = +Lx +� +xi=1 +X1(⃗vi), +W4 = +Ly +� +yi=1 +X0(⃗vi) +W5 = +lcm(Lx,N) +� +xi=1 +� +X1(⃗vi + ˆx) +�xi−x0� +X0(⃗vi) +�yi−y0−1, +W6 = +lcm(Ly,N) +� +yi=1 +� +X2(⃗vi + ˆy) +�yi−y0� +X0(⃗vi) +�xi−x0−1, +(4.11) +after the projection. +At first it seems the number of dis- +tinct holonomy actions is N 4gcd(Lx, N)gcd(Ly, N) since +W1 through W4 has (W)N = 1 but +(W5)gcd(Lx,N) = (W6)gcd(Ly,N) = 1, +at odds with the GSD formula in Eq. (1.1). A delicate con- +sideration is required to see that the number of independent +actions among W1 and W4, which are both products of X0’s, +is not N 2 but Ngcd(Lx, Ly, N). +One begins with labeling the holonomy (W1)n1(W4)n4 by +(n1, n4) and invoking the identity 8 +(W1)Ly|GS⟩ = (W4)Lx|GS⟩. +(4.12) +7 This has been the sole purpose of keeping the arbitrary constants in the +definition of super-operators. +8 The proof of the identity is simple. Since W1(y) = �Lx +xi=1 X0(xi, y) +and W4(x) = �Ly +yi=1 X0(x, yi), it then follows that �Ly +y=1 W1(y) = +�Lx +x=1 W4(x). Furthermore, since the actions of W1(y) and W4(x) do +not actually depend on y and x, respectively, when acting on the ground +states, we obtained the claimed relation. +This implies the equivalence relation (n1, n4) ∼ (n1 + +Ly, n2 − Lx) among the holonomies. We need to carefully +figure out how the points (n1, n4) become equivalent by Eq. +(4.12) when the ZN nature is considered at the same time. +Invoking the two winding numbers cx, cy defined in Eq. +(4.3), +(W1)cyLy|GS⟩ = (W4)cyLx|GS⟩ = |GS⟩, +(W1)cxLy|GS⟩ = (W4)cyLx|GS⟩ = |GS⟩. +(4.13) +Applying the Euclidean argument for identifying the gcd of +two integers, we conclude +(W1)Ny|GS⟩ = (W4)Nx|GS⟩ = |GS⟩ +(4.14) +where +Nx ≡gcd(cyLx, N), +Ny ≡gcd(cxLy, N). +(4.15) +From Eqs. (4.12) and (4.15) we deduce the equivalence rela- +tion +n1 ∼ n1 + Ly ∼ n1 + Ny, +n2 ∼ n2 + Lx ∼ n2 + Nx. +(4.16) +Invoking the Euclidean argument again, the number of in- +equivalent integers n1 for fixed n4 becomes gcd(Ly, Ny), and +the number of inequivalent (n1, n4) equals Nxgcd(Ly, Ny). +It can be simplified further to +Nxgcd(Ly, Ny) = gcd (Lxcy, N) gcd (Ly, N) += Ngcd (Lx, Ly, N) , +(4.17) +by employing several number-theoretic identities +gcd(a, gcd(b, c)) = gcd(gcd(a, b), c) = gcd(a, b, c), +mgcd(a, b) = gcd(ma, mb). +(4.18) +The number of independent holonomy actions among W1 and +W4 is Ngcd(Lx, Ly, N). +The GSD formula in Eq. +(1.1) +breaks down to +N 2 · (Ngcd(Lx, Ly, N)) · gcd(Lx, N) · gcd(Ly, N). +(4.19) +Here, +the +first +N 2 +are +coming +from +W2 +and +W3, +Ngcd(Lx, Ly, N) from W1 and W4, and gcd(Lx, N) · +gcd(Ly, N) from W5 and W6, respectively. +As for the Z-holonomies, projection of the pre-projection + +12 +FIG. 4. Pictorial representation of the holonomies. (first row) Six X-holonomies as creation/annihilation of e dipole-antidipole pairs oriented +either horizontally or vertically (W1 through W4), and of e monopole-antimonopole pairs (W5 and W6). (second row) Six Z-holonomies as +creation/annihilation of mx or my monopole-antimonopole pairs (� +W1 through � +W4), and of m dipole-antidipole pairs (� +W5 and � +W6 Monopole +braiding processes are accompanied by the motion of auxiliary dipoles to preserve the total dipole moment, but they are omitted from the figure +for the sake of clarity. +Z-holonomies of Eqs. (4.4)-(4.6) leads to +� +W1 = +lcm(Ly,N) +� +yi=1 +Z0(⃗vi − ˆy) +� +Z1(⃗vi)Z1(⃗vi + ˆx)−1�yi−y0, +� +W2 = +Lx +� +xi=1 +Z2(⃗vi) +� +Z2(⃗v)Z2(⃗vi + ˆy)−1�yi−y0, +� +W3 = +Ly +� +yi=1 +Z1(⃗vi) +� +Z1(⃗vi)Z1(⃗vi + ˆx)−1�xi−x0, +� +W4 = +lcm(Lx,N) +� +xi=1 +Z0(⃗vi − ˆx) +� +Z2(⃗vi)Z2(⃗vi + ˆy)−1�xi−x0, +� +W5 = +Ly +� +yi=1 +Z1(⃗vi + ˆx)Z1(⃗vi)−1, +� +W6 = +Lx +� +xi=1 +Z2(⃗vi + ˆy)Z2(⃗vi)−1. +(4.20) +One can check the following non-trivial commutators among +the post-projection holonomies. +� +� +W1, W1 +� += ωcy, +� +� +W4, W4 +� += ωcx, +� +� +W2, W2 +� += ω, +� +� +W3, W3 +� += ω, +� +� +W5, W5 +� += ωcx, +� +� +W6, W6 +� += ωcy. +(4.21) +The commutator here means [A, B] = ABA−1B−1. There +exist four more nontrivial commutations relations: +� +� +W1, W5 +� +=ωcxcy[y′ +0−y0− 1 +2 (N−gcd(Ly,N))] +� +� +W2, W6 +� +=ωcy[y′ +0−y0+ 1 +2 (N−gcd(Ly,N))] +� +� +W3, W5 +� +=ωcx[x′ +0−x0+ 1 +2 (N−gcd(Lx,N))] +� +� +W4, W6 +� +=ωcxcy[x′ +0−x0− 1 +2 (N−gcd(Lx,N))], +(4.22) +which is nothing but the projection of Eq. (4.9). Hence, apply- +ing the condition in Eq. (4.10) removes this nontrivial phase +factors as well. +One can read off the GSD from the Heisenberg al- +gebra of the holonomies. +For instance W1 acting on +a ground state changes the eigenvalues of � +W1 by ωcy, +generating in total N/cy += gcd(Ly, N) distinct ground +states. +Naively applying the reasoning to the first two +pairs of commutators (1 and 4) gives the GSD equal to +(N/cx)(N/cy) += +gcd(Lx, N)gcd(Ly, N), the next two +pairs (2 and 3) yields N 2, and the final two pairs (5 +and 6) yields another gcd(Lx, N)gcd(Ly, N). +In total, +this gives the GSD count N 2[gcd(Lx, N)gcd(Ly, N)]2 that +is less than the correct GSD formula, Eq. +(1.1), by +gcd(Lx, N)gcd(Ly, N)/Ngcd(Lx, Ly, N). In other words, +the holonomies constructed above underspans the space of +ground states. +The deficiency comes from the fact that � +W4 given in +Eq. (4.20) is not the most minimal choice of the holonomy. +The correct holonomy expression can be found by referring to + +Wi +W2 +W3 +W4 +Ws +W6 +le +ellé +é +e +e +le el +el +le +e +e +e +aTe +e +mx +omy +T4 +mx +mx +★ +my +my +my +mx +W1 +W +W +W +W +W +4 +5 +613 +Ref. [10]: +� +W min +4 += +nxLx +� +xi=1 +Z0(⃗vi − ˆx) +� +Z2(⃗vi)Z2(⃗vi + ˆy)−1�xi−x0, +⊗ +nyLy +� +yi=1 +Z0(⃗vi − ˆy) +� +Z1(⃗vi)Z1(⃗vi + ˆx)−1�yi−y0, +(4.23) +where the integers nx, ny are given by +nx = +gcd(Lx, N) +gcd(Lx, Ly, N), +ny =lcm(Lx, gcd(Ly, N)) + kN +Ly +. +(4.24) +Here k is a minimal integer that makes ny an integer [10]. +With the new definition of � +W4 → � +W min +4 +we obtain a new +commutator +� +� +W min +4 +, W4 +� += ωnx. +(4.25) +The GSD coming from this sector equals N/nx += +Ngcd(Lx, Ly, N)/gcd(Lx, N) and indeed, we recover the +full GSD simply from the Heisenberg algebra, with the mod- +ified � +W4. Replacing � +W4 by � +W min +4 +gives us an orthogonal set +of six X-holonomies {W1, · · · , W6} and six Z-holonomies +{� +W1, · · · , � +W6} that fully span the ground states of R2TC. +In making physical interpretations of the � +W4 holonomy, +though, we will continue to adopt the simpler (albeit slightly +incorrect) representation as the horizontal braiding of my +quasiparticle as shown in Fig. 4. The interpretation of � +W min +4 +involves a mix of the horizontal braiding of my and the ver- +tical braiding of mx, as can be seen from its definition in +Eq. (4.23). +With the explicit construction of the holonomies, we can +check the quantum numbers of the TN ground state wave +function we have constructed in Sec. III A. Following the sim- +ilar procedure as in Eqs. (3.15) and (3.16), one can verify that +our TN wave function |ψ⟩ on the torus is the simultaneous +eigenstate of the four X-holonomies W1 through W4, as well +as two Z-holonomies � +W5 and � +W6, with eigenvalue +1. The +remaining six holonomies, � +W1 through � +W4 and W5, W6, then +act to shift the ground state into orthogonal ground states. One +can also construct a TN wave function for the eigenstate of all +six of the X-holonomies W1 through W6, but it requires a +‘double layer structure’ of TN that goes beyond the present +construction and will be presented elsewhere [45]. +C. +Physical interpretation of the holonomies +It is well known that the X- and Z-holonomies in the +R1TC has a concise physical picture as the creation and sub- +sequent annihilation of a pair of ee or mm (bar denotes the +anti-particle) anyons after one anyon is braided around one +of the non-contractible paths of the torus. +A total of four +holonomies form two conjugate pairs and span the N 2 de- +generate ground states. To account for the GSD of R2TC, +which reaches the maximum value of N 6, one requires a to- +tal of twelve holonomies breaking up into two groups. Six of +them bear obvious physical interpretations as the braiding of +mx, my, e particles around either of the two circumferences +of the torus. We provide the physical interpretations of the +remaining six holonomies. +Each action of X-holonomies corresponds to the winding +of the three electric quantities that are conserved. Figure 4 +illustrates these processes. The physical action of W1 (W2) +among the X-holonomies in Eq. +(4.11) is to create a y- +oriented e-dipole and its anti-dipole, then to move one of the +dipoles along the horizontal (vertical) non-contractible path of +the torus. For W3 (W4), it is x-oriented e-dipole braided hori- +zontally (vertically). The W5 (W6), on the other hand, moves +the e monopole horizontally (vertically). Note that an auxil- +iary dipole is attached to the e monopole during its adiabatic +motion, to ensure the total dipole moment conservation in the +process, and disappears at the end of completing the loop. We +omit the auxiliary dipoles from Fig. 4 for simplicity. +Each action of X-holonomies corresponds to the winding +of the three magnetic quantities that are conserved. The ac- +tion of the first four Z-holonomies in Eq. (4.20) is to create +a monopole and anti-monopole pair of either mx or my and +braid them. Specifically, � +W1, � +W2 (� +W3, � +W4) braid mx (my) +along the y- and x-oriented non-contractible loops. To en- +force the total dipole moment ymx + xmy = 0, some aux- +iliary dipoles are attached during the vertical motion of mx +as well as the horizontal motion of my [11]. The last two +Z-holonomies, � +W5 and � +W6, correspond to the winding of m- +dipole along the y and x non-contractible loops of the torus. +The list of holonomies and their physical interpretations are +summarized in Table I. +(W1, � +W1) +(e-dipole, h) +(mx-monopole, v) +(W2, � +W2) +(e-dipole, v) +(my-monopole, h) +(W3, � +W3) +(e-dipole, h) +(my-monopole, v) +(W4, � +W4) +(e-dipole, v) +(mx-monopole, h) +(W5, � +W5) (e-monopole, h) +(my-dipole, v) +(W6, � +W6) (e-monopole, v) +(mx-dipole, h) +TABLE I. (left) Pair of holonomies (logical operators) with non- +trivial commutation relations. (middle) nature of e-excitations and +the direction of braiding associated with a given holonomy W. +(right) nature of m-excitations and the direction of braiding asso- +ciated with a given holonomy � +W. (h=horizontal, v=vertical) +D. +Field-theoretic derivation of the holonomies +The holonomy construction thus far proceeded from a +known microscopic Hamiltonian, i.e., R2TC model whose +quasiparticle excitations are well-explored. Historically, the +holonomies engendered by the Wilson line operators manifest +the global flux sectors to which the ground state on a torus +belongs. Building on this line of thinking, we show how to + +14 +obtain the Wilson operators pertinent to the R2TC from the +underlying rank-2 gauge theory. +For higher-rank gauge theories, the Wilson operators cre- +ating immobile quasiparticle excitations turn out to be richer +and more diverse than in the conventional ZN gauge theory +for the following reasons. 1) Due to the restricted mobility of +the quasiparticles, some of the Wilson lines need to be straight +and geometrically oriented in a specific direction[46, 47]. 2) +There might exist other Wilson operators defined on a non- +contractable manifold, such as membrane, cage, or fractal, +that are responsible for the holonomies of higher-rank gauge +theories [47–49]. 3) Different Wilson operators that are paral- +lel to each other may not render the same value, as opposed to +the conventional ZN gauge theory whose Wilson line opera- +tors are invariant under translation. For higher-rank gauge the- +ory, the dipole and quadruple moments transform nontrivially +under translation, and so does the global flux sector. Conse- +quently, two parallel flux lines might return different values. +Recall that in the usual 2D ZN gauge theory, the magnetic +flux is given by m = ∂xAy − ∂yAx and the total flux on the +half cylinder A with boundaries at x = x0 and x = xn is +characterized by parallel Wilson line operators +� +mdV = +� +Ay(xn, y)dy − +� +Ay(x0, y)dy = 0, +with the integral +� +going around the full circumference of the +cylinder. The net flux condition ( +� +mdV = 0) implies that the +two parallel Wilson lines render the same value. Since the two +Wilson lines are spatially separated while the Hamiltonian is +local, each +� +Ay(x, y)dy must commute with all local terms +in the Hamiltonian and can be treated as a global flux operator +that characterizes the holonomy. One obtains another Wilson +line operator along the y-direction from the charge sector, i.e. +� +Ey(x, y)dy. These two comprise all possible Wilson lines +along the y-loop. +Now we apply this protocol to R2TC. Begin with the def- +inition of three monopole charges given in Eq. (2.10) in the +continuum limit, +mx = ∂xAyy − ∂yAxy, +my = ∂xAxy − ∂yAxx, +e = ∂2 +xExx + ∂2 +yEyy + ∂x∂yExy. +(4.26) +As noted in Sec. II D, the magnetic charges mx, my demon- +strate a number of conservation laws +� +mxdV = +� +mydV = +� +(xmy + ymx)dV = 0. (4.27) +The first two yields +� +mxdV = +� +Ayy(xn, y)dy − +� +Ayy(x0, y)dy = 0, +� +mydV = +� +Axy(xn, y)dy − +� +Axy(x0, y)dy = 0. +(4.28) +Following the aforementioned argument, one can define two +Wilson line operators, +W2(x) = +� +Ayy(x, y)dy, +W4(x) = +� +Axy(x, y)dy. +(4.29) +Due to the flux conservation law, Eq. (4.28), they are both +uniform along the x-coordinate: ∂xW2(x) = ∂xW4(x) = 0. +The subscripts 2, 4 are intended to match the definitions of +post-projection Wilson operators in Eq. (4.11) 9 +In addition, we have +� +A +(ymx + xmy)dV = 0 = +� +yAyy(xn, y)dy +− +� +yAyy(x0, y)dy + +� xn +x0 +�� +Axy(x, y)dy +� +dx += +� +yAyy(xn, y)dy − +� +yAyy(x0, y)dy + (xn − x0)W4. +(4.30) +In arriving at the last equality we used the fact that the Wilson +line operator W4 = +� +Axy(x, y)dy is uniform in x. We arrive +at another Wilson line operator, +W6(x) = +� +yAyy(x, y)dy + xW4, ∂xW6(x) = 0, (4.31) +which matches the definition of W6 in Eq. (4.11) after Higgs- +ing. +As the theory is rotationally symmetric, the other set of Wil- +son line operators follow as integrals along the x-loop: +W1 = +� +Axy(x, y)dx, +W3 = +� +Axx(x, y)dx, +W5 = +� +xAxx(x, y)dx + yW1, +(4.32) +with matching definitions in Eq. (4.11) after Higgsing. Their +coordinate independence follows readily. +The previous holonomies W1 through W6 were derived on +the basis of conservation laws of the magnetic charges. Al- +ternatively, the holonomies can be derived from the electric +charge conservation, +e = ∂2 +xExx + ∂2 +yEyy + ∂x∂yExy, +� +e dV = +� +xe dV = +� +ye dV = 0, +(4.33) +and hence +� +A +e dV = +� +∂xExx(x0, y)dy − +� +∂xExx(xn, y)dy = 0. +(4.34) +9 Note that we use the same symbol W1 · · · W6 for the holonomies in +both continuum and the ZN theories although, strictly speaking, the ZN +holonomies are obtained by raising the continuum holonomies to an expo- +nential. + +15 +This yields the first holonomy +� +W5(x) = +� +∂xExx(x, y)dy, ∂x� +W5(x) = 0. +(4.35) +From the other two conservation laws we find +� +ye dV = +� +(y∂yExy + y∂xExx)(xn, y)dy +− +� +(y∂yExy + y∂xExx)(x0, y)dy, +� +A +xe dV = +� +Exx(x0, y)dy − +� +Exx(xn, y)dy ++ (xn − x0)� +W5. +(4.36) +We arrive at two additional Wilson line operators +� +W1 = − +� +(y∂yExy + y∂xExx)dy += +� +(Exy − y∂xExx)dy, +� +W3 = − +� +Exxdy + x� +W5. +(4.37) +The other three Wilson line operators are obtained from rota- +tional symmetry: +� +W2 = − +� +Eyydx + y� +W6, +� +W4 = +� +(Exy − x∂yEyy)dy, +� +W6 = +� +∂yEyydx. +(4.38) +Coordinate independence of all Wilson operators can be ver- +ified easily. After Higgsing, � +W1 through � +W6 match the six +Z-holonomies of Eq. (4.20). Physical interpretation of the +holonomy operators has been given in Table I. +For completeness we briefly mention that in a theory with +vector-electric and scalar-magnetic charges such that +ex = ∂xExx + ∂yExy +ey = ∂xExy + ∂yEyy +m = ∂2 +xAyy + ∂2 +yAxx − ∂x∂yAxy, +(4.39) +we can construct the relevant holonomies based on a different +set of conservation laws +� +exdV = +� +eydV = +� +(yex − xey)dV = 0, +� +mdV = +� +xmdV = +� +ymdV = 0. +(4.40) +As the derivation in this subsection clearly shows, the con- +struction of holonomies are firmly rooted in the conservation +laws such as Eqs. (4.27) and (4.33). The existence of dipole- +like conservations in addition to the usual charge conserva- +tions for mx, my, e monopoles plays a crucial role in con- +structing the full set of holonomies for the rank-2 gauge theory +as well as its Higgs descendant, which is the R2TC. We sus- +pect that a similar scheme can be exploited for the holonomy +construction in other rank-2 gauge theories. +FIG. 5. (a) Braiding charge e around the flux m in the conventional +ZN gauge theory. The trajectory of the braiding loop corresponds to +the total flux inside the enclosed area. (b) Braiding charge e around +the flux mx or my in R2TC. The trajectory of the braiding loop cor- +responds to the total dipolar flux inside the enclosed area. +E. +Understanding the position-dependent braiding +The seemingly puzzling feature of R2TC was the position- +dependent statistical phase obtained when one quasiparticle is +braided around another [9–11]. While various elaborate argu- +ments for why this should be so has been given already [9–11], +it turns out the field-theoretic holonomies just constructed can +provide a simple picture for it. +To do so, we first review how the adiabatic braiding process +relates to the statistical phase. We begin with the prominent +ZN gauge theory example where the charge e and flux m have +nontrivial statistics. Creating a pair of m flux excitations is +implemented at the two endpoints of an open string ei +� x +0 Exdx. +To braid the charge around the flux, we create a pair of charge +(e) and anti-charge (e) connected by an open string, wind the +e particle around m and annihilate it with the anti-charge as +shown in Fig. 5. The trajectory of the e particle is associated +with the Aharonov-Bohm (AB) phase exp +� +i +� ⃗A · d⃗r +� +which +corresponds to the total flux +� +mdV (m = ∇ × A) inside the +area enclosed by the loop. As a result, the braiding of charge +excitation creates a flux loop that measures the total flux inside +so their braiding phase is just the AB phase. +Now let us go back to the R2TC theory with vector- +magnetic and scalar-electric charges as in Eq. (2.6). +The +flux 10 mx or my excitations are created by open-string op- +erators such as +� +W open +1 +∼ exp +� +i +� ym +0 +(Exy − y∂xExx) dy +� +or +� +W open +3 +∼ exp +� +i +� ym +0 +(x∂xExx − Exx) dy +� +, +respectively. They are none other than open-ended versions of +the holonomies constructed in Sec. +IV D and have physical +10 Here we use the word magnetic ‘flux’ interchangeably with the magnetic +‘charge’. + +(a) +(b) +0 +y=yi +m +or m +:=2 +P [xA x + yA*] dx +DA*dx16 +interpretations of creating a mxmx or a mymy pair separated +along the vertical direction as shown in Fig. 5. +To braid the m flux, we create a pair of charge e and anti- +charge ¯e connected by an open string shown as the horizontal +blue segment in Fig. 5(b), wind the e particle around mx or +my as shown by two horizontal dashed lines in Fig. 5(b), and +annihilate it with the anti-charge. The trajectory of the e par- +ticle is associated with the phase factor +W5 ∼ ei +� +[(xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2)]dx. +As one can see from Table I, W5 is associated with the hori- +zontal braiding of e particle. +For simplicity, we choose the braiding trajectory consisting +of two parallel lines along the x-direction above and below +the m flux, i.e. at y = y1 and y = y2, y1 < ym < y2, and +ym indicating the y-coordinate of the m flux. We can further +simplify the braiding operator as +ei +� � +(xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2) +� +dx += ei +� � +(xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2) +� +dx +× ei +� y2 +y1 +� +(yAyy+xAxy)(x1,y)−(yAyy+xAxy)(x2,y) +� +dy += ei +� +(ymx+xmy)dV . +(4.41) +In the second line we inserted some y-oriented integrals that +cancel each other due to the periodic boundary condition and +x2 = x1 + Lx 11. Now one can understand the braiding oper- +ation as the line integral of the vector field +(xAxx + yAxy, xAxy + yAyy). +(4.42) +The third line in Eq. (4.41) follows from Stokes’ theorem and +the definition of mx, my in Eq. (4.26). It shows that the braid- +ing operation measure not the flux, but the ‘dipolar flux’ that +depends on the x-position of my and the y-position of mx that +the e particle braids around. The statistical phase becomes ac- +cordingly position-dependent. Dipolar braiding among other +quasiparticles can be understood in similar ways. The deriva- +tion of dipolar braiding statistics in terms of field-theoretic +Wilson lines given here has some overlap with earlier con- +sideration [9, 11] of the dipolar braiding, but here we give a +more clarified picture of how this seeminingly peculiar braid- +ing statistics arises rather naturally in rank-2 gauge theories. +It also suggests that the dipolar braiding phase is not unique to +R2TC, but may be a general feature of rank-2 gauge theories +and its Higgsed descendants. +To put it in broader perspective, we comment that a +position-dependent braiding process is also present elsewhere. +Indeed, while typically not emphasized, even in Wen’s Z2 Pla- +quette 2D model [16] where there is a single type of stabilizer +and, according to the terminology used here, one quasipar- +ticle species whose self-statistics depends on its initial posi- +tion. Another 2D topologically ordered example, one more +11 We assume periodicity of the fields: Axy(x1 + L, y) = Axy(x1, y), etc. +Further, +� y2 +y1 Axy(x, y)dy = 2πZ/Lx is assumed. +Z1 +Z2 +Z0 +X1 +X2 +X0 +𝔞 +† +† +† +† +𝔟x +𝔟y +† +† +† +†† +† +Z +X +† +† +ac0 +† +† +bc2 +a) +b) +FIG. 6. Graphical representations of (a) the ac0 and bc2 operators in +the R1TC Hamiltonian Eq. (5.1) and (b) the a, by, and bx operators +in the R2TC Hamiltonian Eq. (5.12). The disks are color-coded to +represent Xi and Zi operators, according to the legend. Furthermore, +disks with a † represent the Hermitian conjugate of the corresponding +operator. +complicated than Wen’s plaquette model yet simpler than the +R2TC, is the model considered by Delfino et al. in Ref. 17. In +these 2D topologically ordered examples, a general reason for +position-dependent braiding is that lattice translations induce +nontrivial automorphisms on the anyon lattice [50]. Conse- +quentially the anyon types are labeled by their position which +causes their braiding to become position-dependent [10]. In +3D, position-dependent braiding has been discovered in frac- +ton models [47, 51]. In particular, for 3D twisted fracton the- +ory, the flux excitations denoted as lineons, with restricted mo- +bility along 1D lines, only exhibit nontrivial braiding statistics +between the lineons on adjacent planes. That says that if we +shift the braiding trajectory of the lineon between the layers, +the resultant Berry phase from statistical angles can change. +V. +GENERALIZED SYMMETRIES +The holonomies constructed in Sec. IV are a piece of a +more general structure present in the R2TC: its generalized +symmetries. +The generalized symmetries of some rank-2 +gauge theories have been discussed previously in the litera- +ture [12, 14, 52, 53]. Given the rich properties of the R2TC, +the exactly solvable point of scalar charge rank-2 ZN gauge +theory, it is interesting to wonder what its generalized symme- +tries are. In this section, we will identify its symmetries and +discuss them in the context of spontaneous symmetry break- +ing and ’t Hooft anomalies. We will consider the general N +case. This requires defining a branching and framing structure +of the lattice, which we review in appendix Sec. A. + +17 +A. +Reviewing the 1-form symmetries of the R1TC +Let us first review the generalized symmetries in the ZN +R1TC on a spatial square lattice. The (2 + 1)D ZN R1TC +Hamiltonian can be written as +H = − +� +c0 +Ac0 − +� +c2 +Bc2, +Ac0 = 1 +N +N +� +j=1 +(ac0)j, +Bc2 = 1 +N +N +� +j=1 +(bc2)j, +(5.1) +where ac0 and bc2 are the star and plaquette operators +ac0 = +� +c1∈δc0 +Zc1, +bc2 = +� +c1∈∂c2 +Xc1. +(5.2) +We denote the square lattice’s sites as c0, its edges as c1, and +its plaquettes as c2. In the definitions of ac0 and bc2, δc0 de- +notes the coboundary of c0—an oriented sum of edges whose +boundary includes c0—and ∂p denotes the oriented boundary +of c2. The precise definitions of δ and ∂ are given by Eqs. (A3) +and (A2), respectively. Graphical representations of ac0 and +bc2 are shown in Fig. 6a, from which it is clear that they com- +mute for all c0 and c2. We note that these expressions for ac0 +and bc2 are equivalent to Eq. (2.12). +There are two independent operators that commute with the +R1TC Hamiltonian Eq. (5.1), each corresponding to a symme- +try. First consider the unitary +U(γ) = +� +c1∈γ +Xc1, +(5.3) +where γ is an oriented closed loop made of the lattice’s edges +(e.g., γ1 and γ2 in Fig. 7) and Xc1 satisfies X−c1 = X† +c1. +U(γ) trivially commutes with bc2 for all γ and c2. Further- +more, U(γ) commutes with ac0 since for each site c0, γ is +made up of an even number of elements of δc0 with relative +orientations such that all phases ei2π/N cancel. Therefore, +[U(γ), H] = 0 for all loops γ. Next, consider the unitary +�U(ˆγ) = +� +ˆc1∈ˆγ +Z∗ ˆc1, +(5.4) +where ˆγ is now an oriented closed loop made of the dual lat- +tice’s edges (e.g., ˆγ1 and ˆγ2 in Fig. 7), ˆc1 is a dual lattice edge, +and ∗ ˆc1 is the edge of the direct lattice that crosses ˆc1 (up to +a differing sign, see Eq. (A5)). �U(ˆγ) trivially commutes with +ac0 for all ˆγ and c0. Furthermore, �U(ˆγ) commutes with bc2 +since for each plaquette c2, ˆγ is made up of an even number of +elements of ∂c2 with relative orientations such that all phases +ei2π/N cancel. Therefore, [�U(ˆγ), H] = 0 for all loops ˆγ. +Since U and �U commute with H, and since they transform +the qubits nontrivially, they correspond to symmetries. In- +deed, they generate the transformations +U(γ) Zc1 U †(γ) = ω#(c1,γ) Zc1, +(5.5) +�U(ˆγ) Xc1 �U †(ˆγ) = ω−#(c1,ˆγ) Xc1, +(5.6) +γ1 +γ2 +̂γ1 +̂γ2 +Z +X +† +† +† +† +† +† +† +† +† +FIG. 7. The symmetry operators U(γ) and �U(ˆγ) of the R1TC act on +closed loops of the direct and dual lattice. Examples of U(γ) (see +Eq. (5.3)) acting on loops of the direct lattice γ1 and γ2 are shown +in green while examples of �U(ˆγ) (see Eq. (5.4)) acting on loops of +the dual lattice ˆγ1 and ˆγ2 are in red. γ1 and ˆγ1 are contractible loops +while, assuming periodic boundary conditions, γ2 and ˆγ2 are non- +contractible. +where ω = ei2π/N +and, +for instance, +#(c1, γ) is the +signed +intersection +number +of +c1 +and +γ. +Because +[U(γ)]N = [�U(ˆγ)]N = 1, +they are the generators of a +ZN × ZN symmetry. However, this is not quite an ordinary +global symmetry since U and �U act on closed loops instead of +the entire lattice. Instead, they correspond to non-topological +ZN 1-form symmetries. Physically, this symmetry reflects the +lack of dynamics of e and m anyons in the R1TC is absent +throughout the rest of the deconfined phase of ZN gauge the- +ory. +In the ground state sub-Hilbert space, the operators ac0 and +bc2 obey the constraints ⟨ac0⟩gs = 1 and ⟨bc2⟩gs = 1, where +⟨ ⟩gs denotes the expectation value with respect to the ground +states. Consequentially, when γ and ˆγ are contractible loops, +⟨U(γ)⟩gs = 1 and ⟨�U(ˆγ)⟩gs = 1, which follows from +U(γ = ∂M) = +� +c2∈M +bc2, +(5.7) +�U(ˆγ = ∂ ˆ +M) = +� +ˆc2∈ ˆ +M +a† +∗ ˆc2. +(5.8) +In fact, U(γ) and �U(ˆγ) are so-called topological operators +in the ground state sub-Hilbert space, since their vacuum ex- +pectation values depend only on the topology—the homology +class—of γ and ˆγ, respectively. In other words, in the ground +state sub-Hilbert space, the symmetry operators are nontrivial +only when γ and ˆγ are noncontractible loops. Furthermore, +one can choose γ and ˆγ such that U and �U are the R1TC +“holonomies” discussed in Sec. IV A. +In the ground state sub-Hilbert space, Xc1 and Zc1 are not +allowed operators since they excite e and m anyons, respec- + +18 +tively, violating the ⟨ac1⟩ = 1 and ⟨bc2⟩ = 1 constraints. The +allowed operators are, instead, U(γ) and �U(ˆγ). The afore- +mentioned generalized ZN × ZN symmetry transformations, +Eqs. (5.5) and (5.6), in the ground state sub-Hilbert space are +replaced with +U(γ) �U(ˆγ) U †(γ) = ω#(ˆγ,γ) �U(ˆγ), +(5.9) +�U(ˆγ) U(γ) �U †(ˆγ) = ω−#(γ,ˆγ) U(γ). +(5.10) +These now correspond to ZN 1-form—Z(1) +N —symmetries +since their symmetry operators are topological operators sup- +ported on codimension 1 closed subspaces and their charged +operators are supported on 1-dimensional closed subspaces. +In fact, this Z(1) +N × Z(1) +N symmetry is also a symmetry [54] of +the topological quantum field theory description of the R1TC +ground states [55]. Unlike the non-topological ZN × ZN 1- +form symmetry of Eqs. (5.5) and (5.6), the Z(1) +N × Z(1) +N sym- +metry exists as an emergent symmetry in the ground state sub- +Hilbert space throughout the entire deconfined phase of ZN +gauge theory [56, 57]. +Just like ordinary global symmetries, 1-form symmetries +can spontaneously break [26, 58]. The order parameter of +a 1-form symmetry spontaneous breaking is the vacuum ex- +pectation value of its charged operator supported on a con- +tractible loop. +Recall that ⟨U(γ)⟩gs = 1 and ⟨�U(ˆγ)⟩gs = 1 +when γ and ˆγ are contractible loops. +Since �U is charged +under the Z(1) +N +symmetry generated by U (see Eq. (5.9)) +and vice versa, the R1TC ground states spontaneously break +the Z(1) +N × Z(1) +N symmetry. This reproduces the well known +property that there is a ground state degeneracy depending +on the topology—the 1st cohomology—of the spatial lattice. +In fact, the Z(1) +N × Z(1) +N +symmetry is anomalous, meaning +both Z(1) +N symmetries cannot be simultaneously gauged. The +ground state degeneracy (GSD) arising when this anomalous +Z(1) +N × Z(1) +N symmetry is spontaneously broken is GSD = N 2 +for the square lattice with periodic boundary conditions. +A manifestation of this mixed ’t Hooft anomaly is that the +symmetry operators obey the Heisenberg algebra12 +�U(ˆγ)U †(γ) = (ei2π/N)#(ˆγ,γ) U †(γ)�U(ˆγ), +(5.11) +and therefore U and �U form a projective representation of +Z(1) +N × Z(1) +N . The mixed ’t Hooft anomaly ensures that the +ground state cannot be a trivial product state, and instead the +R1TC must be in either a gapless or an SSB phase. There- +fore, the mixed ’t Hooft anomaly protects the spontaneous +12 Gauging a symmetry U is the procedure of adding additional degrees of +freedom such that the theory becomes invariant under the gauged sym- +metry operator Ugauged. Ugauged acts on both open and closed subspaces +and physical states must satisfy Ugauged |ψ⟩ = |ψ⟩. A contradiction arises +when different Ugauged no longer commute, reflecting an obstruction to +gauging the symmetry (a ’t Hooft anomaly). +For example, consider +U(1) +gaugeU(2) +gauge = −U(2) +gaugeU(1) +gauge. Since U(1,2) +gauge |ψ⟩ = |ψ⟩, this leads to the +contradiction |ψ⟩ = − |ψ⟩. +symmetry breaking pattern and, therefore, the ZN topologi- +cal order. Furthermore, the mixed ’t Hooft anomaly is also +present at higher energies, affecting the non-topological ZN +1-form symmetries. Its manifestation Eq. (5.11) gives rise to +nontrivial mutual statistics between e and m anyons [40]. +B. +Symmetries of the R2TC +Having summarized the symmetries in the R1TC, let us +now consider the R2TC. It is convenient to choose a slightly +different, but physically equivalent, square lattice where the +(X1, Z1) and (X2, Z2) ZN spins reside on horizontal links +while the (X0, Z0) ZN spins reside on vertical links. In fact, +this is the lattice Λ2 in Fig. 1. The ZN R2TC Hamiltonian is +then given by +H = − +� +c(h) +1 +Ac(h) +1 +− +� +c0 +Bx +c0 − +� +c2 +By +c2, +(5.12) +Bx +c0 = 1 +N +N +� +j=1 +(bx +c0)j, +By +c2 = 1 +N +N +� +j=1 +(by +c2)j, +Ac(h) +1 += 1 +N +N +� +j=1 +(ac(h) +1 )j, +where c(h) +1 +denotes a horizontal link, c0 a lattice site, and c2 a +plaquette. Fig. 6b shows graphical representations of the op- +erators a, bx, and by, which are also defined in Eqs. (2.21) +and (2.19), respectively. From Fig. 6b, it is clear these oper- +ators are mutually commuting, and therefore the ground state +satisfies a = 1, bx = 1, and by = 1. +The R2TC Hamiltonian operators a, bx, and by have a rich +and complicated structure. Consequently, the theory can have +many interesting generalized symmetries. We will construct +its symmetries in V B 1, which will include mostly technical +details. Afterwards, in V B 2, we will discuss these symme- +try operators, analyzing how the R2TC’s interesting properties +can be interpreted from a symmetry point of view and com- +paring the symmetry operators to conventional 1-form sym- +metries. +1. +Construction of symmetry operators +Let us first identify the symmetries which are generated by +operators built out of only X0, X1, and X2. To do so, we +define the lattice vector fields X1 and X2 which are related to +X0, X1, and X2 by +Xi +1,c0 = (Xx +1,c0, Xy +0,c0) +(5.13) +Xi +2,ˆc0 = (Xy +0,| ∗ ˆc0|+ˆx, Xx +2,| ∗ ˆc0|+ˆy). +(5.14) +Notice that while X1 is specified by the links of the direct +lattice, X2 is instead specified by the links of the dual lattice. +As elaborated on in appendix section A, the position of | ∗ ˆc0| +is related to a direct lattice site c0 by c0 = | ∗ ˆc0| − ˆx/2 − ˆy/2, + +19 +X1 +X2 +X0 +† +† +† +† +† +† +† +† +† +γ +̂γ +FIG. 8. The R2TC symmetry operators U1(γ) and U2(ˆγ), defined +by Eqs. (5.15) and (5.16) respectively, act on closed loops of the +direct and dual lattice. Here we show graphical representation of an +example of U1(γ) acting on a loop of the direct lattice γ (drawn in +green) and of U2(ˆγ) acting on a loop of the dual lattice ˆγ (drawn in +red). +where | ∗ ˆc0| is just the absolute value of ∗ ˆc0. Using X1 and +X2, we construct the unitary operators +U1(γ) = +� +c1∈γ +X1,c1, +(5.15) +U2(ˆγ) = +� +ˆc1∈ˆγ +X2,ˆc1, +(5.16) +where γ and ˆγ are oriented loops on the direct and dual lat- +tice, respectively (see Fig. 8). U1 and U2 trivially commute +with Bx +c0 and By +c2 in the R2TC Hamiltonian Eq. (5.12). U1 +and U2 also commute with Ac(h) +1 , which can be confirmed +directly or simply by comparing the graphical representa- +tions shown in Figs. 6b and 8. Therefore, for all γ and ˆγ, +[U1(γ), H] = [U2(ˆγ), H] = 0, and U1 and U2 correspond to +symmetry operators. +When γ and ˆγ are contractible, the symmetry operators U1 +and U2 can be written as +U1(γ = ∂M) = +� +c2∈M +by +c2, +(5.17) +U2(ˆγ = ∂ ˆ +M) = +� +ˆc2∈ ˆ +M +bx +| ∗ ˆc2|. +(5.18) +Consequentially, in the ground state subspace where bx,y = 1, +U1 and U2 are topological operators, depending only on the +homology class of γ and ˆγ. Since U N +1 = U N +2 = 1, we there- +fore find that these symmetry operators generate an emergent +Z(1) +N × Z(1) +N +symmetry in the IR. We note that when γ is a +loop of links in the x-direction (y-direction), U1 becomes the +“holonomy” W3 (W4) from Eq. (4.11). Similarly, when ˆγ is a +loop of dual links in the x-direction (y-direction), U2 becomes +the holonomy W1 (W2) from Eq. (4.11). +There is one more symmetry operator which can be con- +structed from the X operators. +Let us define the operator +X3 which acts only on the horizontal links c(h) +1 . X3 is in- +terpreted as a lattice vector field whose x component acts on +X1 +X2 +Γ(s) +† +† +† +† +† +† +† +† +† +† † +† +† +† +† +FIG. 9. +The R2TC symmetry operator U3(Γ(s)) defined by +Eq. (5.20) acts on closed loops Γ(s) of the Vvh lattice. Here we show a +graphical representation of U3(Γ(s)) acting on a particular loop Γ(s) +drawn in blue with N = 3. The Vvh lattice sites belonging to the s +sublattice are denoted by gray squares, and we sometimes include +the operators (X1)3 and (X2)3 despite them being the identity. +the horizontal links of c(h) +1 +but whose y component acts on +the vertical links of the dual lattice ∗ ˆc(v) +1 += c(h) +1 . However, +the horizontal links form their own square lattice Vvh whose +sites v ≡ (vx, vy) are squares in Fig. 1. We will formulate +this symmetry on the Vvh lattice where it turns out to be most +naturally defined. However, this can also be formulated on the +direct lattice if the framing structure is utilized, which makes +the following symmetry a so-called framed-symmetry [52]. +X3 is related to X1 and X2 by +Xi +3,v = (X1,c(h) +1 , X2,c(h) +1 ), +(5.19) +where the Vvh site v on the left hand side is the center of the +edge c(h) +1 +on the right hand side. +Using X3, we can construct a unitary which commutes with +the R2TC Hamiltonian. To do so, we first reconsider the Vvh +square lattice as a Bravais lattice with a basis. The conven- +tional unit cell is an N × N square surrounding N 2 lattice +sites, each of which belong to their own sublattice. We intro- +duce the index s ∈ {1, 2, · · · , N 2 − 1, N 2} which labels each +sublattice. Let us denote a generic oriented closed loop of the +Vvh lattice as Γ, and specify loops made of only length N seg- +ments connecting sites of the sublattice s as Γ(s). With this +set up, we now consider the unitary operator +U3(Γ(s)) = +� +v∈Γ(s) +(X3,r)(vx−r(s) +x )+(vy−r(s) +y +), +(5.20) +where v is a Vvh lattice site13 and r(s) is the basis vector (in +the crystallography sense) of sublattice s. Fig. 9 shows an +example U3(Γ(s)) for N = 3. +13 In Eq. (5.20), the notation v ∈ Γ(s) simply means all Vvh lattice sites v +which the loop Γ(s) crosses. Here, Γ(s) should not be considered as a 1- +cycle—an integer sum of 1-chains in the kernel of the boundary operator +∂1. We will commit similar abuses of notation throughout this section. + +20 +The operator U3 trivially commutes with Bx +c0 and By +c2 in +the R2TC Hamiltonian Eq. (5.12). +Furthermore, U3(Γ(s)) +also commutes with Ac(h) +1 +for all Γ(s), which can be con- +firmed by direct computation or simply from inspecting the +graphical representations shown in Figs. 6b and 9. Therefore, +[U3(Γ(s)), H] = 0 for all Γ(s), and U3 indeed corresponds to a +symmetry operator. When Γ(s) is contractible, the symmetry +operator U3 can be written as +U3(Γ(s) = ∂M)= +� +c0∈M +(bx +c0)(c0)y � +c2∈M +(by +c2)(c2)x. +(5.21) +Here, (c0)y is the distance of c0 from Γ(s) in the −y-direction. +Similarly, (c2)x is the distance of c2 from Γ(s) in the −x- +direction. Since bx +c0 = by +c2 = 1 in the ground state subspace, +U3(Γ(s)) is a topological operator and corresponds to a 1-form +symmetry in the IR. However, this is not an ordinary 1-form +symmetry since Γ(s) is not allowed to be any loop on the Vvh +lattice. As a result U3(Γ(s)) is not a fully topological operator +on the Vvh lattice, but is on the s sublattice. Therefore, we +refer to U3(Γ(s)) as a sublattice 1-form symmetry. +The precise nature of this sublattice 1-form symmetry de- +pends on both the topology and geometry of the lattice in +a sensitive way. Without periodic boundary conditions, this +is a sublattice ZN 1-form symmetry. With periodic bound- +ary conditions, the previous N × N conventional unit cell +shrinks to a gcd(Lx, N) × gcd(Ly, N) unit cell (but Γ(s) is +stll made of only length N segments). Consequently, Γ(s) +must wrap around system N/ gcd(Li, N) times in the i- +direction in order to close. Therefore, on a torus, U3 is a +Z(1) +gcd(Lx,N) × Z(1) +gcd(Ly,N) sublattice 1-form symmetry, where +the noncontractible Γ(s) of the Z(1) +gcd(Li,N) sublattice symme- +try is understood winding only in the i-direction. We note +that the Z(1) +gcd(Lx,N) and Z(1) +gcd(Ly,N) symmetry operators are +related to the W5 and W6 “holonomies,” respectively, from +Eq. (4.11). +We now move on to discuss the symmetry operators con- +structed from only Z0, Z1, and Z2. We will find that there are +three symmetry operators, two of which correspond two sub- +lattice 1-form symmetries and one is a conventional 1-form +symmetry. +To construct the first symmetry operator, we must recon- +sider unit cell of the lattice as a N × 1 unit cell with a basis +labeled by s ∈ {1, · · · , N}. A loop of the dual lattice made of +only length N segments in the horizontal direction connecting +the sites of sublattice s is denoted as ˆγ(s). We then introduce +the lattice vector field Z1 acting on the links of the dual lattice, +ˆc1. It is related to the Z0, Z1, and Z2 operators by +Z1,ˆc1 = (Z0,∗ ˆc1(Z† +2,∗ ˆc1+ˆx/2+ˆy/2Z2,∗ ˆc1+ˆx/2−ˆy/2)x−x(s), +Z1,∗ ˆc1) +(5.22) +where x is the x-coordinate of the dual lattice site in the x- +direction of ˆc1 and x(s) is the x-coordinate of the basis vector +for sublattice s. With this defined, we can then consider the +Z1 +Z2 +Z0 +̂γ(s) +† +† +† +† +† +† +† +† +† +† +† +† +† † † +† +† +† +† +† +† +† † † +† +† † +FIG. 10. +The R2TC symmetry operator �U1(ˆγ(s)) defined by +Eq. (5.23) acts on closed loops ˆγ(s) of the dual lattice. Here we +show a graphical representation of �U1(ˆγ(s)) acting on a particular +loop ˆγ(s) drawn in red with N = 3. The dual lattice sites belonging +to the s sublattice are denoted by gray squares, and we sometimes +include the operator (Z2)3 despite it being the identity. +Z1 +Z2 +Z0 +† +†† +† +† +† +γ(s) +† +† +† +† +† +† +† +† +FIG. 11. +The R2TC symmetry operator �U2(γ(s)) defined by +Eq. (5.25) acts on closed loops γ(s) of the direct lattice. Here we +show a graphical representation of �U2(γ(s)) acting on a particular +loop γ(s) drawn in green with N = 3. The direct lattice sites be- +longing to the s sublattice are denoted by gray squares. +unitary +�U1(ˆγ(s)) = +� +ˆc1∈ˆγ(s) +Z1,ˆc1, +(5.23) +an example of which is shown in Fig. 10 for N = 3. It is +straightforward to check that for all ˆγ(s), �U1 commutes with +a, bx, and by, and therefore [�U1, H] = 0. +The second symmetry operator is rather similar to the first +one. Now we instead consider a 1 × N units cell with a basis +again labeled by s ∈ {1, · · · , N}. A Loop of the direct lat- +tice made of only length N segments in the vertical direction +connecting the sites of sublattice s is denoted as γ(s). We then +introduce the lattice vector field Z2 acting on the links of the +direct lattice, c1. It is related to the Z0, Z1, and Z2 operators + +21 +by +Z2,c1 = (Z1,c1, +Z† +0,c1(Z† +2,c1−ˆx/2+ˆy/2Z2,c1+ˆx/2+ˆy/2)y−y(s)) +(5.24) +where y is the y-coordinate of the lattice site in the y-direction +of c1 and y(s) is the y-coordinate of the basis vector for sub- +lattice s. With this defined, we can then consider the unitary +�U2(γ(s)) = +� +c1∈γ(s) +Z2,c1, +(5.25) +an example of which is shown in Fig. 11 for N = 3. It is +straightforward to check that for all γ(s), �U2 commutes with +a, bx, and by, and therefore [�U2, H] = 0. +Both unitary operators �U1(ˆγ(s)) and �U1(ˆγ(s)) correspond +to symmetry operators. When acting on contractible loops, +they can be written as +�U1(ˆγ(s) = ∂ ˆ +M) = +� +c(h) +1 +∈ ˆ +M +(ac(h) +1 )(c(h) +1 +)x, +(5.26) +�U2(γ(s) = ∂M) = +� +c(h) +1 +∈M +(ac(h) +1 )(c(h) +1 +)y, +(5.27) +where (c(h) +1 )x is the distance of c(h) +1 +from ˆγ(s) in the −x- +direction and (c(h) +1 )y is the distance of c(h) +1 +from γ(s) in the +−y-direction. Evidently, both the symmetries generated by +�U1 and �U2 are sublattice 1-form symmetries, defined on their +respective sublattices. Like for the sublattice 1-form symme- +try U3, the details of the symmetry are sensitive to both the +geometry and topology of the lattice. Indeed, with periodic +boundary conditions the N × 1 (1 × N) unit cell defined for +the �U1 (�U2) symmetry operator becomes a gcd(Lx, N) × 1 +(1 × gcd(Ly, N)) unit cell (γ(s) and ˆγ(s) are still made of +length N segments in the y and x directions, respectively). +Note that when �U1 (�U2) is supported on a non-contractible +loop in the y (x) direction, it is related to � +W3 (� +W2) in +Eq. (4.20). Similarly, when �U1 (�U2) is supported on a non- +contractible loop in the x (y) direction, it becomes � +W4 (� +W1) +in Eq. (4.20) +Let us now construct the final symmetry operator. We de- +fine the operator Z3 which acts only on the vertical links c(v) +1 +of the lattice. Z3 is interpreted as a lattice vector field whose +x component acts on the plaquette c2 but whose y component +acts on the horizontal links of the dual lattice ˆc(h) +1 += ∗ c(v) +1 . It +turns out it is most natural to formulate this symmetry oper- +ator on the previously mentioned Vvh lattice. We will denote +the sites of the dual Vvh as ˆv and note that they are shown in +Fig. 1 as discs. Z3 is related to Z0, Z1, and Z2 by +Z3,ˆv = (Z0,ˆvZ† +0,ˆv+ˆxZ† +2,ˆv+ˆx/2+ˆy/2Z2,ˆv+ˆx/2−ˆy/2, +Z† +2,ˆv−ˆx/2+ˆy/2Z2,ˆv+ˆx/2+ˆy/2). +(5.28) +With Z3 defined, we can then consider the unitary operator +�U3(ˆΓ) = +� +ˆv∈ˆΓ +Z3,ˆv +(5.29) +† +† +† +†† +† +† +†† +† +† +† +† +† +† +† +† +† +† +† +Z1 +Z2 +Z0 +̂Γ +FIG. 12. The R2TC symmetry operator �U3(ˆΓ) defined by Eq. (5.29) +acts on closed loops Γ of the dual Vvh lattice. Here we show a graph- +ical representation of �U3(Γ) acting on a particular loop ˆΓ drawn in +orange. +where ˆΓ is an oriented closed loop on the dual Vvh lattice (see +Fig. 12 for an example). +It is straight forward to check that �U3 commutes with the +Hamiltonian for all ˆΓ and therefore corresponds to a symme- +try operator. When �U3 is supported on a contractible loop, it +can be written as +�U3(ˆΓ = ∂M) = +� +c(h) +1 +∈M +ac(h) +1 . +(5.30) +Since ac(h) +1 += 1 in the ground state subspace, �U3 is a topo- +logical operator. Therefore, in the IR, �U3 is the symmetry +operator of a Z(1) +N +symmetry. In fact, when supported on a +non-contractible loop winding around the system in the x (y) +direction, �U3 becomes � +W6 (� +W5) from Eq. (4.20). +2. +Analysis and discussion of R2TC symmetries +Having identified the generalized symmetries of the R2TC, +let us now use them to interpret the model’s interesting prop- +erties from a symmetry point of view. Recall that the six sym- +metry operators are supported on loops and commute with the +R2TC Hamiltonian for all respective loops. Their expecta- +tion values with respect to excited states depend on more than +just the topology of the loops, so in this sense these micro- +scopic (UV) symmetries are non-topological 1-form symme- +tries. Their existence reflects the lack of dynamics for e and +⃗m anyons. Throughout the rest of the deconfined phase of ZN +rank-2 gauge theory, away from the R2TC point, these non- +topological 1-form symmetries are explicitly broken. +The symmetry operators of the R2TC are much richer and +more complex than those in the R1TC, which were reviewed + +22 +in section V A. As demonstrated in Fig. 7, the R1TC symme- +try operators are nicely defined on 1-cycles of the direct and +dual lattice (so, they admit a straightforward description using +cellular homology). Furthermore, for a given symmetry oper- +ator, each edge of the 1-cycle was acted on by the same X or +Z operator (up to taking the hermitian conjugate, which arises +from the 1-cycles orientation and lattice’s branching struc- +ture). The R2TC symmetry operators, examples of which are +shown in Figs. 8–12, go beyond all of these convenient sim- +plicities. For example, they include the following features, +absent from the R1TC’s symmetries: +1. The symmetry operators �U1(ˆγ(s)), �U2(γ(s)), and �U3(ˆΓ) +act on both the spins on the loops ˆγ(s), γ(s), and ˆΓ, +respectively, and the spins near the loops. +2. For all symmetry operators, the operators acting on/near +the loop’s edges depend on whether the loop is parallel +to the x or y direction. For example, as shown in Fig. 8, +U1(γ) has X0 act on edges when γ is parallel to the y- +direction but has X1 act on edges when γ is parallel to +the x-direction. +3. The operators acting on the spins for symmetry oper- +ators U3(Γ(s)), �U1(ˆγ(s)), and �U2(γ(s)), depend on the +position of those spins. +4. The symmetry operators U3 and �U3 act on loops of the +direct/dual Vvh lattice instead of the direct/dual (Λ2) lat- +tice. In terms of the Λ2 lattice, these operators act on +loops defined on both the direct/dual lattice and there- +fore require additional framing structure, which makes +them framed 1-form symmetries [52]. +5. The symmetry operator �U3(ˆΓ) has operators which only +act on the corners of the loop ˆΓ while absent from other +parts of the loop (i.e., Z0 in Fig. 12). From the Λ2 lattice +point of view, these corners coincides with where the +framing structure connects the direct and dual lattices’ +loops to create ˆΓ. +Unlike the expectation values with respect to excited states +mentioned previously, the vacuum expectation values of the +symmetry operators depend only on the topology of these +loops. Thus, in the ground state sub-Hilbert space—the IR— +of the R2TC, all six of the generalized symmetries are 1-form +symmetries. Three of these (U1, U2, and �U3) were conven- +tional 1-form symmetries. However, the other three (U3, �U1, +and �U2) were not conventional 1-form symmetries since their +symmetry operators relied on an underlying sublattice struc- +ture. These nonconventional 1-form symmetries were called +sublattice 1-form symmetries in the previous section to em- +phasize this additional structure. The lattice symmetries gen- +erally mix these sublattices and act nontrivially on sublat- +tice 1-form symmetries. Therefore, the total symmetry group +of the R2TC is (1-form symmetries)⋊(lattice symmetries). +This interplay between the sublattice 1-form and spatial +symmetries can also be noticed by the R2TC’s symmetry- +enriched topological order, where position-dependent excita- +tions [10] reflect the existence of sublattice 1-form symme- +tries. Throughout the rest of the deconfined phase of ZN rank- +2 gauge theory, away from the R2TC point, we expect that all +of these generalized symmetries are exact emergent IR sym- +metries [57]. This means that despite being emergent symme- +tries, explicitly broken in the microscopic Hamiltonian, they +constrain the IR in the thermodynamic limit as if they were +exact microscopic symmetries. +Since all of the R2TC’s generalized symmetries are 1-form +symmetries, they are sensitive to the topology of the spatial +lattice. The sublattice 1-form symmetries, however, also de- +pend on the geometry of the lattice. Indeed, as we discussed +in the previous section, with periodic boundary conditions the +size of their underlying sublattices depends on the system size. +Furthermore, the sublattices in U3’s, �U1’s, and �U2’s respec- +tive definitions are unique to the square lattice, so the R2TC +on a different lattice would generally have different sublattice +1-form symmetries. Therefore, the sublattice 1-form symme- +tries give rise to UV/IR mixing in the R2TC [10]. The emer- +gent IR symmetries depending on the UV lattice is a general +diagnosis for UV/IR mixing and, in fact, may be a unified +mechanism for UV/IR mixing in all topological and fracton +phases. +The R2TC’s symmetry operators satisfy the algebra +U1(γ) �U1(ˆγ(s1)) = ω#(γ,ˆγ(s1)) �U1(ˆγ(s1)) U1(γ), +(5.31) +U2(ˆγ) �U2(γ(s2)) = ω#(ˆγ,γ(s2)) �U2(γ(s2)) U2(ˆγ), +(5.32) +U3(Γ(s3)) �U3(ˆΓ) = ω#(Γ(s3),ˆΓ) �U3(ˆΓ) U3(Γ(s3)), +(5.33) +where ω ≡ ei2π/N and #( , ) is the signed intersection +number. +We thus see that the Ui (�Ui) symmetry oper- +ator transforms nontrivially under the �Ui (Ui) symmetry +transformation—Ui (�Ui) is a charged operator of the �Ui (Ui) +symmetry. Recall from the previous section that when sup- +ported on contractible loops, ⟨Ui⟩gs = ⟨�Ui⟩gs = 1. The fact +that ⟨Ui⟩gs = 1 (⟨�Ui⟩gs = 1) for contractible loops means that +the �Ui (Ui) symmetry charges are condensed in the R2TC +ground state, and the �Ui (Ui) symmetry is spontaneously bro- +ken. Therefore, the R2TC ground state spontaneously breaks +all six of the 1-form symmetries. +Discrete symmetries spontaneously breaking always gives +rise to a ground state degeneracy. +The GSD is computed +by finding the smallest faithful representation of the sponta- +neously broken symmetry operators. This exact calculation +was done in section IV B, where the holonomies Wi and � +Wi +are the generators of the spontaneously broken symmetries, +and yields the correct GSD Eq. (1.1). Therefore, the GSD +is system size dependent because some of the spontaneously +broken symmetries are sublattice 1-form symmetries that en- +code geometrical information of the lattice. +The algebra Eqs. (5.31)-(5.33) also reveals that the R2TC +realizes these generalized symmetries in a projective repre- +sentation. This prevents the 1-form symmetries from being + +23 +gauged, and is thus a manifestation of an ’t Hooft anomaly14. +In particular, there is a mixed ’t Hooft anomaly between +the Ui and �Ui symmetries. Like in the R1TC discussed in +Sec. V A, mixed ’t Hooft anomalies for 1-form symmetries +realized through projective representations are physically re- +flected through the nontrivial braiding statistics of anyons. +Since some of the R2TC’s anomalous symmetries are sublat- +tice 1-form symmetries, the braiding statistics will generally +depend on the sublattice the anyon resides on. However, this +is precisely the position dependent-braiding discussed in sec- +tion IV E. +VI. +SUMMARY AND OUTLOOK +We have applied the idea of coupled-layer construction, +previously invented to understand the emergence of fracton +models out of toric codes in three dimensions [20, 21], to shed +light on the appearance of symmetric rank-2 gauge fields in +two dimensions and from there the rank-2 toric code through +Higgsing. +Condensation of gauge fields can take place in +either one of the two conjugate gauge fields A and E, and +leads to theories with either vector-electric or vector-magnetic +charges that are ultimately dual to each other. +Construction of holonomy (Wilson line) operators for the +rank-2 toric code follows rather naturally in this approach, +as one can start by identifying the holonomy operators in the +Hilbert space before the condensation took place. We thus ar- +rive at the picture of holonomies as the creation/annihilation +of magnetic and electric charge-anti-charge pairs, and of their +dipole-anti-dipole pairs. The dependence of the ground state +degeneracy on the system size (the UV/IR mixing) can be +thoroughly understood from analysis of the Wilson loop oper- +ators thus obtained. We further suggest an easy-to-implement, +heuristic derivation of the holonomies based on the rank-2 +gauge theory. This may well have applications in the holon- +omy construction of other, rank-2 gauge theories and the cor- +responding stabilizer models. +Furthermore the exact tensor network expression of the +ground state of the rank-2 toric code is derived starting from +two copies of the rank-1 toric code’s ground state wave func- +tions, by sewing them together with an isometry operation that +faithfully reflects the condensation of the gauge fields. This, +too, may have application in the construction of other rank-2 +based stabilizer ground state wave functions. For one thing, +analyzing entanglement entropy becomes easy with the ten- +sor network wave function at hand. Additionally, the tensor +network projection of R2TC provides a clear picture of how +coupled toric code layers engender higher-rank gauge theory +in terms of anyon condensation. This also sheds light on ex- +ploring phase transitions between conventional gauge theory +and R2TC, where one can replace the tensor projection proce- +dure with an additional parameter in the tensor element. We +will explore these issues in a future study [45]. +14 See footnote 12. +The anyon condensation idea can lead to a number of pow- +erful applications. As an example we showed how the Levin- +Gu semionic topological model [24] can undergo a similar +condensation procedure to result in a new model. The notion +of generalized symmetry is a new and powerful description +of the topological order in the toric code, and we have dis- +cussed how the notion applies to the rank-2 toric code. We +believe the ‘generalization’ of the generalized symmetry idea +to other rank-2 based models can find interesting applications +in the future. Futhermore, it would be interesting to investi- +gate if general rank-N symmetric tensor gauge theories, with +N > 2, can be constructed from many copies of rank-1 theo- +ries in a particular condensed phase. +ACKNOWLEDGMENTS +Y.-T.O. +was +supported +by +National +Research +Foundation (NRF) +of +Korea +under +Grant +NRF- +2022R1I1A1A01065149. +S.D.P. is supported by the +National Science Foundation Graduate Research Fellowship +under Grant No. +2141064 and by the Henry W. Kendall +Fellowship. +J.H.H. was supported by Grant No. +NRF- +2019R1A6A1A10073079. +He also acknowledges financial +support from EPIQS Moore theory centers at MIT and +Harvard. Y.Y. is supported by Northeastern University COS +start-up grant. H.-Y.L. was supported by NRF of Korea under +Grant No. NRF-2020R1I1A3074769. H.-Y.L. and Y.-T.O. +were supported by the Basic Science Research Program +funded by the Ministry of Education (2014R1A6A1030732). +J.H.H. acknowledges informative discussion with T. Hughes, +B. Kang, H. T. Lam, Z. X. Luo, N. Tantivasadakarn, and +X.-G. Wen. +Appendix A: Review of discrete differential geometry for +d-dimensional cubic lattices +In this appendix section, we review relevant parts of dis- +crete differential geometry (in a non-rigorous fashion) used +in Sec. V of the main text. Consider a cubic lattice in d- +dimensional space with periodic boundary conditions, de- +noted by Md. While a Bravais lattice is a collection of lattice +sites x ∈ Zd, it is useful to view it as also formed by higher- +dimensional objects, like links, plaquettes, cubes, etc. We call +a p-dimensional object a p-cell, with 0 ≤ p ≤ d. So, a 0-cell is +a lattice site, a 1-cell is a link, a 2-cell is a plaquette, etc. This +does not add additional structures to the lattice, but instead is +just a useful way of organizing the lattice sites. Indeed, de- +noting a p-cell associated with site x as cp(x)µ1µ2···µp, where +µ1 < µ2 < · · · < µp and µi ∈ {1, 2, · · · , d}, a p-cell of the + +24 +p = 1 +p = 2 +p = 1 +p = 2 +p = 3 +FIG. 13. The p-cells of the d-dimensional cubic lattice are equiv- +alently the 0-cells—the sites—of some other d-dimensional lattice. +Shown here are examples of this equivalent lattice (drawn in pink) +embedded in the conventional unit cell of the cubic lattice (drawn +in black). +(First row) In 2 dimensions, the 1-cells form another +square lattice, rotated by 45 degrees, whose lattice constant is 1/ +√ +2 +times that of the original square lattice. +The 2-cells also form +another square lattice, which is the original shifted by the vector +(ˆµ1 + ˆµ2)/2. (Second row) In 3 dimensions, both the 1-cells and +also the 2-cells form a lattice of corner-sharing octahedra with a lat- +tice constant that is 1/ +√ +2 times the cubic lattice’s. When p = 1, the +octagons are centered at the cubic lattice’s 0-cells. When p = 2, the +octagons are centered at the cubic lattices 3-cells. Lastly, the 3-cells +form another cubic lattice of the same size, but shifted by the vector +(ˆµ1 + ˆµ2 + ˆµ3)/2. +cubic lattice is the set of 2p lattice sites15 +cp(x)µ1µ2···µp= {x} ∪ {x + ˆµi | 1 ≤ i ≤ p} +∪ {x + ˆµi + ˆµj | 1 ≤ i < j ≤ p} +∪ · · · ∪ {x + ˆµ1 + . . . + ˆµp}, +(A1) +where +ˆµi is the unit vector in the µi-direction. +It +is often convenient to drop the requirement that the +indices are canonically ordered (i.e., +that they satisfy +µ1 < µ2 < · · · < µp < ν) and instead let cp(x)µ1µ2···µp obey +the relation cp(x)···µ1µ2··· = −cp(x)···µ2µ1···. The p-cells of +the d-dimensional cubic lattice are equivalently viewed as the +0-cells of some other lattice in d-dimensions, as demonstrated +for d = 2 and 3 in Fig. 13. +Introducing the concept of p-cells is strictly unnecessary +but very convenient because “sewing” p-cells together gives a +natural way to form p-dimensional subspaces of the lattice. +Furthermore these subspaces can also be given an orienta- +tion by defining an orientation structure to the lattice. A nice +local scheme for the lattice orientation is a branching struc- +ture, where the orientation on each 1-cell is chosen such that +a collection of 1-cells cannot form an oriented closed loop. +A canonical orientation on all other p-cells then follows from +the branching structure. We use the branching structure where +each 1-cell c1(x)µ has an arrow pointing in the ˆµ direction +(see Fig. 14). However, it is important to note that the choice +of lattice orientation is a formal convention, and choosing dif- +15 We adopt the discrete differential geometry and exterior calculus notations +and conventions used in Ref. 59. +̂x +̂y +̂z +FIG. 14. Example of the branching structure used for a chunk of the +cubic lattice in three-dimensional space. +ferent branching structures does not affect the physics16. +A p-cell can be related to (p − 1) cells using the boundary +operator ∂. The boundary operator acting on a p-cell—∂cp— +is the oriented sum of (p−1)-cells on the boundary of cp. For +the branching structure we use, it is given by +∂cp(x)µ1···µp= +p +� +k=1 +(−1)k+1� +cp−1(x + ˆµk)µ1··· +oµk···µp +−cp−1(x)µ1··· +oµk···µp +� +, +(A2) +where the notation +oµk indicates that the µk index is omitted. +From its definition, the boundary operator satisfies ∂2cp = 0 +for any p-cell. Furthermore, as there are no (−1)-cells, the +boundary operator acting on a 0-cell is defined to be zero. +On the other hand, a p-cell can be related to (p + 1)-cells +using the coboundary operator δ. The coboundary operator +acting on a p-cell—δcp—is an oriented sum of all (p + 1)- +cells whose boundary includes cp. For the branching structure +we use, it is given by +δcp(x)µ1···µp = +� +ν +cp+1(x)νµ1...µp − cp+1(x − ˆν)νµ1...µp. +(A3) +From +its +definition, +the +coboundary +operator +satisfies +δ2cp = 0 for any p-cell. Furthermore, as there are no (d + 1)- +cells, the coboundary operator acting on a d-cell is defined to +be zero. +Lastly, the lattice has an associated dual lattice. The dual +lattice has its lattice sites centered at the d-cells of the direct +lattice. For the cubic lattice, one choice of framing that relates +a dual lattice site ˆx to a direct lattice site x is by ˆx = x + 1 +2 ˆr +with ˆr = � +i ˆµi. +Each p-cell cp on the direct lattice is associated with a +(d − p)-cell ˆcd−p on the dual lattice. This is implemented +by the dual operator ∗. For this choice of framing, a p-cell +cp(x)µ1···µp (with canonical ordering µ1 < · · · < µp) and a +(d − p)-cell of the dual lattice ˆcd−p(ˆx)µ1···µd−p (with canon- +16 However, according to a conjecture from Ref. 60, observables are indepen- +dent of the branching structure only if the continuum effective field theory +is free of a framing anomaly [61]. + +25 +ical ordering µ1 < · · · < µd−p) are related to one another by +∗ cp(x)µ1···µp = ϵµ1···µpµp+1···µd +(A4) +× ˆcd−p(ˆx − ˆµp+1 − . . . − ˆµd)µp+1···µd, +∗ ˆcp(ˆx)µ1...µp = ϵµ1···µpµp+1···µd +(A5) +× cd−p(x + ˆµ1 + . . . + ˆµp)µp+1···µd, +where summation is not implied on the right hand side. 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Phys. 121, 351 (1989). + diff --git a/NtE3T4oBgHgl3EQfwwva/content/tmp_files/load_file.txt b/NtE3T4oBgHgl3EQfwwva/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d6246d9f726cd400cb6eefcc62da5c3cab611ae --- /dev/null +++ b/NtE3T4oBgHgl3EQfwwva/content/tmp_files/load_file.txt @@ -0,0 +1,1648 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf,len=1647 +page_content='Aspects of ZN rank-2 gauge theory in (2 + 1)D: construction schemes, holonomies, and sublattice one-form symmetries Yun-Tak Oh,1 Salvatore D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Pace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2 Jung Hoon Han,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3 Yizhi You,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' ∗ and Hyun-Yong Lee1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' † 1Division of Display and Semiconductor Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Sejong 30019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Massachusetts Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Massachusetts 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Sungkyunkwan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Suwon 16419,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 360 Huntington Ave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' MA 02115,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' USA 5Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Graduate School,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Sejong 30019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea 6Interdisciplinary Program in E·ICT-Culture-Sports Convergence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Sejong 30019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Korea (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2023) Rank-2 toric code (R2TC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' a prototypical archetype of the discrete rank-2 symmetric gauge theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' has prop- erties that differ from those of the standard toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Specifically, it features a blending of UV and IR in its ground state, restricted mobility of its quasiparticles, and variations in the braiding statistics of its quasiparticles based on their position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this paper, we investigate various aspects of ZN rank-2 gauge theory in (2 + 1)- dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Firstly, we demonstrate that U(1) rank-2 gauge theory can arise from U(1) × U(1) rank-1 gauge theory after condensing the gauge charges in a specific way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This construction scheme of U(1) rank-2 gauge theory carries over to the ZN case simply by Higgsing U(1) to ZN, after which the resulting rank-2 gauge theory can be tuned to the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The holonomy operators of R2TC are readily identified using this scheme and are given clear physical interpretation as the pair creation/annihilation of various monopoles and dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Explicit tensor network construction of the ground states of R2TC are given as two copies of the ground states of Kitaev’s toric code that are ‘sewn together’ according to the condensation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In addi- tion, through a similar anyon condensation protocol, we present a double semion version of rank-2 toric code whose flux excitations exhibit restricted mobility and semionic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Finally, we identify the generalized discrete symmetries of the R2TC, which are much more complex than typical 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' They in- clude conventional and unconventional 1-form symmetries, such as framed 1-form symmetries and what we call sublattice 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Using these, we interpret the R2TC’s unique properties (UV/IR mixing, position- dependent braiding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=') from the modern perspective of generalized spontaneous symmetry breaking and ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Introduction 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation Scheme 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation of rank-2 U(1) lattice gauge fields 3 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation of stabilizers and holonomies 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Higher-order instanton and confinement in the rank-2 U(1) gauge theory 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Conservation laws 6 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Applications of the Condensation Scheme 7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Tensor network representation of ZN R2TC wavefunctions 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Twisted rank-2 gauge theory from anyon condensation 8 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Holonomy Construction 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Pre-projection holonomies 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Post-projection holonomies 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physical interpretation of the holonomies 13 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Field-theoretic derivation of the holonomies 13 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Understanding the position-dependent braiding 15 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Generalized Symmetries 16 ∗ Electronic address: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='you@northeastern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='com † Electronic address: hyunyong@korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='kr A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Reviewing the 1-form symmetries of the R1TC 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Symmetries of the R2TC 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Construction of symmetry operators 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Analysis and discussion of R2TC symmetries 21 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Summary and Outlook 23 Acknowledgments 23 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Review of discrete differential geometry for d-dimensional cubic lattices 23 References 25 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' INTRODUCTION Long-range entangled phases of quantum matter are com- monly described by fractionalized quasiparticles and emer- gent gauge fields which provide an effective description cap- turing the phase’s universal properties [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Indeed, canoni- cal examples include fractional quantum Hall liquids [2] and quantum spin liquids [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Unsurprisingly, long-range entan- gled quantum matter with increasingly exotic properties is de- scribed by increasingly rich generalizations of conventional gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A particular example is abelian gauge theories whose gauge fields are symmetric tensor fields instead of vec- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='04706v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='str-el] 11 Jan 2023 2 tor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These higher-rank gauge theories1 have attracted substantial interest recently in the study of fracton phases [4– 8] and topological order [7–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One of the simplest archetype of discrete gauge theories can be obtained from Higgsing the U(1) theory into Z2 by con- densing charge-2 gauge charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Following this protocol, we can obtain the rank-2 Z2 gauge theory starting from a rank- 2 U(1) gauge theory and Higgsing U(1) down to Z2 [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the zero-correlation length limit, the resultant gauge the- ory can be interpreted as an exactly solvable Hamiltonian, so- called rank-2 toric code (R2TC) [9–11, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC features several interesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One of them is the sensitive dependence of the ground state degener- acy (GSD) on the system size Lx × Ly against N, the Hilbert space dimension of the local spin state |s⟩ (s = 0, · · · , N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The GSD varying from N 3 to N 6 was first discovered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [9] and was soon clarified as a rigorous formula [10] GSD = N 3gcd(Lx, N)gcd(Ly, N)gcd(Lx, Ly, N), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) where gcd stands for the greatest common divisor among the two or three integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The fact that the GSD, a macro- scopic property, depends sensitively on the number of unit cells, a microscopic property, is a manifestation of what’s known as UV/IR mixing [10, 12, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The braiding pro- cess between a pair of quasiparticles showed interesting po- sition dependence that is not seen in Kitaev’s toric code, and requires a new form of field theory called the dipolar BF the- ory (dBF) [11] for comprehensive understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A differ- ent interpretation of the dipolar braiding in terms of multi- component mutual Chern-Simons theory was given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Lastly, quasiparticle excitations in the model showed restricted mobility such as the ability to hop only in multiples of N lattice sites in certain directions [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We note that many of these features were already apparent in the plaquette model of Wen [16], and more recently several models sharing similar features were proposed [12, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The restricted mobility exhibited by quasiparticle excita- tions in R2TC is clearly shared in a more rigorous way in fracton models such as the X-cube model [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A standard way of constructing these fracton models is to use the net- work construction scheme first proposed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In it, one starts with layers of 2D toric codes with fully mobile quasiparticle excitations, and produces a fractonic model with immobile excitations by imposing constraints among the lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is a natural question then if a similar scheme does exist to construct R2TC -a model based on rank-2 gauge theory - from the R1TC which is rooted in conventional rank-1 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This paper answers this question in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In accomplish this, we get to exploit the idea of coupling two gauge theories together through constraint, in a process often called the anyon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Several past works have 1 We will generally denote abelian symmetric tensor gauge theory simply as higher-rank gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This is not to be confused with higher-form gauge theories whose gauge fields are differential forms—antisymmetric tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' exploited the condensation idea to produce the X-cube model from layers of 2D toric codes [20, 21] (hereafter referred to as rank-1 toric code, or R1TC for short), or to produce rank-2 gauge theories from rank-1 theories [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Our condensa- tion scheme share the similar spirit as these works, but dif- fers greatly in details of how we implement the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In particular, we make a clear comparison between the con- densation scheme of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [23] and our own in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II B in an effort to emphasize the consequences of various conden- sation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the past, R2TC was obtained by Higgsing the symmetric rank-2 gauge field [7, 9] but the origin of this higher-rank gauge field was left obscure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We show here that it emerges naturally in the course of constraining the two copies of rank-1 gauge fields in a certain way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, the GSD of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) is closely related to the existence of six independent Wilson line operators in the model, which have been identified previously in the spin oper- ator [9] and the field theory language [10], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Still lacking was a clear physical picture accompanying these Wil- son operators, such as the creation/annihilation of electric and magnetic quasiparticle pairs in the case of R1TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It turns out that the condensation scheme provides a helpful guide in con- structing the full set of holonomies 2 needed to fully account for the GSD, which are also amenable to physically appealing interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In addition to the braiding of charges, braid- ing of dipoles play an important role in accounting for the degeneracy of the R2TC ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As an added benefit of the condensation picture, we find some useful applications in explicitly constructing the first- quantized ground state wave function of R2TC in tensor net- work (TN) form, as two copies of R1TC ground states sewn together through some constraining tensor that directly re- flects the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As another example we construct the rank-2 generalization of a model with semionic flux statis- tics [24] by coupling two copies of the pristine double semion model through anyon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As a final topic of the paper, we explore the generalized symmetries of the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Modern generalizations of sym- metry [25–31] have opened up an exciting frontier for the discovery of new phases of quantum matter [32–38] and in the conceptual organization of both known and new quan- tum phases [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For instance, these generalizations have allowed topological order to be understood in a symmetry framework [29, 34, 39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is therefore natural to wonder if the interesting properties of the R2TC can be understood in this unifying, modern point of view of topological quan- tum matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here, we construct all of the symmetry operators for the R2TC for general N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the ground state sub-Hilbert space, the symmetries we identify are all 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, they are not all conventional 1-form symmetries: some rely on a framing structure of the lattice (framed 1-form symmetries) and others on a sublattice structure of the lattice (sublattice 1-form symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, these symme- tries have a rich mixed ’t Hooft anomaly structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We show 2 In this paper, we use the words holonomies interchangeably with Wilson line operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Illustration of two possible condensation processes leading to the appearance of new (a) magnetic and (b) electric flux opera- tors given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There are two inter-penetrating sublattices Λ1, Λ2 shown by dashed and solid lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The dark squares (circles) represent Vvh (Vhv) sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The coordinate ⃗vi refers to the Vvh sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The pink and blue arrows represent rank-1 gauge field patterns from the sublattice Λ1 and Λ2, respectively, that combine to give new flux patterns in the rank-2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' that the R2TC ground state spontaneously breaks all of these 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This allows us to interpret the unconven- tional properties of the R2TC (position-dependent braiding, UV/IR mixing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=') all in terms of these symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Organization of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II we out- line the condensation scheme that leads one from two copies of rank-1 lattice gauge theory to the rank-2 lattice gauge the- ory and ultimately to the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' III we discuss two applications of the condensation idea in the construction of the ground state of rank-2 toric code out of those of the rank- 1 toric code, and the construction of ‘twisted’ rank-2 gauge theory resulting in a new model with semionic flux statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV we carefully go through the procedure by which all the holonomies in the R2TC can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physical in- terpretation of the holonomies thus constructed is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' V, after first reviewing the generalized symmetries of the R1TC, we discuss the R2TC from the point of view of general- ized symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Additional themes such as instanton effects in rank-1 gauge theories in 2+1D (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II C), field-theoretic understanding of the holonomy and the position-dependent braiding (Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV D and IV E) are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The summary and outlook follows in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' CONDENSATION SCHEME We show how rank-2 gauge theoires can emerge from two copies of rank-1 gauge theory through the condensation of certain components of the gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We outline this proce- dure first from the perspective of U(1) gauge theory, followed by that of ZN gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Discussion of instanton suppres- sion in the rank-2 gauge theory is given as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation of rank-2 U(1) lattice gauge fields Consider two interpenetrating square lattices denoted Λ1 (dashed lines) and Λ2 (solid lines) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Each square lattice has gauge degrees of freedom (Aµ a, Eµ a ) residing at the µ = x, y-oriented links of the respective square sublattice labeled by a = 1, 2, satisfying the canonical commutation [Aµ a, Eµ′ a′ ] = iδaa′δµµ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The two square lattices are super- posed in such a way that horizontal bonds in Λ1 and verti- cal bonds in Λ2 intersect at one set of sites belonging to Vhv, while vertical bonds of Λ1 and horizontal bonds of Λ2 cross at sites belonging to Vvh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1, sites belonging to Vvh and Vhv are designated by dark squares and circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The coordinate ⃗vi = xiˆx + yiˆy, or sometimes just i, is used to label the Vvh sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (Note that what we call sites are the links in the individual sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=') To reduce the notational clutter, we will also use i to label the vertex position ⃗r1,i = ⃗vi−ˆy/2 of the Λ1 lattice and the vertex position ⃗r2,i = ⃗vi−ˆx/2 of the Λ2 lat- tice as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' According to this notation scheme, the y-oriented fields (Ay 1,i, Ey 1,i) and the x-oriented fields (Ax 2,i, Ex 2,i) both reside on the same site ⃗vi = xiˆx + yiˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The sites in Vhv and the fields defined on them are then assigned appropriate coordinates in reference to those given to Vvh sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Each square lattice Λ1, Λ2 hosts its own gauge-invariant quantities (a = 1, 2), Ga(⃗ra,i) = (∇ · Ea)i, Ba(⃗ra,i) = (∇ × Aa)i, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) which are the lattice divergence and lattice curl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Suppose that we impose the constraint Ex 1,i+ˆy = Ey 2,i+ˆx at half the links, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' on the Vhv sites (dark circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In other words, we write the combined Hilbert spaces of the two lattice gauge theory’s as |Ψ⟩ = |ψ1⟩ ⊗ |ψ2⟩ where |ψa⟩ belongs to the Hilbert space of Λa, and insist that only the subset of Hilbert spaces obeying the following constraint survives: (Ex 1,i+ˆy − Ey 2,i+ˆx)|ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) To be clear, i refers to all the Vhv sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Such constraint nec- essarily precludes operators that do not commute with it, such as (∇ × Aa)i, while (∇ · Ea)i is still allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In their place, a new operator that commutes with the constraint can be con- structed by noting that [Ex 1,i+ˆy − Ey 2,i+ˆx, Ax 1,i+ˆy + Ay 2,i+ˆx] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We find Bi = ∆x(∇ × A1)i − ∆y(∇ × A2)i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) indeed involves only the combination Ax 1 + Ay 2 at all the Vhv sites where the constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) is imposed, and therefore commutes with it 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The meaning of discrete derivatives ∆x and ∆y is clear from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Due to the constraint, one must identify Ex 1,i+ˆy = Ey 2,i+ˆx as one gauge field and accordingly introduce a new label (Ex 2,i, Ey 1,i, Ex 1,i+ˆy = Ey 2,i+ˆx) → (Exx i , Eyy i , Exy i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4) 3 It is easy to convince that no simpler operator exists that commutes with the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (b) a4 A similar re-labeling (Ax 2,i, Ay 1,i, Ax 1,i+ˆy + Ay 2,i+ˆx) → (Axx i , Ayy i , Axy i ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) yields symmetric rank-2 gauge fields (Aa i , Ea i ) (a = xx, xy, yy) obeying the canonical relation [Aa i , Eb j] = iδijδab4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There are two electric charges (ex i , ey i ) and one vector charge mi in the projected Hilbert space obeying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) given by ex i =Exx i+ˆx − Exx i + Exy i − Exy i−ˆy, ey i =Exy i − Exy i−ˆx + Eyy i+ˆy − Eyy i , mi =Axx i+ˆy + Axx i−ˆy − 2Axx i + Ayy i+ˆx + Ayy i−ˆx − 2Ayy i − Axy i + Axy i−ˆx + Axy i−ˆy − Axy i−ˆx−ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) Here mi is simply the re-writing of Bi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The sym- metric rank-2 gauge fields as well as the new mutually com- muting generators formed by them emerge naturally from the condensation process just outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Upon Higgsing, the three charge operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) become the three commuting spin operators of R2TC [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Among the tensor gauge fields, xx and yy components reside at the Vvh sites where no con- densation has taken place, and the xy component resides at the Vhv sites where condensation reduces the degrees of freedom from two to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The constraint expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) is by no means the unique one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Instead of condensing E, one can condense the A fields through the constraint (Ax 1,i+ˆy − Ay 2,i+ˆx)|Ψ⟩ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) at the Vhv sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this case, (∇·Ea)i is no longer an allowed operator but a new quantity Gi = ∆x(∇ · E2)i + ∆y(∇ · E1)i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8) emerges as a viable operator in the constrained Hilbert space see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' After re-labeling (Ax 2,i, Ay 1,i, Ax 1,i+ˆy = Ay 2,i+ˆx) → (Axx i , Ayy i , Axy i ) (Ex 2,i, Ey 1,i, Ex 1,i+ˆy + Ey 2,i+ˆx) → (Exx i , Eyy i , Exy i ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) one arrives at two magnetic charges (mx i , my i ) and one electric charge ei defined by mx i =Ayy i+ˆx − Ayy i − Axy i + Axy i−ˆy, my i =Axy i − Axy i−ˆx − Axx i+ˆy + Axx i , ei =Eyy i+ˆy + Eyy i−ˆy − 2Eyy i + Exx i+ˆx + Exx i−ˆx − 2Exx i + Exy i − Exy i−ˆx − Exy i−ˆy + Exy i−ˆx−ˆy, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) where ei is a mere re-writing of Gi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Higging them leads to R2TC with a scalar electric charge and vector mag- netic charges, which is dual to the theory with vector electric and scalar magnetic charge [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the ensuing discussion, we will adopt this version of R2TC that has scalar-electric and vector-magnetic charges unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 4 This follows from the original gauge fields obeying the canonical relation [Aµ a,i, Eν b,j] = iδabδijδµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation of stabilizers and holonomies The previous subsection showed which operators survive under the projection (condensation) of two rank-1 lattice gauge theory’s to the constrained Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The oper- ators that become the stabilizers in the R2TC emerged nat- urally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this subsection we elaborate how the condensation idea plays out for the various spin operators and stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To be specific, we first construct the stabilizers and holonomies in the pre-projected Hilbert space consisting of two copies of R1TCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Then we examine which of these operators survive, or become modified, under the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Stabilizers of the R2TC are recovered once again in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Although at first sight this discussion seems redundant in light of the aforemen- tioned projection scheme outlined in the context of U(1) gauge fields, there is a nice benefit to the present discussion in that it paves the way for the efficient identification and construction of holonomy operators of R2TC in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The insight gained in this subsection will also be pivotal in the construction of TN wave functions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As before we consider two interpenetrating square lattices Λ1 and Λ2, and place ZN spins on the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There are gener- alized Pauli operators satisfying ZX = ωXZ (ω = e2πi/N) at the links of each sublattice, which follow from the Higgsing formula [9]: X = e2πiA, Z = e2πiE/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) We place R1TC on each of the sublattices Λ1 and Λ2, with the star (aa(⃗ra,i)) and the plaquette (ba(⃗ra,i)) operators defined, respectively, by (a = 1, 2) aa,i =Za,x(⃗ra,i)Za,x(⃗ra,i−ˆx)−1Za,y(⃗ra,i)Za,y(⃗ra,i−ˆy)−1, ba,i =Xa,x(⃗ra,i)Xa,x(⃗ra,i+ˆy)−1Xa,y(⃗ra,i)−1Xa,y(⃗ra,i+ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) Here, the subscript i indicate the vertex of square lattice ⃗ra,i, and the extra subscripts x, y in the X, Z operators indicates the direction of the bond on which the operators are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' States in each R1TC are denoted as |ψ⟩1 and |ψ⟩2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The eigenstates of X operator are X|n⟩ = ωn|n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The constraint, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7), implies Ax 1,i+ˆy|ψ⟩ = Ay 2,i+ˆx|ψ⟩ or, after Higgsing, X1,x(⃗r1,i + ˆy)|ψ⟩ = X2,y(⃗r2,i + ˆx)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='13) In other words, only the following product of states in the pre- projection Hilbert space survives the projection, |n⟩1 ⊗ |n⟩2 P −→ |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14) Besides, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) states that Ex 1,i+ˆy + Ey 2,i+ˆx must be identi- fied with Exy i as well, which in the ZN language means Z1,x(⃗r1,i + ˆy)Z2,y(⃗r2,i + ˆx)|ψ⟩ = Z(⃗vi)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This constraint can be expressed in the Z-basis Z|m⟩ = ωm|m⟩ as the projection |m1⟩1 ⊗ |m2⟩2 P −→ |m1 + m2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) 5 In both Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) the mapping acts only at the Vhv sites where the gauge field constraint has been imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can think of the operator projection as follows: P [O|Ψ⟩] = O′ [P|Ψ⟩] = O′|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16) Here |Ψ⟩ and O refer to the pre-projected state and the oper- ator, respectively, while |ψ⟩ and O′ are their post-projection counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Based on the above consideration, one can iden- tify the operator mapping X1,x(⃗r1,i + ˆy) P −→ X0(⃗vi), X2,y(⃗r2,i + ˆx) P −→ X0(⃗vi), Z1,x(⃗r1,i + ˆy)Z2,y(⃗r2,i + ˆx) P −→ Z0(⃗vi), Z1,x(⃗r1,i + ˆy) P −→ 0, Z2,y(⃗r2,i + ˆx) P −→ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='17) where the new subscript 0 indicates the condensed sites Vhv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Operators at the Vvh sites are not affected by the projection and are simply re-labeled as X1,y(⃗r1,i) P −→ X2(⃗vi), X2,x(⃗r2,i) P −→ X1(⃗vi), Z1,y(⃗r1,i) P −→ Z2(⃗vi), Z2,x(⃗r2,i) P −→ Z1(⃗vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='18) The post-projected X, Z operators are defined with respect to the site ⃗ri, and carry three internal indices 0,1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The pre- projection plaquette operators b1(⃗r1,i) and b2(⃗r2,i), with sup- ports on Λ1 and Λ2 respectively, survive the projection P and become, after some re-labeling, b1,i P −→ bx i =X2(⃗vi)−1X2(⃗vi + ˆx)X0(⃗vi)−1X0(⃗vi − ˆy), b2,i P −→ by i =X1(⃗vi)X1(⃗vi + ˆy)−1X0(⃗vi)X0(⃗vi − ˆx)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='19) Despite the re-labeling, they are the same stabilizers from the two underlying R1TCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' On the other hand, the pre-projection star operators a1,i and a2,i from Λ1 and Λ2 become zero under the projection as they contain only Z1,x or Z2,y, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To survive the projec- tion, Z1,x(⃗r1,i + ˆy) and Z2,y(⃗r2,i + ˆx) must appear simulta- neously, as in the following operator ai =a1,ia−1 1,i−ˆxa2,ia−1 2,i−ˆy, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) which becomes, under the projection ai P −→ ai, ai =Z0(⃗vi)Z0(⃗vi−ˆx)−1Z0(⃗vi−ˆy)−1Z0(⃗vi−ˆx−ˆy) ⊗ Z2(⃗vi−ˆy)Z2(⃗vi)−2Z2(⃗vi+ˆy) ⊗ Z1(⃗vi−ˆx)Z1(⃗vi)−2Z1(⃗vi+ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='21) The three post-projection stabilizers ai, bx i , and by i are mutu- ally commuting, and are none other than the stabilizers of the R2TC Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' So far the discussion seems limited to the recovery of stabilizers that make up the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Importantly, though, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The superoperator � Xx(⃗ri) (left panel) is defined at x-bond (blue oval) with respect to ⃗ri, and � Xy(⃗ri) (right panel) at the y-bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' there is an additional stabilizer one can identify in the pre- projected Hilbert space that is not given as a mere product of ai, b1,i, b2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is given by b3,i = � Xx(⃗vi) � Xy(⃗vi + ˆx) � Xx(⃗vi + ˆy)−1 � Xy(⃗vi)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='22) The super-operators ˜Xx, ˜Xy are defined by � Xx(⃗vi) = � X1,x(⃗r1,i + ˆy) �yi−y0−1� X2,x(⃗r2,i + ˆx) �xi−x0 � Xy(⃗vi) = � X1,y(⃗r1,i + ˆy) �yi−y0� X2,y(⃗r2,i + ˆx) �xi−x0−1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23) and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The arbitrary constants x0 and y0 are kept here to simplify certain algebraic relations among the holonomies, and do not serve other purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Other stabiliz- ers b1,i, b2,i, ai have the matching lattice gauge theory expres- sions given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As for b3,i, the corresponding gauge field expression is the lattice curl m′ i = (A′)x i − (A′)x i+ˆy − (A′)y i + (A′)y i+ˆx, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='24) where (A′)x i =(xi − x0)Axx i+ˆx + (yi − y0 − 1)Axy i , (A′)y i =(yi − y0)Ayy i+ˆy + (xi − x0 − 1)Axy i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='25) Despite the apparent complexity of the definition of b3,i, the virtue of this choice is that it allows us to express the product of b3,i as a product of boundary operators and thereby leads naturally to the new holonomies, as discussed thoroughly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In fact, there is another choice, namely b3,i = X1,x(⃗r1,i + ˆy)X2,y(⃗r2,i + ˆx)−1, which is composed of op- erators from both sublattices and commutes with ai, b1,i, b2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Such a choice amounts to the condensation scheme adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This choice, however, does not allow the transfor- mation of the bulk product to the boundary product, hence no new holonomy operators can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The new field m′ i commutes with mx i , my i , ei and may seem to constitute the fourth charge in the rank-2 theory, but one can show that, after projection, b3,i becomes b3,i P −→ � bx i+ˆy �yi−y0 � by i+ˆx �xi−x0 - a composite of existing stabiliz- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The main use of identifying the stabilizer b3,i is that, through it, we come to identify the two super- � X operators as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Naively, two copies of R1TC will gen- erate only four holonomies, made of products of X1,i or X2,i 6 along horizontal and vertical directions of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The exis- tence of super-operators allows the construction of two addi- tional holonomies, as products of � Xx along the x- and of � Xy along the y-direction of the torus, and in total account for the six holonomies generating the GSD of R2TC, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Higher-order instanton and confinement in the rank-2 U(1) gauge theory While this paper focuses mainly on the ZN gauge theory on a lattice, it is instructive to touch upon the physics of U(1) rank-2 compact gauge theory in the continuum for compari- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The Maxwell theory for the gauge fields of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) is given by the effective Lagrangian, L = � (Exx)2 + (Eyy)2 + 2(Exy)2� − 1 2g B2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='26) with a quadratic dispersion ω ∼ k2 due to the fact that B is given by second spatial derivatives, B = ∂2 yAxx + ∂2 xAxx − ∂x∂yAxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For a compact gauge theory with gapless fluctua- tions, the key question is whether the theory becomes confined due to the proliferation of instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To delineate the instan- ton event, we consider the pure gauge theory in the charge- neutral sector ex = ey = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) that allows the solu- tion Exx = ∂2 yh, Eyy = ∂2 xh, Exy = −∂x∂yh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='27) The h field can be viewed as the height operator that is canonically conjugate with the flux [B(⃗r), h(⃗r′)] = iδ(⃗r −⃗r′) so that the instanton operator ei2πh creates a 2π flux [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Such an instanton event, once proliferated, can potentially lead to a confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The low energy effective theory of the height field can be obtained by integrating out the gaussian fluctuation of B, Lh = −g(∂th)2 + (∇2h)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='28) The quantum theory of h is defined in 2+1D space-time with a quadratic dispersion reminiscent of the Rokhsar-Kivelson point in 2D compact gauge theory, suggesting that the instan- ton operator has a power-law decay correlation whose opera- tor dimension depends on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The relevance of 2π flux tunnel- ing event and the proliferation of topological defects depends on the parameters of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' On the other hand, there exists another kind of higher-order instanton events that are more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For instance the in- stanton operator ei∂xh creating a flux-dipole - a pair of 2π and −2π fluxes spatially separated along the x-link - has the cor- relator 5 e−(∂xh(0)∂xh(⃗r)) r→∞ −−−→ Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='29) 5 The importance of flux-dipole tunneling events in governing the phase of matter was pointed out in the context of one-dimensional dipolar boson Hubbard model recently [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These higher-order instanton terms creating flux-dipole tun- neling events display long-range order and thus can prolif- erate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As a result, the theory would be confined due to the proliferation of instanton-dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This unique feature is due to the fact that the dipole flux is conserved in our higher-rank gauge theory and thus the 2π flux tunneling event must appear in a quadrupolar process, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', creating a pair of opposite flux- dipoles from the vacuum and separating them apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The cor- relation function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='29) implies the interaction between flux-dipoles are short-ranged so they will proliferate and gap out the low-energy modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Conservation laws Before the explicit construction of holonomies, it is useful to identity the full content of conserved charges in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physically, it is the braiding of one of these conserved charges around the non-contractible loop of the torus that defines the holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The discussion is most conveniently carried out in the continuum language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The three expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) can be cast in the con- tinuum as mx = ∂xAyy − ∂yAxy, my = ∂xAxy − ∂yAxx, e = ∂2 xExx + ∂2 yEyy + ∂x∂yExy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='30) The three charge densities obey the continuity equations as derived recently [11], ∂tmx + ∂xJxx m + ∂yJxy m = 0, ∂tmy + ∂xJxy m − ∂yJyy m = 0, ∂te + ∂2 xJxx e + ∂x∂yJxy e + ∂2 yJyy e = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='31) where Jab m and Jab e (a, b = x, y) are symmetric rank-2 current densities for the magnetic and electric charges, respectively 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' By assuming vanishing currents at the boundary, one can show that all three monopole charges are conserved: ∂t � edV = ∂t � mxdV = ∂t � mydV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='32) In addition, we have three dipole conservation laws ∂t � xedV =− � d2r x � ∂2 xJxx e + ∂x∂yJxy e + ∂2 yJyy e � = 0, ∂t � yedV =− � d2r y � ∂2 xJxx e + ∂x∂yJxy e + ∂2 yJyy e � = 0, ∂t � (xmy + ymx)dV = − � d2r [x∂xJxy m − y∂yJxy m ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='33) 6 The continuity equations derived in [11] were for the vector electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we are dealing with the dual theory with vector magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The two theories are dual to each other [9], and the continuity equations for the quasiparticles have the same structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7 Altogether we have the conservation of three monopoles and three dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will now construct the six magnetic and six electric holonomies associated with the x- and y-winding around the torus of the six conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' APPLICATIONS OF THE CONDENSATION SCHEME Two useful applications of the condensation idea are con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One is the explicit construction of the tensor network wave function for the ground state of R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The second is the construction of the rank-2 version of the double-semion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Tensor network representation of ZN R2TC wavefunctions In this section, we show that ZN R2TC wave function can be obtained by stacking two copies of ZN R1TC wave func- tion followed by a certain isometric operation that reflects the gauge-field constraint of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To this end, we begin with the tensor network (TN) representation of the R1TC ground state wave function that is composed of two types of tensors g and T as below: , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) where gm ij = δi,jδj,m, Tlurd = δr+u,l+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) The delta function in the second line is implemented mod N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The physical index m represents the qudit state in the Z-basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', Z|m⟩ = ωm|m⟩, and all subscripts denote the virtual indices of dimension N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can easily show that g and T tensors satisfy the following relations: [Zn]mm′gm′ ij = [Zn′]ii′[Zn−n′]jj′gm i′j′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) [Xn]mm′gm′ ij = [X−n]ii′[X−n]jj′gm i′j′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4) [Zn]ll′[Z−n]uu′[Z−n]rr′[Zn]dd′Tl′u′r′d′ = Tlurd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) and [Xnl]ll′[Xnu]uu′[Xnr]rr′[Xnd]dd′Tl′u′r′d′ = Tlurd, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) if (nr−nl+nu−nd) mod N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Graphical representations of the above equations are the following: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) Note that the T-tensor generates the string-net configura- tions corresponding to the domain wall configurations of the N-state Potts model on the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For example, Z2 R1TC wave function is depicted as a superposition of closed- loop configurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', the domain wall of the Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Using the above relations, one can easily verify that the TN wave function |ψ⟩, obtained by contracting all the virtual in- dices, is the ground state of the ZN R1TC Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', ai|ψ⟩ = |ψ⟩ and bi|ψ⟩ = |ψ⟩, or graphically as below, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8) Now, we consider the square lattice (Λ1) and its dual (Λ2) together, and accommodate the ZN R1TC wave function on each lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', |R1TC⟩Λ1 ⊗ |R1TC⟩Λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Then, there are two types of vertices in the system: Vhv (vh) at which horizon- tal (vertical) bonds in Λ1 and vertical (horizontal) bonds in Λ2 cross each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Generally, two unentangled qudits live on the vertex Vhv ⊕ Vvh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now we impose the following isome- try on the two qudits labeled by quantum numbers (m1, m2) residing on the Vvh vertices: P m m1m2 = δm,m1+m2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) where the delta function is implemented mod N, and P m m1m2P m′ m1m2 = δmm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The two-qudit state is mapped to a single-qudit state through isometry and, furthermore, the re- sulting TN exactly represents the ground state of ZN R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The TN wave function thus constructed is written in the Z- basis, Z|m⟩ = ωm|m⟩, and the constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) is faithfully reflected through the isometry tensor P m m1m2 = δm,m1+m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The TN representation for the R2TC ground state is illus- m m u r=T7n 7n-n n n n Yni n n8 trated below: , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) where the square lattice (dual lattice) in solid (dotted) line de- notes Λ1 (2), and the gray square stands for the T-tensor given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The isometry P satisfies the relations, [Zn]mm′P m′ m1m2 = [Zn]m1m′ 1[Zn]m2m′ 2P m m′ 1m′ 2, [Xn]mm′P m′ m1m2 = [X−n′]m1m′ 1[Xn′−n]m2m′ 2P m m′ 1m′ 2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) or graphically .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12), it is straightforward to derive the following relation: [Xn]mm′P m′ m1m2gm1 ud gm2 lr = P m m1m′ 2gm1 ud [X−n]ll′[X−n]rr′gm2 l′r′, [Zn]mm′P m′ m1m2gm1 ud gm2 lr = P m m1m2gm1 u′dgm2 l′r [Zn]uu′[Zn]ll′ = P m m1m2gm1 u′dgm2 lr′ [Zn]uu′[Zn]rr′ = P m m1m2gm1 ud′gm2 lr′ [Zn]dd′[Zn]rr′ = P m m1m2gm1 ud′gm2 l′r [Zn]dd′[Zn]ll′, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='13) or graphically .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14) Now, using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14), we show that the above TN wave function, |ψ⟩, is the ground state of the ZN R2TC Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', bx i |ψ⟩ = |ψ⟩, by i |ψ⟩ = |ψ⟩ as below , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) and ai|ψ⟩ = |ψ⟩ in the following way, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16) This completes the proof that the TN ground state wave function of R2TC is given as two copies of those of R1TC with an additional isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To summarize, the ground state wave function of the R1TC is constructed using the well- known tensors given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Two copies of such TN wave functions are introduced one for each of the two in- terpenetrating square lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Then the isometry operation P m m1m2 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) acts on half of the overlapping sites (the Vvh sites) to reduce the two qudits (m1, m2) to a single qudit m = m1 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There are a large number of ground states given by the GSD formula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1), and our TN construction captures only one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The rest of the states can be generated by applying holonomy operators, to be derived in the next section, to the existing TN wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Twisted rank-2 gauge theory from anyon condensation In this section, we utilize the coupled layer construction protocol to build a twisted rank-2 gauge theory in 2D with dipole conservation, whose gauge flux turns out to have semionic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The strategy is to combine two intersect- ing Z2 twisted gauge theories from string-net models [24] and implement anyon condensation to impose restricted mobility for quasiparticle excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To avoid technical complexities, the discussion in this section is limited to N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To construct the “semionic” version of Z2 gauge theory from commuting projectors, we will need to start from triva- lent 2D lattices such as the Fisher lattice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A small diamond shape is added at each vertex of the square lat- tice so that every vertex is connected to three links with Z2 qubits living on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7u m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' m u m om m m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=" 0 mmo aXn Zn Zn Vn'-nh n Zn Th79 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (a) The double semion model on the Fisher lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The flux operator � i∈Y Zi is defined on the vertex with three Z operators on the adjoining links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The charge operators � X are defined on the diamond and the octagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (b) The string operator in the double semion model (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (c) Intersecting bilayers of the Fisher lattice, illustrated as one solid and one dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The cir- cles are the intersection between x-link from the first layer and the y-link from the second layer where we put a strong coupling term −JzZ1Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (d) The charge operator after perturbation contains the product of four octagon operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now we begin with the conventional double semion model that manifests a twisted Z2 gauge theory [24] on the Fisher lattice with the Hamiltonian, H = − � Y � i∈Y Zi − � O F0 � i∈O Xi − � D F1 � i∈D Xi F0 = � i∈VO Si, F1 = � i∈VD Si, Si = � 1 0 0 i � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='17) Here Y is any set of three co-planar links entering a vertex, O is the octagon, and D is the diamond on the Fisher lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' VO (VD) refers to the eight (four) links pointing outward from an octagon (diamond), and are indicated by S in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the Z basis, the first term in H imposes a condition that the parity of the gauge flux entering any vertex be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The remaining two terms in H provide dynamics to the gauge field while preserving this parity at each vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Specifically, they effectively bind the charge, given by � X, to the gauge flux as measured by F0 and F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This charge-flux binding has important consequences for the braiding statistics, as creating a flux excitation would si- multaneously generate half Z2 charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The string operator L that creates a pair of flux is Lm = � a∈red Xa � a∈blue Sa � a∈green (−1)Ga, Ga = 1 4(1 − Za), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='18) where the X operator along the red links in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3(b) flips the spins along the string, analogous to the flux operator in the toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The red string has a product of X operators while the operators on blue (green) lines living on either side of the string embellish it with an additional sign structure that endows the semionic statistics between the flux, as well as ensuring that Lm commutes with the Hamiltonian except near the endpoints of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Two Lm operators anti-commute with each other when they intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The extra sign structure embellished a half-charge with the flux, and, as a result, the flux excitation carries a half gauge charge and thus displays semion statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now we take two intersecting layers of the Fisher lattice with the x-link from the first layer intersecting the y-link from the second layer and vice versa, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Links that form the diamonds do not overlap between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We then strongly couple the qubits from distinct layers on the cir- cled intersections in Fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3 through the interaction, − JzZ1 i Z2 i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='19) with Jz ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this strong coupling limit, the vertex oper- ators on Y -junctions and the charge operators � X on dia- monds are unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, charge operators around oc- tagons do not commute with the Jz term, and instead, a prod- uct of four such terms appearing at the third order in pertur- bation theory does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3(d), we illustrated the new charge operator after perturbation, composed of the product of four octagon operators from Jz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The resulting Hamiltonian takes a form very similar to the R2TC, except that the dia- mond plaquette terms and the quadrupolar octagon terms are supplemented with a product of S operators over outward- pointing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It can be checked that all the terms in the new Hamiltonian commute with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' What is the excitation structure of this semionic version of R2TC?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The charge excitations share a similar character as the scalar charge theory in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The novelty comes from the vector flux excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For example, consider a 1D parti- cle moving in the x-direction which creates a flux mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Sup- pose that the line on which this 1D excitation move intersects that of another flux mx moving in the y- direction, which can only hop on the even-numbered sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The string operators associated with the two 1D particles anti-commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This anti- commutation of string operators is related to the fact that two such flux excitations can undergo a full braiding, so their mu- tual statistics is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this case, the two types of flux have mutual statistics θ = π, which contrasts with trivial mutual statistics θ = 0 in the original R2TC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (b) (a) S S S X X S (c) (d) X S C10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' HOLONOMY CONSTRUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Pre-projection holonomies There are six pre-projection holonomies consisting of the product of X operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The first four holonomies are taken directly from those of two independent R1TC’s, W pp 1 (yi) = Lx � xi=1 X1,x(⃗r1,i), W pp 2 (xi) = Ly � yi=1 X1,y(⃗r1,i), W pp 3 (yi) = Lx � xi=1 X2,x(⃗r2,i), W pp 4 (xi) = Ly � yi=1 X2,y(⃗r2,i) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) for a torus of size Lx × Ly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here, ⃗r1,i = ⃗vi − ˆy/2, and ⃗r2,i = ⃗vi − ˆx/2, where ⃗vi = (xi, yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' All of them commute with the pre-projection stabilizers ai, b1,i, b2,i and b3,i introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The two additional holonomies are constructed from the super-operators � Xx and � Xy: W pp 5 (yi) = lcm(Lx,N) � xi=1 � Xx(⃗vi), W pp 6 (xi) = lcm(Ly,N) � yi=1 � Xy(⃗ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) Note that in these two cases the product over xi (yi) goes around the torus multiple times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' by cx =lcm(Lx, N)/Lx = N/gcd(Lx, N), cy =lcm(Ly, N)/Ly = N/gcd(Ly, N), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) to ensure that the holonomy action on a ground state returns another ground state with no residual excitations [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As a consequence we have [W pp 5 ]gcd(Lx,N) = [W pp 6 ]gcd(Ly,N) = 1 while for other holonomies it is (W)N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The six holonomies are seemingly coordinate-dependent, but this de- pendence goes away when their actions on the ground state are examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We omit the proof, which is purely technical, since the result is well-anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The holonomies we constructed can be motivated in a dif- ferent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can show that the product of b1,i or b2,i over all sites inside a rectangle S = [x1, x2] × [y1, y2] is equal to the product of X’s or X−1’s along its four boundaries as all terms in the interior cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These boundary operators precisely take the form of W pp 1 through W pp 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In a similar fashion, product of b3,i over a closed area leads to the cancel- lation of all terms in the interior, leaving only the product of super-X operators along the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These boundary op- erators motivate the W pp 5 , W pp 6 holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Collectively we refer to the six logical operators in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) as X-holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The six Z-holonomies are constructed by following a sim- ilar reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can show that the product of ai in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) inside a closed region reduces to the boundary product, which motivates the two Z-holonomies: � W pp 5 (xi) = Ly � yi=1 Z2,x(⃗r2,i + ˆx)Z2,x(⃗r2,i)−1, � W pp 6 (yi) = Lx � xi=1 Z1,y(⃗r1,i + ˆy)Z1,y(⃗r1,i)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4) Hereafter we drop the explicit coordinate dependence from the holonomy operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Taking the product of (ai)yi−y′ 0 on a closed region and extracting the boundary terms gives two other Z-holonomies: � W pp 1 = lcm(Ly,N) � yi=1 �Z1,x(⃗vi), � W pp 2 = Lx � xi=1 �Z1,y(⃗vi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) Finally, the product of (ai)xi−x′ 0 gives � W pp 3 = Ly � yi=1 �Z2,x(⃗vi), � W pp 4 = lcm(Lx,N) � xi=1 �Z2,y(⃗vi), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) Various super-Z operators appearing in the holonomies are �Z1,x(⃗vi) =Z1,x(⃗r1,i)Z2,y(⃗r2,i +ˆx − ˆy) ⊗ � Z2,x(⃗r2,i)Z2,x(⃗r2,i + ˆx)−1�yi−y′ 0−2 , �Z1,y(⃗vi) =Z1,y(⃗r1,i) � Z1,y(⃗r1,i)Z1,y(⃗r1,i + ˆy)−1�yi−y′ 0−1 , �Z2,x(⃗vi) =Z2,x(⃗r2,i) � Z2,x(⃗r2,i)Z2,x(⃗r2,i + ˆx)−1�xi−x′ 0−1 , �Z2,y(⃗vi) =Z2,y(⃗r2,i)Z1,x(⃗r1,i − ˆx + ˆy) ⊗ � Z1,y(⃗r1,i)Z1,y(⃗r1,i + ˆy)−1�xi−x′ 0−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) Note that � W1 and � W4 involve the product of super-operators over the circumference of the torus cy and cx times, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Arbitrary constants x′ 0, y′ 0 are introduced for generality and for simplifying certain aspect of the holonomy algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Invoking ZX = ωXZ, one can verify the following Heisenberg algebra among the X- and Z-holonomies: � � W pp 1 , W pp 1 � = ωcy, � � W pp 2 , W pp 2 � = ω, � � W pp 3 , W pp 3 � = ω, � � W pp 4 , W pp 4 � = ωcx, � � W pp 5 , W pp 5 � = ωcx, � � W pp 6 , W pp 6 � = ωcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8) The commutator here means [A, B] = ABA−1B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In addi- tion, the following set of holonomies show nontrivial commu- tation:� � W pp 1 , W pp 5 � =ωcxcy[y′ 0−y0− 1 2 (N−gcd(Ly,N))] � � W pp 2 , W pp 6 � =ωcy[y′ 0−y0+ 1 2 (N−gcd(Ly,N))] � � W pp 3 , W pp 5 � =ωcx[x′ 0−x0+ 1 2 (N−gcd(Lx,N))] � � W pp 4 , W pp 6 � =ωcxcy[x′ 0−x0− 1 2 (N−gcd(Lx,N))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) 11 However, the following choice removes the non-trivial phase factors among them 7, y′ 0 =y0 + 1 2 � N − gcd(Ly, N) � , x′ 0 =x0 + 1 2 � N − gcd(Lx, N) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) and the nontrivial Heisenberg algebra is spanned entirely by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The holonomies of the R2TC are then obtained by projection of the pre-projection holonomies constructed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Post-projection holonomies As one can see from the projection schemes, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='17)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='18), the X-operators remain intact through the projection except some re-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The pre-projection X-holonomies of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) become, after re-labeling, W1 = Lx � xi=1 X0(⃗vi), W2 = Ly � yi=1 X2(⃗vi) W3 = Lx � xi=1 X1(⃗vi), W4 = Ly � yi=1 X0(⃗vi) W5 = lcm(Lx,N) � xi=1 � X1(⃗vi + ˆx) �xi−x0� X0(⃗vi) �yi−y0−1, W6 = lcm(Ly,N) � yi=1 � X2(⃗vi + ˆy) �yi−y0� X0(⃗vi) �xi−x0−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) after the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' At first it seems the number of dis- tinct holonomy actions is N 4gcd(Lx, N)gcd(Ly, N) since W1 through W4 has (W)N = 1 but (W5)gcd(Lx,N) = (W6)gcd(Ly,N) = 1, at odds with the GSD formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A delicate con- sideration is required to see that the number of independent actions among W1 and W4, which are both products of X0’s, is not N 2 but Ngcd(Lx, Ly, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One begins with labeling the holonomy (W1)n1(W4)n4 by (n1, n4) and invoking the identity 8 (W1)Ly|GS⟩ = (W4)Lx|GS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) 7 This has been the sole purpose of keeping the arbitrary constants in the definition of super-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 8 The proof of the identity is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since W1(y) = �Lx xi=1 X0(xi, y) and W4(x) = �Ly yi=1 X0(x, yi), it then follows that �Ly y=1 W1(y) = �Lx x=1 W4(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, since the actions of W1(y) and W4(x) do not actually depend on y and x, respectively, when acting on the ground states, we obtained the claimed relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This implies the equivalence relation (n1, n4) ∼ (n1 + Ly, n2 − Lx) among the holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We need to carefully figure out how the points (n1, n4) become equivalent by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) when the ZN nature is considered at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Invoking the two winding numbers cx, cy defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3), (W1)cyLy|GS⟩ = (W4)cyLx|GS⟩ = |GS⟩, (W1)cxLy|GS⟩ = (W4)cyLx|GS⟩ = |GS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='13) Applying the Euclidean argument for identifying the gcd of two integers, we conclude (W1)Ny|GS⟩ = (W4)Nx|GS⟩ = |GS⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14) where Nx ≡gcd(cyLx, N), Ny ≡gcd(cxLy, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) we deduce the equivalence rela- tion n1 ∼ n1 + Ly ∼ n1 + Ny, n2 ∼ n2 + Lx ∼ n2 + Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16) Invoking the Euclidean argument again, the number of in- equivalent integers n1 for fixed n4 becomes gcd(Ly, Ny), and the number of inequivalent (n1, n4) equals Nxgcd(Ly, Ny).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It can be simplified further to Nxgcd(Ly, Ny) = gcd (Lxcy, N) gcd (Ly, N) = Ngcd (Lx, Ly, N) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='17) by employing several number-theoretic identities gcd(a, gcd(b, c)) = gcd(gcd(a, b), c) = gcd(a, b, c), mgcd(a, b) = gcd(ma, mb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='18) The number of independent holonomy actions among W1 and W4 is Ngcd(Lx, Ly, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The GSD formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) breaks down to N 2 · (Ngcd(Lx, Ly, N)) · gcd(Lx, N) · gcd(Ly, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='19) Here, the first N 2 are coming from W2 and W3, Ngcd(Lx, Ly, N) from W1 and W4, and gcd(Lx, N) · gcd(Ly, N) from W5 and W6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As for the Z-holonomies, projection of the pre-projection 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Pictorial representation of the holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (first row) Six X-holonomies as creation/annihilation of e dipole-antidipole pairs oriented either horizontally or vertically (W1 through W4), and of e monopole-antimonopole pairs (W5 and W6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (second row) Six Z-holonomies as creation/annihilation of mx or my monopole-antimonopole pairs (� W1 through � W4), and of m dipole-antidipole pairs (� W5 and � W6 Monopole braiding processes are accompanied by the motion of auxiliary dipoles to preserve the total dipole moment, but they are omitted from the figure for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Z-holonomies of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) leads to � W1 = lcm(Ly,N) � yi=1 Z0(⃗vi − ˆy) � Z1(⃗vi)Z1(⃗vi + ˆx)−1�yi−y0, � W2 = Lx � xi=1 Z2(⃗vi) � Z2(⃗v)Z2(⃗vi + ˆy)−1�yi−y0, � W3 = Ly � yi=1 Z1(⃗vi) � Z1(⃗vi)Z1(⃗vi + ˆx)−1�xi−x0, � W4 = lcm(Lx,N) � xi=1 Z0(⃗vi − ˆx) � Z2(⃗vi)Z2(⃗vi + ˆy)−1�xi−x0, � W5 = Ly � yi=1 Z1(⃗vi + ˆx)Z1(⃗vi)−1, � W6 = Lx � xi=1 Z2(⃗vi + ˆy)Z2(⃗vi)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) One can check the following non-trivial commutators among the post-projection holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' � � W1, W1 � = ωcy, � � W4, W4 � = ωcx, � � W2, W2 � = ω, � � W3, W3 � = ω, � � W5, W5 � = ωcx, � � W6, W6 � = ωcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='21) The commutator here means [A, B] = ABA−1B−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There exist four more nontrivial commutations relations: � � W1, W5 � =ωcxcy[y′ 0−y0− 1 2 (N−gcd(Ly,N))] � � W2, W6 � =ωcy[y′ 0−y0+ 1 2 (N−gcd(Ly,N))] � � W3, W5 � =ωcx[x′ 0−x0+ 1 2 (N−gcd(Lx,N))] � � W4, W6 � =ωcxcy[x′ 0−x0− 1 2 (N−gcd(Lx,N))], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='22) which is nothing but the projection of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Hence, apply- ing the condition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) removes this nontrivial phase factors as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can read off the GSD from the Heisenberg al- gebra of the holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For instance W1 acting on a ground state changes the eigenvalues of � W1 by ωcy, generating in total N/cy = gcd(Ly, N) distinct ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Naively applying the reasoning to the first two pairs of commutators (1 and 4) gives the GSD equal to (N/cx)(N/cy) = gcd(Lx, N)gcd(Ly, N), the next two pairs (2 and 3) yields N 2, and the final two pairs (5 and 6) yields another gcd(Lx, N)gcd(Ly, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In total, this gives the GSD count N 2[gcd(Lx, N)gcd(Ly, N)]2 that is less than the correct GSD formula, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1), by gcd(Lx, N)gcd(Ly, N)/Ngcd(Lx, Ly, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In other words, the holonomies constructed above underspans the space of ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The deficiency comes from the fact that � W4 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) is not the most minimal choice of the holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The correct holonomy expression can be found by referring to Wi W2 W3 W4 Ws W6 le ellé é e e le el el le e e e aTe e mx omy T4 mx mx ★ my my my mx W1 W W W W W 4 5 613 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' [10]: � W min 4 = nxLx � xi=1 Z0(⃗vi − ˆx) � Z2(⃗vi)Z2(⃗vi + ˆy)−1�xi−x0, ⊗ nyLy � yi=1 Z0(⃗vi − ˆy) � Z1(⃗vi)Z1(⃗vi + ˆx)−1�yi−y0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23) where the integers nx, ny are given by nx = gcd(Lx, N) gcd(Lx, Ly, N), ny =lcm(Lx, gcd(Ly, N)) + kN Ly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='24) Here k is a minimal integer that makes ny an integer [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With the new definition of � W4 → � W min 4 we obtain a new commutator � � W min 4 , W4 � = ωnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='25) The GSD coming from this sector equals N/nx = Ngcd(Lx, Ly, N)/gcd(Lx, N) and indeed, we recover the full GSD simply from the Heisenberg algebra, with the mod- ified � W4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Replacing � W4 by � W min 4 gives us an orthogonal set of six X-holonomies {W1, · · · , W6} and six Z-holonomies {� W1, · · · , � W6} that fully span the ground states of R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In making physical interpretations of the � W4 holonomy, though, we will continue to adopt the simpler (albeit slightly incorrect) representation as the horizontal braiding of my quasiparticle as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The interpretation of � W min 4 involves a mix of the horizontal braiding of my and the ver- tical braiding of mx, as can be seen from its definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With the explicit construction of the holonomies, we can check the quantum numbers of the TN ground state wave function we have constructed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Following the sim- ilar procedure as in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16), one can verify that our TN wave function |ψ⟩ on the torus is the simultaneous eigenstate of the four X-holonomies W1 through W4, as well as two Z-holonomies � W5 and � W6, with eigenvalue +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The remaining six holonomies, � W1 through � W4 and W5, W6, then act to shift the ground state into orthogonal ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One can also construct a TN wave function for the eigenstate of all six of the X-holonomies W1 through W6, but it requires a ‘double layer structure’ of TN that goes beyond the present construction and will be presented elsewhere [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physical interpretation of the holonomies It is well known that the X- and Z-holonomies in the R1TC has a concise physical picture as the creation and sub- sequent annihilation of a pair of ee or mm (bar denotes the anti-particle) anyons after one anyon is braided around one of the non-contractible paths of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A total of four holonomies form two conjugate pairs and span the N 2 de- generate ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To account for the GSD of R2TC, which reaches the maximum value of N 6, one requires a to- tal of twelve holonomies breaking up into two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Six of them bear obvious physical interpretations as the braiding of mx, my, e particles around either of the two circumferences of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We provide the physical interpretations of the remaining six holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Each action of X-holonomies corresponds to the winding of the three electric quantities that are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Figure 4 illustrates these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The physical action of W1 (W2) among the X-holonomies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) is to create a y- oriented e-dipole and its anti-dipole, then to move one of the dipoles along the horizontal (vertical) non-contractible path of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For W3 (W4), it is x-oriented e-dipole braided hori- zontally (vertically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The W5 (W6), on the other hand, moves the e monopole horizontally (vertically).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Note that an auxil- iary dipole is attached to the e monopole during its adiabatic motion, to ensure the total dipole moment conservation in the process, and disappears at the end of completing the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We omit the auxiliary dipoles from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 4 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Each action of X-holonomies corresponds to the winding of the three magnetic quantities that are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The ac- tion of the first four Z-holonomies in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) is to create a monopole and anti-monopole pair of either mx or my and braid them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Specifically, � W1, � W2 (� W3, � W4) braid mx (my) along the y- and x-oriented non-contractible loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To en- force the total dipole moment ymx + xmy = 0, some aux- iliary dipoles are attached during the vertical motion of mx as well as the horizontal motion of my [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The last two Z-holonomies, � W5 and � W6, correspond to the winding of m- dipole along the y and x non-contractible loops of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The list of holonomies and their physical interpretations are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (W1, � W1) (e-dipole, h) (mx-monopole, v) (W2, � W2) (e-dipole, v) (my-monopole, h) (W3, � W3) (e-dipole, h) (my-monopole, v) (W4, � W4) (e-dipole, v) (mx-monopole, h) (W5, � W5) (e-monopole, h) (my-dipole, v) (W6, � W6) (e-monopole, v) (mx-dipole, h) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (left) Pair of holonomies (logical operators) with non- trivial commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (middle) nature of e-excitations and the direction of braiding associated with a given holonomy W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (right) nature of m-excitations and the direction of braiding asso- ciated with a given holonomy � W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (h=horizontal, v=vertical) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Field-theoretic derivation of the holonomies The holonomy construction thus far proceeded from a known microscopic Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', R2TC model whose quasiparticle excitations are well-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Historically, the holonomies engendered by the Wilson line operators manifest the global flux sectors to which the ground state on a torus belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Building on this line of thinking, we show how to 14 obtain the Wilson operators pertinent to the R2TC from the underlying rank-2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For higher-rank gauge theories, the Wilson operators cre- ating immobile quasiparticle excitations turn out to be richer and more diverse than in the conventional ZN gauge theory for the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1) Due to the restricted mobility of the quasiparticles, some of the Wilson lines need to be straight and geometrically oriented in a specific direction[46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2) There might exist other Wilson operators defined on a non- contractable manifold, such as membrane, cage, or fractal, that are responsible for the holonomies of higher-rank gauge theories [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3) Different Wilson operators that are paral- lel to each other may not render the same value, as opposed to the conventional ZN gauge theory whose Wilson line opera- tors are invariant under translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For higher-rank gauge the- ory, the dipole and quadruple moments transform nontrivially under translation, and so does the global flux sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Conse- quently, two parallel flux lines might return different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Recall that in the usual 2D ZN gauge theory, the magnetic flux is given by m = ∂xAy − ∂yAx and the total flux on the half cylinder A with boundaries at x = x0 and x = xn is characterized by parallel Wilson line operators � mdV = � Ay(xn, y)dy − � Ay(x0, y)dy = 0, with the integral � going around the full circumference of the cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The net flux condition ( � mdV = 0) implies that the two parallel Wilson lines render the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since the two Wilson lines are spatially separated while the Hamiltonian is local, each � Ay(x, y)dy must commute with all local terms in the Hamiltonian and can be treated as a global flux operator that characterizes the holonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' One obtains another Wilson line operator along the y-direction from the charge sector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' � Ey(x, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These two comprise all possible Wilson lines along the y-loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now we apply this protocol to R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Begin with the def- inition of three monopole charges given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) in the continuum limit, mx = ∂xAyy − ∂yAxy, my = ∂xAxy − ∂yAxx, e = ∂2 xExx + ∂2 yEyy + ∂x∂yExy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='26) As noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' II D, the magnetic charges mx, my demon- strate a number of conservation laws � mxdV = � mydV = � (xmy + ymx)dV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='27) The first two yields � mxdV = � Ayy(xn, y)dy − � Ayy(x0, y)dy = 0, � mydV = � Axy(xn, y)dy − � Axy(x0, y)dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='28) Following the aforementioned argument, one can define two Wilson line operators, W2(x) = � Ayy(x, y)dy, W4(x) = � Axy(x, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='29) Due to the flux conservation law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='28), they are both uniform along the x-coordinate: ∂xW2(x) = ∂xW4(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The subscripts 2, 4 are intended to match the definitions of post-projection Wilson operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) 9 In addition, we have � A (ymx + xmy)dV = 0 = � yAyy(xn, y)dy − � yAyy(x0, y)dy + � xn x0 �� Axy(x, y)dy � dx = � yAyy(xn, y)dy − � yAyy(x0, y)dy + (xn − x0)W4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='30) In arriving at the last equality we used the fact that the Wilson line operator W4 = � Axy(x, y)dy is uniform in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We arrive at another Wilson line operator, W6(x) = � yAyy(x, y)dy + xW4, ∂xW6(x) = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='31) which matches the definition of W6 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) after Higgs- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As the theory is rotationally symmetric, the other set of Wil- son line operators follow as integrals along the x-loop: W1 = � Axy(x, y)dx, W3 = � Axx(x, y)dx, W5 = � xAxx(x, y)dx + yW1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='32) with matching definitions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) after Higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Their coordinate independence follows readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The previous holonomies W1 through W6 were derived on the basis of conservation laws of the magnetic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Al- ternatively, the holonomies can be derived from the electric charge conservation, e = ∂2 xExx + ∂2 yEyy + ∂x∂yExy, � e dV = � xe dV = � ye dV = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='33) and hence � A e dV = � ∂xExx(x0, y)dy − � ∂xExx(xn, y)dy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='34) 9 Note that we use the same symbol W1 · · · W6 for the holonomies in both continuum and the ZN theories although, strictly speaking, the ZN holonomies are obtained by raising the continuum holonomies to an expo- nential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 15 This yields the first holonomy � W5(x) = � ∂xExx(x, y)dy, ∂x� W5(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='35) From the other two conservation laws we find � ye dV = � (y∂yExy + y∂xExx)(xn, y)dy − � (y∂yExy + y∂xExx)(x0, y)dy, � A xe dV = � Exx(x0, y)dy − � Exx(xn, y)dy + (xn − x0)� W5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='36) We arrive at two additional Wilson line operators � W1 = − � (y∂yExy + y∂xExx)dy = � (Exy − y∂xExx)dy, � W3 = − � Exxdy + x� W5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='37) The other three Wilson line operators are obtained from rota- tional symmetry: � W2 = − � Eyydx + y� W6, � W4 = � (Exy − x∂yEyy)dy, � W6 = � ∂yEyydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='38) Coordinate independence of all Wilson operators can be ver- ified easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' After Higgsing, � W1 through � W6 match the six Z-holonomies of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physical interpretation of the holonomy operators has been given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For completeness we briefly mention that in a theory with vector-electric and scalar-magnetic charges such that ex = ∂xExx + ∂yExy ey = ∂xExy + ∂yEyy m = ∂2 xAyy + ∂2 yAxx − ∂x∂yAxy, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='39) we can construct the relevant holonomies based on a different set of conservation laws � exdV = � eydV = � (yex − xey)dV = 0, � mdV = � xmdV = � ymdV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='40) As the derivation in this subsection clearly shows, the con- struction of holonomies are firmly rooted in the conservation laws such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='27) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The existence of dipole- like conservations in addition to the usual charge conserva- tions for mx, my, e monopoles plays a crucial role in con- structing the full set of holonomies for the rank-2 gauge theory as well as its Higgs descendant, which is the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We sus- pect that a similar scheme can be exploited for the holonomy construction in other rank-2 gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (a) Braiding charge e around the flux m in the conventional ZN gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The trajectory of the braiding loop corresponds to the total flux inside the enclosed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (b) Braiding charge e around the flux mx or my in R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The trajectory of the braiding loop cor- responds to the total dipolar flux inside the enclosed area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Understanding the position-dependent braiding The seemingly puzzling feature of R2TC was the position- dependent statistical phase obtained when one quasiparticle is braided around another [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' While various elaborate argu- ments for why this should be so has been given already [9–11], it turns out the field-theoretic holonomies just constructed can provide a simple picture for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To do so, we first review how the adiabatic braiding process relates to the statistical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We begin with the prominent ZN gauge theory example where the charge e and flux m have nontrivial statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Creating a pair of m flux excitations is implemented at the two endpoints of an open string ei � x 0 Exdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To braid the charge around the flux, we create a pair of charge (e) and anti-charge (e) connected by an open string, wind the e particle around m and annihilate it with the anti-charge as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The trajectory of the e particle is associated with the Aharonov-Bohm (AB) phase exp � i � ⃗A · d⃗r � which corresponds to the total flux � mdV (m = ∇ × A) inside the area enclosed by the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As a result, the braiding of charge excitation creates a flux loop that measures the total flux inside so their braiding phase is just the AB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now let us go back to the R2TC theory with vector- magnetic and scalar-electric charges as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The flux 10 mx or my excitations are created by open-string op- erators such as � W open 1 ∼ exp � i � ym 0 (Exy − y∂xExx) dy � or � W open 3 ∼ exp � i � ym 0 (x∂xExx − Exx) dy � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' They are none other than open-ended versions of the holonomies constructed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV D and have physical 10 Here we use the word magnetic ‘flux’ interchangeably with the magnetic ‘charge’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (a) (b) 0 y=yi m or m :=2 P [xA x + yA*] dx DA*dx16 interpretations of creating a mxmx or a mymy pair separated along the vertical direction as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To braid the m flux, we create a pair of charge e and anti- charge ¯e connected by an open string shown as the horizontal blue segment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5(b), wind the e particle around mx or my as shown by two horizontal dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5(b), and annihilate it with the anti-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The trajectory of the e par- ticle is associated with the phase factor W5 ∼ ei � [(xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2)]dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As one can see from Table I, W5 is associated with the hori- zontal braiding of e particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For simplicity, we choose the braiding trajectory consisting of two parallel lines along the x-direction above and below the m flux, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' at y = y1 and y = y2, y1 < ym < y2, and ym indicating the y-coordinate of the m flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We can further simplify the braiding operator as ei � � (xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2) � dx = ei � � (xAxx+yAxy)(x,y1)−(xAxx+yAxy)(x,y2) � dx × ei � y2 y1 � (yAyy+xAxy)(x1,y)−(yAyy+xAxy)(x2,y) � dy = ei � (ymx+xmy)dV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='41) In the second line we inserted some y-oriented integrals that cancel each other due to the periodic boundary condition and x2 = x1 + Lx 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now one can understand the braiding oper- ation as the line integral of the vector field (xAxx + yAxy, xAxy + yAyy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='42) The third line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='41) follows from Stokes’ theorem and the definition of mx, my in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It shows that the braid- ing operation measure not the flux, but the ‘dipolar flux’ that depends on the x-position of my and the y-position of mx that the e particle braids around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The statistical phase becomes ac- cordingly position-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Dipolar braiding among other quasiparticles can be understood in similar ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The deriva- tion of dipolar braiding statistics in terms of field-theoretic Wilson lines given here has some overlap with earlier con- sideration [9, 11] of the dipolar braiding, but here we give a more clarified picture of how this seeminingly peculiar braid- ing statistics arises rather naturally in rank-2 gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It also suggests that the dipolar braiding phase is not unique to R2TC, but may be a general feature of rank-2 gauge theories and its Higgsed descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To put it in broader perspective, we comment that a position-dependent braiding process is also present elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Indeed, while typically not emphasized, even in Wen’s Z2 Pla- quette 2D model [16] where there is a single type of stabilizer and, according to the terminology used here, one quasipar- ticle species whose self-statistics depends on its initial posi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Another 2D topologically ordered example, one more 11 We assume periodicity of the fields: Axy(x1 + L, y) = Axy(x1, y), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Further, � y2 y1 Axy(x, y)dy = 2πZ/Lx is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Z1 Z2 Z0 X1 X2 X0 𝔞 † † † † 𝔟x 𝔟y † † † †† † Z X † † ac0 † † bc2 a) b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Graphical representations of (a) the ac0 and bc2 operators in the R1TC Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) and (b) the a, by, and bx operators in the R2TC Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The disks are color-coded to represent Xi and Zi operators, according to the legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, disks with a † represent the Hermitian conjugate of the corresponding operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' complicated than Wen’s plaquette model yet simpler than the R2TC, is the model considered by Delfino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In these 2D topologically ordered examples, a general reason for position-dependent braiding is that lattice translations induce nontrivial automorphisms on the anyon lattice [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Conse- quentially the anyon types are labeled by their position which causes their braiding to become position-dependent [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In 3D, position-dependent braiding has been discovered in frac- ton models [47, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In particular, for 3D twisted fracton the- ory, the flux excitations denoted as lineons, with restricted mo- bility along 1D lines, only exhibit nontrivial braiding statistics between the lineons on adjacent planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' That says that if we shift the braiding trajectory of the lineon between the layers, the resultant Berry phase from statistical angles can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' GENERALIZED SYMMETRIES The holonomies constructed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV are a piece of a more general structure present in the R2TC: its generalized symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The generalized symmetries of some rank-2 gauge theories have been discussed previously in the litera- ture [12, 14, 52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Given the rich properties of the R2TC, the exactly solvable point of scalar charge rank-2 ZN gauge theory, it is interesting to wonder what its generalized symme- tries are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In this section, we will identify its symmetries and discuss them in the context of spontaneous symmetry break- ing and ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will consider the general N case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This requires defining a branching and framing structure of the lattice, which we review in appendix Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 17 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Reviewing the 1-form symmetries of the R1TC Let us first review the generalized symmetries in the ZN R1TC on a spatial square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The (2 + 1)D ZN R1TC Hamiltonian can be written as H = − � c0 Ac0 − � c2 Bc2, Ac0 = 1 N N � j=1 (ac0)j, Bc2 = 1 N N � j=1 (bc2)j, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1) where ac0 and bc2 are the star and plaquette operators ac0 = � c1∈δc0 Zc1, bc2 = � c1∈∂c2 Xc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='2) We denote the square lattice’s sites as c0, its edges as c1, and its plaquettes as c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the definitions of ac0 and bc2, δc0 de- notes the coboundary of c0—an oriented sum of edges whose boundary includes c0—and ∂p denotes the oriented boundary of c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The precise definitions of δ and ∂ are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (A3) and (A2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Graphical representations of ac0 and bc2 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6a, from which it is clear that they com- mute for all c0 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We note that these expressions for ac0 and bc2 are equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There are two independent operators that commute with the R1TC Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1), each corresponding to a symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' First consider the unitary U(γ) = � c1∈γ Xc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3) where γ is an oriented closed loop made of the lattice’s edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', γ1 and γ2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7) and Xc1 satisfies X−c1 = X† c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' U(γ) trivially commutes with bc2 for all γ and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Further- more, U(γ) commutes with ac0 since for each site c0, γ is made up of an even number of elements of δc0 with relative orientations such that all phases ei2π/N cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, [U(γ), H] = 0 for all loops γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Next, consider the unitary �U(ˆγ) = � ˆc1∈ˆγ Z∗ ˆc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4) where ˆγ is now an oriented closed loop made of the dual lat- tice’s edges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', ˆγ1 and ˆγ2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7), ˆc1 is a dual lattice edge, and ∗ ˆc1 is the edge of the direct lattice that crosses ˆc1 (up to a differing sign, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (A5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' �U(ˆγ) trivially commutes with ac0 for all ˆγ and c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, �U(ˆγ) commutes with bc2 since for each plaquette c2, ˆγ is made up of an even number of elements of ∂c2 with relative orientations such that all phases ei2π/N cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, [�U(ˆγ), H] = 0 for all loops ˆγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since U and �U commute with H, and since they transform the qubits nontrivially, they correspond to symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In- deed, they generate the transformations U(γ) Zc1 U †(γ) = ω#(c1,γ) Zc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) �U(ˆγ) Xc1 �U †(ˆγ) = ω−#(c1,ˆγ) Xc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6) γ1 γ2 ̂γ1 ̂γ2 Z X † † † † † † † † † FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The symmetry operators U(γ) and �U(ˆγ) of the R1TC act on closed loops of the direct and dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Examples of U(γ) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='3)) acting on loops of the direct lattice γ1 and γ2 are shown in green while examples of �U(ˆγ) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='4)) acting on loops of the dual lattice ˆγ1 and ˆγ2 are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' γ1 and ˆγ1 are contractible loops while, assuming periodic boundary conditions, γ2 and ˆγ2 are non- contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' where ω = ei2π/N and, for instance, #(c1, γ) is the signed intersection number of c1 and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Because [U(γ)]N = [�U(ˆγ)]N = 1, they are the generators of a ZN × ZN symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, this is not quite an ordinary global symmetry since U and �U act on closed loops instead of the entire lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Instead, they correspond to non-topological ZN 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Physically, this symmetry reflects the lack of dynamics of e and m anyons in the R1TC is absent throughout the rest of the deconfined phase of ZN gauge the- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the ground state sub-Hilbert space, the operators ac0 and bc2 obey the constraints ⟨ac0⟩gs = 1 and ⟨bc2⟩gs = 1, where ⟨ ⟩gs denotes the expectation value with respect to the ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Consequentially, when γ and ˆγ are contractible loops, ⟨U(γ)⟩gs = 1 and ⟨�U(ˆγ)⟩gs = 1, which follows from U(γ = ∂M) = � c2∈M bc2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='7) �U(ˆγ = ∂ ˆ M) = � ˆc2∈ ˆ M a† ∗ ˆc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='8) In fact, U(γ) and �U(ˆγ) are so-called topological operators in the ground state sub-Hilbert space, since their vacuum ex- pectation values depend only on the topology—the homology class—of γ and ˆγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In other words, in the ground state sub-Hilbert space, the symmetry operators are nontrivial only when γ and ˆγ are noncontractible loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, one can choose γ and ˆγ such that U and �U are the R1TC “holonomies” discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In the ground state sub-Hilbert space, Xc1 and Zc1 are not allowed operators since they excite e and m anyons, respec- 18 tively, violating the ⟨ac1⟩ = 1 and ⟨bc2⟩ = 1 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The allowed operators are, instead, U(γ) and �U(ˆγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The afore- mentioned generalized ZN × ZN symmetry transformations, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6), in the ground state sub-Hilbert space are replaced with U(γ) �U(ˆγ) U †(γ) = ω#(ˆγ,γ) �U(ˆγ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9) �U(ˆγ) U(γ) �U †(ˆγ) = ω−#(γ,ˆγ) U(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='10) These now correspond to ZN 1-form—Z(1) N —symmetries since their symmetry operators are topological operators sup- ported on codimension 1 closed subspaces and their charged operators are supported on 1-dimensional closed subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In fact, this Z(1) N × Z(1) N symmetry is also a symmetry [54] of the topological quantum field theory description of the R1TC ground states [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Unlike the non-topological ZN × ZN 1- form symmetry of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='6), the Z(1) N × Z(1) N sym- metry exists as an emergent symmetry in the ground state sub- Hilbert space throughout the entire deconfined phase of ZN gauge theory [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Just like ordinary global symmetries, 1-form symmetries can spontaneously break [26, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The order parameter of a 1-form symmetry spontaneous breaking is the vacuum ex- pectation value of its charged operator supported on a con- tractible loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Recall that ⟨U(γ)⟩gs = 1 and ⟨�U(ˆγ)⟩gs = 1 when γ and ˆγ are contractible loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since �U is charged under the Z(1) N symmetry generated by U (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='9)) and vice versa, the R1TC ground states spontaneously break the Z(1) N × Z(1) N symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This reproduces the well known property that there is a ground state degeneracy depending on the topology—the 1st cohomology—of the spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In fact, the Z(1) N × Z(1) N symmetry is anomalous, meaning both Z(1) N symmetries cannot be simultaneously gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The ground state degeneracy (GSD) arising when this anomalous Z(1) N × Z(1) N symmetry is spontaneously broken is GSD = N 2 for the square lattice with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A manifestation of this mixed ’t Hooft anomaly is that the symmetry operators obey the Heisenberg algebra12 �U(ˆγ)U †(γ) = (ei2π/N)#(ˆγ,γ) U †(γ)�U(ˆγ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) and therefore U and �U form a projective representation of Z(1) N × Z(1) N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The mixed ’t Hooft anomaly ensures that the ground state cannot be a trivial product state, and instead the R1TC must be in either a gapless or an SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There- fore, the mixed ’t Hooft anomaly protects the spontaneous 12 Gauging a symmetry U is the procedure of adding additional degrees of freedom such that the theory becomes invariant under the gauged sym- metry operator Ugauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Ugauged acts on both open and closed subspaces and physical states must satisfy Ugauged |ψ⟩ = |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A contradiction arises when different Ugauged no longer commute, reflecting an obstruction to gauging the symmetry (a ’t Hooft anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For example, consider U(1) gaugeU(2) gauge = −U(2) gaugeU(1) gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since U(1,2) gauge |ψ⟩ = |ψ⟩, this leads to the contradiction |ψ⟩ = − |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' symmetry breaking pattern and, therefore, the ZN topologi- cal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, the mixed ’t Hooft anomaly is also present at higher energies, affecting the non-topological ZN 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Its manifestation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11) gives rise to nontrivial mutual statistics between e and m anyons [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Symmetries of the R2TC Having summarized the symmetries in the R1TC, let us now consider the R2TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is convenient to choose a slightly different, but physically equivalent, square lattice where the (X1, Z1) and (X2, Z2) ZN spins reside on horizontal links while the (X0, Z0) ZN spins reside on vertical links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In fact, this is the lattice Λ2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The ZN R2TC Hamiltonian is then given by H = − � c(h) 1 Ac(h) 1 − � c0 Bx c0 − � c2 By c2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12) Bx c0 = 1 N N � j=1 (bx c0)j, By c2 = 1 N N � j=1 (by c2)j, Ac(h) 1 = 1 N N � j=1 (ac(h) 1 )j, where c(h) 1 denotes a horizontal link, c0 a lattice site, and c2 a plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6b shows graphical representations of the op- erators a, bx, and by, which are also defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='19), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6b, it is clear these oper- ators are mutually commuting, and therefore the ground state satisfies a = 1, bx = 1, and by = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC Hamiltonian operators a, bx, and by have a rich and complicated structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Consequently, the theory can have many interesting generalized symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will construct its symmetries in V B 1, which will include mostly technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Afterwards, in V B 2, we will discuss these symme- try operators, analyzing how the R2TC’s interesting properties can be interpreted from a symmetry point of view and com- paring the symmetry operators to conventional 1-form sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Construction of symmetry operators Let us first identify the symmetries which are generated by operators built out of only X0, X1, and X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To do so, we define the lattice vector fields X1 and X2 which are related to X0, X1, and X2 by Xi 1,c0 = (Xx 1,c0, Xy 0,c0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='13) Xi 2,ˆc0 = (Xy 0,| ∗ ˆc0|+ˆx, Xx 2,| ∗ ˆc0|+ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='14) Notice that while X1 is specified by the links of the direct lattice, X2 is instead specified by the links of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As elaborated on in appendix section A, the position of | ∗ ˆc0| is related to a direct lattice site c0 by c0 = | ∗ ˆc0| − ˆx/2 − ˆy/2, 19 X1 X2 X0 † † † † † † † † † γ ̂γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operators U1(γ) and U2(ˆγ), defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16) respectively, act on closed loops of the direct and dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we show graphical representation of an example of U1(γ) acting on a loop of the direct lattice γ (drawn in green) and of U2(ˆγ) acting on a loop of the dual lattice ˆγ (drawn in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' where | ∗ ˆc0| is just the absolute value of ∗ ˆc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Using X1 and X2, we construct the unitary operators U1(γ) = � c1∈γ X1,c1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='15) U2(ˆγ) = � ˆc1∈ˆγ X2,ˆc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='16) where γ and ˆγ are oriented loops on the direct and dual lat- tice, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' U1 and U2 trivially commute with Bx c0 and By c2 in the R2TC Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' U1 and U2 also commute with Ac(h) 1 , which can be confirmed directly or simply by comparing the graphical representa- tions shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6b and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, for all γ and ˆγ, [U1(γ), H] = [U2(ˆγ), H] = 0, and U1 and U2 correspond to symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When γ and ˆγ are contractible, the symmetry operators U1 and U2 can be written as U1(γ = ∂M) = � c2∈M by c2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='17) U2(ˆγ = ∂ ˆ M) = � ˆc2∈ ˆ M bx | ∗ ˆc2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='18) Consequentially, in the ground state subspace where bx,y = 1, U1 and U2 are topological operators, depending only on the homology class of γ and ˆγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since U N 1 = U N 2 = 1, we there- fore find that these symmetry operators generate an emergent Z(1) N × Z(1) N symmetry in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We note that when γ is a loop of links in the x-direction (y-direction), U1 becomes the “holonomy” W3 (W4) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Similarly, when ˆγ is a loop of dual links in the x-direction (y-direction), U2 becomes the holonomy W1 (W2) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' There is one more symmetry operator which can be con- structed from the X operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Let us define the operator X3 which acts only on the horizontal links c(h) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' X3 is in- terpreted as a lattice vector field whose x component acts on X1 X2 Γ(s) † † † † † † † † † † † † † † † FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operator U3(Γ(s)) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) acts on closed loops Γ(s) of the Vvh lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we show a graphical representation of U3(Γ(s)) acting on a particular loop Γ(s) drawn in blue with N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The Vvh lattice sites belonging to the s sublattice are denoted by gray squares, and we sometimes include the operators (X1)3 and (X2)3 despite them being the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' the horizontal links of c(h) 1 but whose y component acts on the vertical links of the dual lattice ∗ ˆc(v) 1 = c(h) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, the horizontal links form their own square lattice Vvh whose sites v ≡ (vx, vy) are squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will formulate this symmetry on the Vvh lattice where it turns out to be most naturally defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, this can also be formulated on the direct lattice if the framing structure is utilized, which makes the following symmetry a so-called framed-symmetry [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' X3 is related to X1 and X2 by Xi 3,v = (X1,c(h) 1 , X2,c(h) 1 ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='19) where the Vvh site v on the left hand side is the center of the edge c(h) 1 on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Using X3, we can construct a unitary which commutes with the R2TC Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To do so, we first reconsider the Vvh square lattice as a Bravais lattice with a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The conven- tional unit cell is an N × N square surrounding N 2 lattice sites, each of which belong to their own sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We intro- duce the index s ∈ {1, 2, · · · , N 2 − 1, N 2} which labels each sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Let us denote a generic oriented closed loop of the Vvh lattice as Γ, and specify loops made of only length N seg- ments connecting sites of the sublattice s as Γ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With this set up, we now consider the unitary operator U3(Γ(s)) = � v∈Γ(s) (X3,r)(vx−r(s) x )+(vy−r(s) y ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) where v is a Vvh lattice site13 and r(s) is the basis vector (in the crystallography sense) of sublattice s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 9 shows an example U3(Γ(s)) for N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 13 In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20), the notation v ∈ Γ(s) simply means all Vvh lattice sites v which the loop Γ(s) crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here, Γ(s) should not be considered as a 1- cycle—an integer sum of 1-chains in the kernel of the boundary operator ∂1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will commit similar abuses of notation throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 20 The operator U3 trivially commutes with Bx c0 and By c2 in the R2TC Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, U3(Γ(s)) also commutes with Ac(h) 1 for all Γ(s), which can be con- firmed by direct computation or simply from inspecting the graphical representations shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 6b and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, [U3(Γ(s)), H] = 0 for all Γ(s), and U3 indeed corresponds to a symmetry operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When Γ(s) is contractible, the symmetry operator U3 can be written as U3(Γ(s) = ∂M)= � c0∈M (bx c0)(c0)y � c2∈M (by c2)(c2)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='21) Here, (c0)y is the distance of c0 from Γ(s) in the −y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Similarly, (c2)x is the distance of c2 from Γ(s) in the −x- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since bx c0 = by c2 = 1 in the ground state subspace, U3(Γ(s)) is a topological operator and corresponds to a 1-form symmetry in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, this is not an ordinary 1-form symmetry since Γ(s) is not allowed to be any loop on the Vvh lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As a result U3(Γ(s)) is not a fully topological operator on the Vvh lattice, but is on the s sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, we refer to U3(Γ(s)) as a sublattice 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The precise nature of this sublattice 1-form symmetry de- pends on both the topology and geometry of the lattice in a sensitive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Without periodic boundary conditions, this is a sublattice ZN 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With periodic bound- ary conditions, the previous N × N conventional unit cell shrinks to a gcd(Lx, N) × gcd(Ly, N) unit cell (but Γ(s) is stll made of only length N segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Consequently, Γ(s) must wrap around system N/ gcd(Li, N) times in the i- direction in order to close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, on a torus, U3 is a Z(1) gcd(Lx,N) × Z(1) gcd(Ly,N) sublattice 1-form symmetry, where the noncontractible Γ(s) of the Z(1) gcd(Li,N) sublattice symme- try is understood winding only in the i-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We note that the Z(1) gcd(Lx,N) and Z(1) gcd(Ly,N) symmetry operators are related to the W5 and W6 “holonomies,” respectively, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We now move on to discuss the symmetry operators con- structed from only Z0, Z1, and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will find that there are three symmetry operators, two of which correspond two sub- lattice 1-form symmetries and one is a conventional 1-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' To construct the first symmetry operator, we must recon- sider unit cell of the lattice as a N × 1 unit cell with a basis labeled by s ∈ {1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A loop of the dual lattice made of only length N segments in the horizontal direction connecting the sites of sublattice s is denoted as ˆγ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We then introduce the lattice vector field Z1 acting on the links of the dual lattice, ˆc1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is related to the Z0, Z1, and Z2 operators by Z1,ˆc1 = (Z0,∗ ˆc1(Z† 2,∗ ˆc1+ˆx/2+ˆy/2Z2,∗ ˆc1+ˆx/2−ˆy/2)x−x(s), Z1,∗ ˆc1) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='22) where x is the x-coordinate of the dual lattice site in the x- direction of ˆc1 and x(s) is the x-coordinate of the basis vector for sublattice s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With this defined, we can then consider the Z1 Z2 Z0 ̂γ(s) † † † † † † † † † † † † † † † † † † † † † † † † † † † FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operator �U1(ˆγ(s)) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23) acts on closed loops ˆγ(s) of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we show a graphical representation of �U1(ˆγ(s)) acting on a particular loop ˆγ(s) drawn in red with N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The dual lattice sites belonging to the s sublattice are denoted by gray squares, and we sometimes include the operator (Z2)3 despite it being the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Z1 Z2 Z0 † †† † † † γ(s) † † † † † † † † FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operator �U2(γ(s)) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='25) acts on closed loops γ(s) of the direct lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we show a graphical representation of �U2(γ(s)) acting on a particular loop γ(s) drawn in green with N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The direct lattice sites be- longing to the s sublattice are denoted by gray squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' unitary �U1(ˆγ(s)) = � ˆc1∈ˆγ(s) Z1,ˆc1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='23) an example of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 10 for N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is straightforward to check that for all ˆγ(s), �U1 commutes with a, bx, and by, and therefore [�U1, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The second symmetry operator is rather similar to the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Now we instead consider a 1 × N units cell with a basis again labeled by s ∈ {1, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A Loop of the direct lat- tice made of only length N segments in the vertical direction connecting the sites of sublattice s is denoted as γ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We then introduce the lattice vector field Z2 acting on the links of the direct lattice, c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is related to the Z0, Z1, and Z2 operators 21 by Z2,c1 = (Z1,c1, Z† 0,c1(Z† 2,c1−ˆx/2+ˆy/2Z2,c1+ˆx/2+ˆy/2)y−y(s)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='24) where y is the y-coordinate of the lattice site in the y-direction of c1 and y(s) is the y-coordinate of the basis vector for sub- lattice s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' With this defined, we can then consider the unitary �U2(γ(s)) = � c1∈γ(s) Z2,c1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='25) an example of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 11 for N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is straightforward to check that for all γ(s), �U2 commutes with a, bx, and by, and therefore [�U2, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Both unitary operators �U1(ˆγ(s)) and �U1(ˆγ(s)) correspond to symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When acting on contractible loops, they can be written as �U1(ˆγ(s) = ∂ ˆ M) = � c(h) 1 ∈ ˆ M (ac(h) 1 )(c(h) 1 )x, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='26) �U2(γ(s) = ∂M) = � c(h) 1 ∈M (ac(h) 1 )(c(h) 1 )y, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='27) where (c(h) 1 )x is the distance of c(h) 1 from ˆγ(s) in the −x- direction and (c(h) 1 )y is the distance of c(h) 1 from γ(s) in the −y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Evidently, both the symmetries generated by �U1 and �U2 are sublattice 1-form symmetries, defined on their respective sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Like for the sublattice 1-form symme- try U3, the details of the symmetry are sensitive to both the geometry and topology of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Indeed, with periodic boundary conditions the N × 1 (1 × N) unit cell defined for the �U1 (�U2) symmetry operator becomes a gcd(Lx, N) × 1 (1 × gcd(Ly, N)) unit cell (γ(s) and ˆγ(s) are still made of length N segments in the y and x directions, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Note that when �U1 (�U2) is supported on a non-contractible loop in the y (x) direction, it is related to � W3 (� W2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Similarly, when �U1 (�U2) is supported on a non- contractible loop in the x (y) direction, it becomes � W4 (� W1) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20) Let us now construct the final symmetry operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We de- fine the operator Z3 which acts only on the vertical links c(v) 1 of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Z3 is interpreted as a lattice vector field whose x component acts on the plaquette c2 but whose y component acts on the horizontal links of the dual lattice ˆc(h) 1 = ∗ c(v) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It turns out it is most natural to formulate this symmetry oper- ator on the previously mentioned Vvh lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will denote the sites of the dual Vvh as ˆv and note that they are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 1 as discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Z3 is related to Z0, Z1, and Z2 by Z3,ˆv = (Z0,ˆvZ† 0,ˆv+ˆxZ† 2,ˆv+ˆx/2+ˆy/2Z2,ˆv+ˆx/2−ˆy/2, Z† 2,ˆv−ˆx/2+ˆy/2Z2,ˆv+ˆx/2+ˆy/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='28) With Z3 defined, we can then consider the unitary operator �U3(ˆΓ) = � ˆv∈ˆΓ Z3,ˆv (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='29) † † † †† † † †† † † † † † † † † † † † Z1 Z2 Z0 ̂Γ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operator �U3(ˆΓ) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='29) acts on closed loops Γ of the dual Vvh lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here we show a graph- ical representation of �U3(Γ) acting on a particular loop ˆΓ drawn in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' where ˆΓ is an oriented closed loop on the dual Vvh lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 12 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is straight forward to check that �U3 commutes with the Hamiltonian for all ˆΓ and therefore corresponds to a symme- try operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When �U3 is supported on a contractible loop, it can be written as �U3(ˆΓ = ∂M) = � c(h) 1 ∈M ac(h) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='30) Since ac(h) 1 = 1 in the ground state subspace, �U3 is a topo- logical operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, in the IR, �U3 is the symmetry operator of a Z(1) N symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In fact, when supported on a non-contractible loop winding around the system in the x (y) direction, �U3 becomes � W6 (� W5) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Analysis and discussion of R2TC symmetries Having identified the generalized symmetries of the R2TC, let us now use them to interpret the model’s interesting prop- erties from a symmetry point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Recall that the six sym- metry operators are supported on loops and commute with the R2TC Hamiltonian for all respective loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Their expecta- tion values with respect to excited states depend on more than just the topology of the loops, so in this sense these micro- scopic (UV) symmetries are non-topological 1-form symme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Their existence reflects the lack of dynamics for e and ⃗m anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Throughout the rest of the deconfined phase of ZN rank-2 gauge theory, away from the R2TC point, these non- topological 1-form symmetries are explicitly broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The symmetry operators of the R2TC are much richer and more complex than those in the R1TC, which were reviewed 22 in section V A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 7, the R1TC symme- try operators are nicely defined on 1-cycles of the direct and dual lattice (so, they admit a straightforward description using cellular homology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, for a given symmetry oper- ator, each edge of the 1-cycle was acted on by the same X or Z operator (up to taking the hermitian conjugate, which arises from the 1-cycles orientation and lattice’s branching struc- ture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC symmetry operators, examples of which are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 8–12, go beyond all of these convenient sim- plicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For example, they include the following features, absent from the R1TC’s symmetries: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The symmetry operators �U1(ˆγ(s)), �U2(γ(s)), and �U3(ˆΓ) act on both the spins on the loops ˆγ(s), γ(s), and ˆΓ, respectively, and the spins near the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For all symmetry operators, the operators acting on/near the loop’s edges depend on whether the loop is parallel to the x or y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 8, U1(γ) has X0 act on edges when γ is parallel to the y- direction but has X1 act on edges when γ is parallel to the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The operators acting on the spins for symmetry oper- ators U3(Γ(s)), �U1(ˆγ(s)), and �U2(γ(s)), depend on the position of those spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The symmetry operators U3 and �U3 act on loops of the direct/dual Vvh lattice instead of the direct/dual (Λ2) lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In terms of the Λ2 lattice, these operators act on loops defined on both the direct/dual lattice and there- fore require additional framing structure, which makes them framed 1-form symmetries [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The symmetry operator �U3(ˆΓ) has operators which only act on the corners of the loop ˆΓ while absent from other parts of the loop (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', Z0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' From the Λ2 lattice point of view, these corners coincides with where the framing structure connects the direct and dual lattices’ loops to create ˆΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Unlike the expectation values with respect to excited states mentioned previously, the vacuum expectation values of the symmetry operators depend only on the topology of these loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Thus, in the ground state sub-Hilbert space—the IR— of the R2TC, all six of the generalized symmetries are 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Three of these (U1, U2, and �U3) were conven- tional 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, the other three (U3, �U1, and �U2) were not conventional 1-form symmetries since their symmetry operators relied on an underlying sublattice struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' These nonconventional 1-form symmetries were called sublattice 1-form symmetries in the previous section to em- phasize this additional structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The lattice symmetries gen- erally mix these sublattices and act nontrivially on sublat- tice 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, the total symmetry group of the R2TC is (1-form symmetries)⋊(lattice symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This interplay between the sublattice 1-form and spatial symmetries can also be noticed by the R2TC’s symmetry- enriched topological order, where position-dependent excita- tions [10] reflect the existence of sublattice 1-form symme- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Throughout the rest of the deconfined phase of ZN rank- 2 gauge theory, away from the R2TC point, we expect that all of these generalized symmetries are exact emergent IR sym- metries [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This means that despite being emergent symme- tries, explicitly broken in the microscopic Hamiltonian, they constrain the IR in the thermodynamic limit as if they were exact microscopic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since all of the R2TC’s generalized symmetries are 1-form symmetries, they are sensitive to the topology of the spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The sublattice 1-form symmetries, however, also de- pend on the geometry of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Indeed, as we discussed in the previous section, with periodic boundary conditions the size of their underlying sublattices depends on the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, the sublattices in U3’s, �U1’s, and �U2’s respec- tive definitions are unique to the square lattice, so the R2TC on a different lattice would generally have different sublattice 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, the sublattice 1-form symme- tries give rise to UV/IR mixing in the R2TC [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The emer- gent IR symmetries depending on the UV lattice is a general diagnosis for UV/IR mixing and, in fact, may be a unified mechanism for UV/IR mixing in all topological and fracton phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The R2TC’s symmetry operators satisfy the algebra U1(γ) �U1(ˆγ(s1)) = ω#(γ,ˆγ(s1)) �U1(ˆγ(s1)) U1(γ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='31) U2(ˆγ) �U2(γ(s2)) = ω#(ˆγ,γ(s2)) �U2(γ(s2)) U2(ˆγ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='32) U3(Γ(s3)) �U3(ˆΓ) = ω#(Γ(s3),ˆΓ) �U3(ˆΓ) U3(Γ(s3)), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='33) where ω ≡ ei2π/N and #( , ) is the signed intersection number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We thus see that the Ui (�Ui) symmetry oper- ator transforms nontrivially under the �Ui (Ui) symmetry transformation—Ui (�Ui) is a charged operator of the �Ui (Ui) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Recall from the previous section that when sup- ported on contractible loops, ⟨Ui⟩gs = ⟨�Ui⟩gs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The fact that ⟨Ui⟩gs = 1 (⟨�Ui⟩gs = 1) for contractible loops means that the �Ui (Ui) symmetry charges are condensed in the R2TC ground state, and the �Ui (Ui) symmetry is spontaneously bro- ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, the R2TC ground state spontaneously breaks all six of the 1-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Discrete symmetries spontaneously breaking always gives rise to a ground state degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The GSD is computed by finding the smallest faithful representation of the sponta- neously broken symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This exact calculation was done in section IV B, where the holonomies Wi and � Wi are the generators of the spontaneously broken symmetries, and yields the correct GSD Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Therefore, the GSD is system size dependent because some of the spontaneously broken symmetries are sublattice 1-form symmetries that en- code geometrical information of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The algebra Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='31)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='33) also reveals that the R2TC realizes these generalized symmetries in a projective repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This prevents the 1-form symmetries from being 23 gauged, and is thus a manifestation of an ’t Hooft anomaly14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' In particular, there is a mixed ’t Hooft anomaly between the Ui and �Ui symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Like in the R1TC discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' V A, mixed ’t Hooft anomalies for 1-form symmetries realized through projective representations are physically re- flected through the nontrivial braiding statistics of anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Since some of the R2TC’s anomalous symmetries are sublat- tice 1-form symmetries, the braiding statistics will generally depend on the sublattice the anyon resides on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, this is precisely the position dependent-braiding discussed in sec- tion IV E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' SUMMARY AND OUTLOOK We have applied the idea of coupled-layer construction, previously invented to understand the emergence of fracton models out of toric codes in three dimensions [20, 21], to shed light on the appearance of symmetric rank-2 gauge fields in two dimensions and from there the rank-2 toric code through Higgsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Condensation of gauge fields can take place in either one of the two conjugate gauge fields A and E, and leads to theories with either vector-electric or vector-magnetic charges that are ultimately dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Construction of holonomy (Wilson line) operators for the rank-2 toric code follows rather naturally in this approach, as one can start by identifying the holonomy operators in the Hilbert space before the condensation took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We thus ar- rive at the picture of holonomies as the creation/annihilation of magnetic and electric charge-anti-charge pairs, and of their dipole-anti-dipole pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The dependence of the ground state degeneracy on the system size (the UV/IR mixing) can be thoroughly understood from analysis of the Wilson loop oper- ators thus obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We further suggest an easy-to-implement, heuristic derivation of the holonomies based on the rank-2 gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This may well have applications in the holon- omy construction of other, rank-2 gauge theories and the cor- responding stabilizer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore the exact tensor network expression of the ground state of the rank-2 toric code is derived starting from two copies of the rank-1 toric code’s ground state wave func- tions, by sewing them together with an isometry operation that faithfully reflects the condensation of the gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This, too, may have application in the construction of other rank-2 based stabilizer ground state wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For one thing, analyzing entanglement entropy becomes easy with the ten- sor network wave function at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Additionally, the tensor network projection of R2TC provides a clear picture of how coupled toric code layers engender higher-rank gauge theory in terms of anyon condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This also sheds light on ex- ploring phase transitions between conventional gauge theory and R2TC, where one can replace the tensor projection proce- dure with an additional parameter in the tensor element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We will explore these issues in a future study [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 14 See footnote 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The anyon condensation idea can lead to a number of pow- erful applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' As an example we showed how the Levin- Gu semionic topological model [24] can undergo a similar condensation procedure to result in a new model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The notion of generalized symmetry is a new and powerful description of the topological order in the toric code, and we have dis- cussed how the notion applies to the rank-2 toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We believe the ‘generalization’ of the generalized symmetry idea to other rank-2 based models can find interesting applications in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Futhermore, it would be interesting to investi- gate if general rank-N symmetric tensor gauge theories, with N > 2, can be constructed from many copies of rank-1 theo- ries in a particular condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' ACKNOWLEDGMENTS Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' was supported by National Research Foundation (NRF) of Korea under Grant NRF- 2022R1I1A1A01065149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' is supported by the National Science Foundation Graduate Research Fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 2141064 and by the Henry W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Kendall Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' was supported by Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' NRF- 2019R1A6A1A10073079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' He also acknowledges financial support from EPIQS Moore theory centers at MIT and Harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' is supported by Northeastern University COS start-up grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' was supported by NRF of Korea under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' NRF-2020R1I1A3074769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' were supported by the Basic Science Research Program funded by the Ministry of Education (2014R1A6A1030732).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' acknowledges informative discussion with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Hughes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Kang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Lam, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Luo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Tantivasadakarn, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Appendix A: Review of discrete differential geometry for d-dimensional cubic lattices In this appendix section, we review relevant parts of dis- crete differential geometry (in a non-rigorous fashion) used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' V of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Consider a cubic lattice in d- dimensional space with periodic boundary conditions, de- noted by Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' While a Bravais lattice is a collection of lattice sites x ∈ Zd, it is useful to view it as also formed by higher- dimensional objects, like links, plaquettes, cubes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We call a p-dimensional object a p-cell, with 0 ≤ p ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' So, a 0-cell is a lattice site, a 1-cell is a link, a 2-cell is a plaquette, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This does not add additional structures to the lattice, but instead is just a useful way of organizing the lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Indeed, de- noting a p-cell associated with site x as cp(x)µ1µ2···µp, where µ1 < µ2 < · · · < µp and µi ∈ {1, 2, · · · , d}, a p-cell of the 24 p = 1 p = 2 p = 1 p = 2 p = 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The p-cells of the d-dimensional cubic lattice are equiv- alently the 0-cells—the sites—of some other d-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Shown here are examples of this equivalent lattice (drawn in pink) embedded in the conventional unit cell of the cubic lattice (drawn in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (First row) In 2 dimensions, the 1-cells form another square lattice, rotated by 45 degrees, whose lattice constant is 1/ √ 2 times that of the original square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The 2-cells also form another square lattice, which is the original shifted by the vector (ˆµ1 + ˆµ2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (Second row) In 3 dimensions, both the 1-cells and also the 2-cells form a lattice of corner-sharing octahedra with a lat- tice constant that is 1/ √ 2 times the cubic lattice’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When p = 1, the octagons are centered at the cubic lattice’s 0-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' When p = 2, the octagons are centered at the cubic lattices 3-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Lastly, the 3-cells form another cubic lattice of the same size, but shifted by the vector (ˆµ1 + ˆµ2 + ˆµ3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' cubic lattice is the set of 2p lattice sites15 cp(x)µ1µ2···µp= {x} ∪ {x + ˆµi | 1 ≤ i ≤ p} ∪ {x + ˆµi + ˆµj | 1 ≤ i < j ≤ p} ∪ · · · ∪ {x + ˆµ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' + ˆµp}, (A1) where ˆµi is the unit vector in the µi-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' It is often convenient to drop the requirement that the indices are canonically ordered (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=', that they satisfy µ1 < µ2 < · · · < µp < ν) and instead let cp(x)µ1µ2···µp obey the relation cp(x)···µ1µ2··· = −cp(x)···µ2µ1···.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The p-cells of the d-dimensional cubic lattice are equivalently viewed as the 0-cells of some other lattice in d-dimensions, as demonstrated for d = 2 and 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Introducing the concept of p-cells is strictly unnecessary but very convenient because “sewing” p-cells together gives a natural way to form p-dimensional subspaces of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore these subspaces can also be given an orienta- tion by defining an orientation structure to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A nice local scheme for the lattice orientation is a branching struc- ture, where the orientation on each 1-cell is chosen such that a collection of 1-cells cannot form an oriented closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A canonical orientation on all other p-cells then follows from the branching structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' We use the branching structure where each 1-cell c1(x)µ has an arrow pointing in the ˆµ direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' However, it is important to note that the choice of lattice orientation is a formal convention, and choosing dif- 15 We adopt the discrete differential geometry and exterior calculus notations and conventions used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' ̂x ̂y ̂z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Example of the branching structure used for a chunk of the cubic lattice in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' ferent branching structures does not affect the physics16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' A p-cell can be related to (p − 1) cells using the boundary operator ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The boundary operator acting on a p-cell—∂cp— is the oriented sum of (p−1)-cells on the boundary of cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For the branching structure we use, it is given by ∂cp(x)µ1···µp= p � k=1 (−1)k+1� cp−1(x + ˆµk)µ1··· oµk···µp −cp−1(x)µ1··· oµk···µp � , (A2) where the notation oµk indicates that the µk index is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' From its definition, the boundary operator satisfies ∂2cp = 0 for any p-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, as there are no (−1)-cells, the boundary operator acting on a 0-cell is defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' On the other hand, a p-cell can be related to (p + 1)-cells using the coboundary operator δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The coboundary operator acting on a p-cell—δcp—is an oriented sum of all (p + 1)- cells whose boundary includes cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For the branching structure we use, it is given by δcp(x)µ1···µp = � ν cp+1(x)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='µp − cp+1(x − ˆν)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' (A3) From its definition, the coboundary operator satisfies δ2cp = 0 for any p-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Furthermore, as there are no (d + 1)- cells, the coboundary operator acting on a d-cell is defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Lastly, the lattice has an associated dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' The dual lattice has its lattice sites centered at the d-cells of the direct lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For the cubic lattice, one choice of framing that relates a dual lattice site ˆx to a direct lattice site x is by ˆx = x + 1 2 ˆr with ˆr = � i ˆµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Each p-cell cp on the direct lattice is associated with a (d − p)-cell ˆcd−p on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' This is implemented by the dual operator ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' For this choice of framing, a p-cell cp(x)µ1···µp (with canonical ordering µ1 < · · · < µp) and a (d − p)-cell of the dual lattice ˆcd−p(ˆx)µ1···µd−p (with canon- 16 However, according to a conjecture from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 60, observables are indepen- dent of the branching structure only if the continuum effective field theory is free of a framing anomaly [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' 25 ical ordering µ1 < · · · < µd−p) are related to one another by ∗ cp(x)µ1···µp = ϵµ1···µpµp+1···µd (A4) × ˆcd−p(ˆx − ˆµp+1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' − ˆµd)µp+1···µd, ∗ ˆcp(ˆx)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content='µp = ϵµ1···µpµp+1···µd (A5) × cd−p(x + ˆµ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' + ˆµp)µp+1···µd, where summation is not implied on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf'} +page_content=' Here ϵ is the Levi-Civita symbol, which takes into account the lat- tice’s and dual lattice’s 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Ferretti,1, ∗ R. Magaña Vsevolodovna,2, † J. Kotila,1, 3, 4, ‡ and E. Santopinto2, § +1Department of Physics, University of Jyväskylä, PO Box 35, FI-40014, Jyväskylä, Finland +2INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova (Italy) +3Finnish Institute for Educational Research, University of Jyväskylä, P.O. Box 35, FI-40014 Jyväskylä, Finland +4Center for Theoretical Physics, Sloane Physics Laboratory, +Yale University, New Haven, Connecticut 06520-8120, USA +(Dated: January 6, 2023) +Here, we study the neutrinoless double-β (0νββ) decay between the ground state and the first +2+ state of 76Ge → 76Se, 82Se → 82Kr, 130Te → 130Xe and 136Xe → 136Ba systems. The relevant +nuclear matrix elements (NMEs) involved in the process are calculated within the formalism of the +microscopic interacting boson model (IBM-2). The IBM-2 has been widely used to obtain predictions +for nuclear observables, such as the spectrum, but also to explore the possible emergence of beyond- +the-Standard Model effects in the weak interactions of nuclei. +Our calculations are carried out +by considering the exchange of a Majorana neutrino between two nucleons (2N-mechanism). In +addition to NMEs, we calculate the associated leptonic phase-space factors (PSFs) using electron +radial wave functions, which are obtained by solving numerically the Dirac equation of a screened +Coulomb potential that takes into account finite nuclear size. By combining our IBM-2 results for +the NMEs with those for the PSFs along with experimental half-life limits, we can set limits on the +⟨λ⟩ and ⟨η⟩ couplings of left-right (L-R) models. +I. +INTRODUCTION +Neutrinos have a long story. Their existence was pos- +tulated by Pauli in 1930 to ensure the conservation of +energy and angular momentum in β-decay [1]. Fermi’s +renowned theory of beta decay dates back to 1933 [2]. +In 1956, neutrinos were first observed at Los Alamos by +Cowan and Reines via the study of inverse beta decay +[3]. Several decades later, neutrinos are still fascinating +and mysterious particles. +Important questions regarding some of their main +properties remain unsolved, +including the unknown +mechanism that generates their masses and a complete +understanding of their mixing mechanism and mass hi- +erarchy [4]. Because of the lack in the standard model +(SM) of a Yukawa coupling between the Higgs boson and +neutrinos, due to the absence of right-handed neutrinos, +the SM has to be extended to provide a neutrino mass +term. Extensions of the SM include the L-R symmetric +[5–7] and SUSY [8–13] models. +Some important issues are directly related to the na- +ture of neutrinos as Fermi- or Majorana-type particles, +a nature which could be directly assessed via the exper- +imental observation of neutrinoless double-beta (0νββ) +decay process [14]. +However, despite of the strenuous +attempts by many experimental groups, e.g., [15–21], +0νββ-decay has not yet been observed. +Several theoretical investigations on 0νββ decay have +∗Electronic address: jacopo.ferretti80@gmail.com +†Electronic address: Ruslan.Magana@ge.infn.it +‡Electronic address: jenni.kotila@jyu.fi +§Electronic address: elena.santopinto@ge.infn.it +been published over the years (for a review see e.g., Refs. +[22, 23]) in order to guide the experimentalists in their +searches. +Owing to the low-energy character of 0νββ +processes, these studies necessarily involve elements of +both particle and nuclear physics. In particular, nuclear +structure models are necessary in order to take care of +the nuclear matrix elements (NMEs) [24–26] entering the +expression of the 0νββ-decay half-life. +Here, we show the results of a calculation of the 0+ to +2+ 0νββ decay of 76Ge, 82Se, 130Te and 136Xe, in which +we consider the exchange of a Majorana neutrino be- +tween two nucleons, the so-called 2N-mechanism, within +an L-R symmetric model [27, 28]. +Specifically, in our +study: +I) we compute the relevant NMEs within the +microscopic interacting boson model (IBM-2) formalism +[29] and compare our results with previous calculations +for the studied nuclei within different nuclear structure +models; II) we calculate the leptonic phase-space factors +(PSFs) by means of electron radial wave functions, ob- +tained by solving numerically the Dirac equation of a +screened Coulomb potential that takes into account finite +nuclear size [30]; and III) by combining the two above ele- +ments, namely the results for the NMEs and the leptonic +PSFs, with the experimental limits on the half-life, we set +limits on the ⟨λ⟩ and ⟨η⟩ couplings of L-R models. Ex- +perimental studies on this decay mode can be found e.g., +in [31–36]. Previous calculations for the 0+ → 2+ decay +rate for the studied nuclei within different nuclear struc- +ture models and via the 2N-mechanism can be found in +Refs. [27, 28, 37–39]. +This article is organized as follows: In Sec. II the im- +portance of 0+ → 2+ 0νββ decay is discussed and some +details on the calculation of 0νββ decay rates in L-R +symmetric models are provided. In Sec. III the calcula- +tion of IBM-2 wave functions is briefly summarized, and +arXiv:2301.02007v1 [nucl-th] 5 Jan 2023 + +2 +in Sec. IV the decay operators needed for the descrip- +tion of 0+ → 2+ 0νββ-decay are presented. In Sec. V the +numerical results for the ingredients needed for the calcu- +lation of the decay rate are given and discussed. Finally, +the conclusions are presented in Sec. VI. +II. +0+ → 2+ 0νββ DECAY IN THE L-R +SYMMETRIC MODEL +If we restrict ourselves only to long-range mechanisms +for 0νββ decay, the most general effective Lagrangian +is the Lorenz-invariant combination of leptonic, jα, and +hadronic, Jα, currents with definite tensor structure and +chirality [40–42], +L = GF cos θc +√ +2 +� +�Uei jµ,i +V −AJ† +V −A,µ + +� +α,β +ϵβ +α,i ji +βJ† +α + h.c. +� +� . +(1) +Here, GF is the Fermi constant; θc is the Cabibbo an- +gle; the hadronic and leptonic currents are defined as +J† +α = ¯uOαd and ji +β = ¯eOβνi, where the index i spans the +neutrino mass eigenstates. The indices α, β are V ∓ A, +S ∓ P, T ∓ T5, where S, P, T and T5 stand for scalar, +pseudo-scalar, tensor, and pseudo-tensor, respectively. In +Eq. (1) we have isolated the standard model contribution +proportional to Uei, where Uei is the PMNS mixing ma- +trix element [43, 44], from non-standard contributions, +which are those proportional to the couplings ϵβ +α,i. +By isolating the V ∓ A currents of the L-R models +from those allowed in other types of mechanisms, such +as SUSY, and performing a non-relativistic reduction of +both the leptonic and hadronic currents, one can obtain +the expression of the 0νββ decay half-life for a 0+ → 0+ +transition in L-R models: +� +τ 0ν +1/2(0+ → 0+) +�−1 += C(0) +mm +� +⟨mν⟩ +me +�2 ++ C(0) +λλ ⟨λ⟩2 ++ C(0) +ηη ⟨η⟩2 + 2C(0) +mλ +⟨mν⟩ +me ⟨λ⟩ ++ 2C(0) +mη +⟨mν⟩ +me ⟨η⟩ + 2C(0) +λη ⟨λ⟩ ⟨η⟩ . +(2) +The above equation is a complicated combination of the +three parameters ⟨mν⟩, ⟨λ⟩, and ⟨η⟩ and their respective +matrix elements and phase-space factors; for details on +the combinations C(0) +ij , i, j = m, λ, η see e.g., Ref. [42]. +The new physics beyond the standard model is en- +closed in the three parameters +⟨mν⟩ = +� +i +U 2 +eimi , +(3a) +⟨λ⟩ = λ +� +j +UejVej , +(3b) +and +⟨η⟩ = η +� +j +UejVej , +(3c) +where ⟨mν⟩ is the average neutrino mass obtained by +summing over mass mi of neutrino species i, and ⟨λ⟩ and +⟨η⟩ are the standard couplings of L-R models [28]. Uej +and Vej are the mixing matrix elements of the PMNS +matrix [43, 44] for the standard and non-standard (L-R) +mechanisms, respectively. Thus, the 0+ → 0+ process +can occur because either of right- or left-handed leptonic +currents. +By contrast, if 0νββ-decay to a 2+ state is observed +then in addition to proving the Majorana character of +neutrinos, the existence of V + A current would also +be established, since as a first approximation, this de- +cay mode is triggered by right-handed leptonic currents +only [39]. To be more specific, the combination of the +lowest electron partial waves for the 0+ → 2+ transition +is the S and P3/2 case since the total angular momen- +tum of the two-electron system should be 2. However, +in order to have a non-zero contribution of the L-L term +that is proportional to the average neutrino mass, ⟨mν⟩, +the leading term requires the combination of S and D3/2 +electron waves, making it negligible compared with the +contributions due to L-R terms, which are those propor- +tional to ⟨λ⟩ and ⟨η⟩). For more details see [27, App. +C]. +Therefore, for the 0+ → 2+ case the half-life can safely +be written without the dependence on the average neu- +trino mass +� +τ 0νββ +1/2 (0+ → 2+) +�−1 += g4 +A +� +G1 (Mλ⟨λ⟩ − Mη⟨η⟩)2 ++ G2 +� +M ′ +η⟨η⟩ +�2 � +, +(4) +where one can factorize the leptonic phase-space factor +(PSF) [28, 30], G, the nuclear matrix elements (NMEs), +M, and the axial vector coupling constant, gA. +Neu- +trinoless double-beta decay to a 2+ state thus provides +information that is different from that one could gather +from the study of 0+ → 0+ 0νββ processes. Moreover, +the observation of a 0+ → 2+ 0νββ-decay may also pos- +sibly rule out those non-standard mechanisms in which +no right-handed gauge bosons or fermions are present. +III. +IBM-2 NUCLEAR WAVE FUNCTIONS +IBM-2 is a nuclear structure model and was originally +introduced as a phenomenological approach to describ- +ing collective excitations in nuclei [45]. Soon afterwards, +its relation with the shell model was established [46–48]. +The starting point of IBM-2 calculations of any nuclear +observable, including weak decay rates, is to obtain the +nuclear wave functions of the nuclei of interest. Realistic +nuclear wave functions are obtained by fitting the IBM-2 +parameters in order to reproduce the experimental en- +ergy levels and other nuclear properties, such as elec- +tromagnetic transition rates, quadrupole, and magnetic +moments etc. [29, 49–54], and the relevant two-body op- +erators are derived in the IBM-2 formalism [46, 47, 55]. + +3 +Nucleus +ϵdν +ϵdπ +κ +χν +χπ +ξ1 +ξ2 +ξ3 +c(0) +ν +c(2) +ν +c(0) +π +c(2) +π +c(4) +π +76Ge [49] +1.20 +1.20 +-0.21 +1.00 +-1.20 +-0.05 +0.10 +-0.05 +76Se [53] +0.96 +0.96 +-0.16 +0.50 +-0.90 +-0.10 +82Se [53] +1.00 +1.00 +-0.28 +1.14 +-0.90 +-0.10 +82Kr [50] +1.15 +1.15 +-0.19 +0.93 +-1.13 +-0.10 +-0.10 +130Te [52] +1.05 +1.05 +-0.20 +0.90 +-1.20 +-0.18 +0.24 +-0.18 +0.30 +0.22 +130Xe [51] +0.76 +0.76 +-0.19 +0.50 +-0.80 +-0.18 +0.24 +-0.18 +0.30 +0.22 +136Xea +1.31 +-0.04 +0.01 +-0.02 +136Ba [51] +1.03 +1.03 +-0.23 +1.00 +-0.90 +-0.18 +0.24 +-0.18 +0.30 +0.10 +aGS parameters fitted to reproduce the spectroscopic data of the +low-lying energy states. +Table I: Hamiltonian parameters employed in the IBM-2 calculation, along with their references. All the values are in MeV, +with the exception of those of χπ and χν, which are dimensionless. The IBM-2 parameters not shown here are set to zero. +The IBM-2 Hamiltonian describing the spectra of even- +even nuclei, which is often used in literature, and which +is general enough for the phenomenological studies, reads +[29, 47] +HB = ϵd(ˆndπ + ˆndν) − κ +� +QB +ν · QB +π +� ++ +1 +2ξ2 +�� +d† +νs† +π − d† +πs† +ν +� +· +� +˜dνsπ − ˜dπsν +�� ++ 1 +2 +� +ρ +� +K=1,3 +ξK +� +d† +ν × d† +π +�(K) · +� +˜dπ × ˜dν +�(K) ++ +� +L=0,2,4 +Cρ +L +� +[d† +ρd† +ρ](L) · [ ˜dρ ˜dρ](L)� +. +(5) +In the previous expression, ˆndρ = d† +ρdρ and +QB +ρ = d† +ρsρ + s† +ρ ˜dρ + χρ[d† +ρ × ˜dρ](2) +(6) +represent the d-boson number operators and the boson +quadrupole operators for the proton (ρ = π) and neu- +tron (ρ = ν) pairs, respectively; s† +ρ and d† +ρ are sρ- and +dρ-boson creation operators, and the modified dρ-boson +annihilation operator satisfies ˜dρ,m = (−1)mdρ,−m. The +third term on RHS of Eq. (5) is the so-called Majorana +term, which is relevant to the proton-neutron mixed sym- +metry, and has been considered, e.g., in the context of +the isovector collective motion of valence shells. The last +term on the RHS of Eq. (5) corresponds to the interac- +tion between like bosons, and consists of L = 0, 2 and 4 +components, respectively. +A detailed description of the IBM-2 Hamiltonian is +given in Refs. [46] and [54]. The Hamiltonian parameters +are taken from the literature [26, 49–53]. The values of +the Hamiltonian parameters, together with the references +from which they are taken, are reported in Table I. +IV. +0νββ DECAY OPERATORS FOR 0+ → 2+ +TRANSITIONS IN THE IBM-2 +In the present study, we focus on the 2N-mechanism +discussed by Tomoda [39] in the context of L-R models +[5–7, 27, 56]. To do so, we need to consider a specific set +of V (2) +i += MiCκi operators, where the index κ refers to +the λ, η and η′ contributions of L-R models. The coeffi- +cients Cκi and the corresponding two-body operators Mi +are enlisted in Table II. +The seven operators in this table can be written +as a combination of three parts: +the relative coor- +dinate, Oi(ˆr12)h(r12), the center-of-mass coordinate, +Oi(ˆr+12)f(r+12), and the spin part, Oi(σ1, σ2), where +the following notation for the coordinates is used: r12 = +r2 − r1, r+12 = r2 + r1, ˆr = r/|r|. +The neutrino potential, which comes from the electron- +Majorana neutrino exchange, is given by [57, 58] +h(r12) = −r12 +∂ +∂r12 +H(r12, A) . +(7) +In the above equation, +A = ⟨EN⟩ − E(0+) + me + 1 +2Qββ(2+) +(8) +is the closure energy, where E(0+) is the energy of the +initial 0+ state, ⟨EN⟩ the average energy of the interme- +diate excited state, and Qββ(2+) is the Q-value of the +0+ → 2+ decay process. The neutrino propagation func- +tion in Eq. (7), H(r12, A), is given by [57, 58] +H(r12, A) = +4π +(2π)3 +� +dp12 +exp(ip12 · r12) +p12(p12 + A) +, +(9) +where p12 is the conjugate momentum to the r12 coor- +dinate. +More details on the neutrino potential can be +found in e.g., [55]. +On introducing a proton (neutron) creation (annihi- +lation) operator π† +nljm(˜νnljm) that acts on the single- +particle state |nljm⟩, the second quantized fermion op- +erator can be written as + +4 +i +Mi +Cλi +Cηi +C′ +ηi +1 +σ1 · σ2[ˆr12 × ˆr12](2)h(r12) +1 +3 +1 +3 +– +2 +[σ1 ⊗ σ2](2)h(r12) +− 2 +3 +− 2 +3 +– +3 +� +[σ1 × σ2](2) × [ˆr12 ⊗ ˆr12](2)�(2) +h(r12) +� +7 +3 +� +7 +3 +– +4 +[ˆr12 × ˆr12](2)h(r12) +(gV/gA)2 +−(gV/gA)2 +– +5 +� +(σ1 + σ2) × [ˆr12 × ˆr12](2)�(2) +h(r12) +− +� +3 +2(gV/gA) +– +– +6 +� +(σ1 − σ2) × [ˆr12 × ˆr+12](1)�(2) r+12 +r12 h(r12) +– +– +� +1 +2(gV/gA) +7 +� +(σ1 − σ2) × [ˆr12 × ˆr+12](2)�(2) r+12 +r12 h(r12) +– +– +− +� +3 +2(gV/gA) +Table II: 0+ → 2+ 0νββ decay via the 2N-mechanism. Here, we enlist the two-body operators, Mi, giving the dominant +contribution, as well as the coefficients Cλi, Cηi and C′ +ηi of the two-body operators [39]. The coefficients which are not given +explicitly are null. +V (λ) +i += −1 +4 +� +j1,j2 +� +j1′,j2′ +� +J,J′ +(−1)J+J′� +1 + (−1)Jδj1,j2 +� +1 + (−1)J′δj1′,j2′ +×Gi(j1, j2, J, j1′, j2′, J′, λ)(π† +j1 × π† +j2)(J)(˜νj′ +1 × ˜νj′ +2)(J′) , +(10) +with J, J′ = 0, 2 for the current study, and for i = 1 − 5 +Gi = +� +2 +3 +� +kk′ +� +k1k2 +ik1−k2+λ2 ˆk2 +1ˆk2 +2 +ˆλ2 +2 +⟨k10k20λ20⟩ vk1,k2;λ2(r1, r2) ˆkˆk′ˆλ1ˆλ2 +� +� +� +� +� +s1 k1 +k +s2 k2 k′ +λ1 λ2 λ +� +� +� +� +� +ˆJˆλ ˆJ′ +� +� +� +� +� +j1 j2 J +j′ +1 j′ +2 J′ +k +k′ +λ +� +� +� +� +� +׈j1ˆkˆj′ +1 +� +� +� +� +� +1 +2 +l1 j1 +1 +2 +l′ +1 j′ +1 +s1 k1 +k +� +� +� +� +� +ˆj2ˆk′ˆj′ +2 +� +� +� +� +� +1 +2 +l2 j2 +1 +2 +l′ +2 j′ +2 +s2 k2 +k +� +� +� +� +� +⟨ 1 +2∥Σ(s1)∥ 1 +2⟩ (−1)k1ˆl1⟨l10k1l′ +10⟩⟨ 1 +2∥Σ(s2)∥ 1 +2⟩(−1)−k2 +׈l2⟨l20k2l′ +20⟩R(k1k2λ2)(n1, l1, n2, l2, n′ +1, l′ +1, n′ +2, l′ +2) . +(11) +In Eq. +(11), Σ(1) += +σ, Σ0 += +1, ⟨ 1 +2∥Σ(s)∥ 1 +2⟩ += +� +2(2s + 1), |li − l′ +i| ≤ ki ≤ li + l′ +i, |j1 − j′ +1| ≤ k ≤ j1 + j′ +1, +and |j2 − j′ +2| ≤ k′ ≤ j2 + j′ +2. Also, an additional factor +of − +√ +3 is needed for G1 and of − +� +3/2 for G2. +G5 is +evaluated in two parts by applying additional factor of +1/3 +� +1/6. In our current study the protons and neutrons +occupy the same major shell and thus the contributions +of G6 and G7 vanish. Regarding the radial integrals, indi- +cated as R(k1k2λ2)(n1, l1, n2, l2, n′ +1, l′ +1, n′ +2, l′ +2) in Eq. (11), +their evaluation follows the procedure discussed in [59]. +The calculation of the nuclear matrix elements of 0νββ +decay would, in principle, proceed by going through all +the virtual intermediate states in the odd-odd nucleus. +However, this is a demanding task, which can be greatly +simplified by treating the sum over the intermediate +states in the closure limit. This is a good approximation +in the case of 0νββ decay, as the energy of the virtual +neutrino exchange between nucleons is much larger than +the typical excitation energy of the intermediate states +[27, 28]. In the closure approximation, one is left with +the calculation of two-body NMEs between even-even ini- +tial and final states. +In order to obtain the bosonic image of the fermionic +0νββ decay operator, we proceed in a similar way to +Ref. [55]. The following step is then to define a map- +ping between the IBM-2 JP = 0+ and 2+ boson cre- +ation operators, s† and d†, and the shell-model creation +operators of collective nucleon pairs, S† = +� +ρ† +j × ρ† +j +�(0) +and D† = +� +ρ† +j × ρ† +j′ +�(2) +M , where the fermion operators +ρ† +j create nucleons (either neutrons, ρ = ν, or protons, +ρ = π) with angular momentum j [46, 47, 55]. This pro- +cedure enables us to find a direct correspondence between +the matrix elements between fermionic states in the SD +shell-model subspace and the matrix elements in the sd +bosonic space of the IBM-2. One has +� +ρ† +j × ρ† +j +�(0) +→ Aρ(j) s† +ρ +(12a) +and +� +ρ† +j × ρ† +j′ +�(2) +M → Bρ(j, j′) d† +ρ,M , +(12b) + +5 +where the mapping coefficients Aρ(j) and Bρ(j, j′) are +obtained by means of the OAI method [47] and depend +on the specific normalization of the nuclear structure co- +efficients that one considers. +Our choice is to use the +conventions for the Aρ(j) and Bρ(j, j′) coefficients re- +ported in Refs. [60, 61], which are based on the procedure +for diagonalizing the Surface Delta Interaction (SDI) of +Ref. [62] and the use of the commutator method of Refs. +[55, 63, 64]. +V. +0+ +1 TO 2+ +1 0νββ DECAY +A. +Nuclear matrix elements +Here, we give results for the NMEs relevant to the +0+ +1 to 2+ +1 0νββ decay processes of 76Ge, 82Se, 130Te and +136Xe. The nuclear matrix elements of the operators Mi +(with i = 1, ..., 7) in Table II are computed in the IBM- +2 formalism [46, 47, 55]. The calculations can be made +more realistic by introducing short-range correlation ef- +fects, i.e. the two-body operators in Table II need to be +multiplied by the short-range correlation function, f(r), +squared. Following Ref. [55], we make use of the Jastrow +function, +f(r) = 1 − ce−ar2(1 − br2) , +(13) +with Argonne parametrization a = 1.59 fm−2, b = 1.45 +fm−2 and c = 0.92 [65]. +The finite size of the nucleon is taken into into account +by substituting the coupling constants gA and gV with +the form factors, +gV(p2 +12) = +gV +� +1 + p2 +12 +M 2 +V +�2 +(14a) +and +gA(p2 +12) = +gA +� +1 + p2 +12 +M 2 +A +�2 . +(14b) +In the above equations, the constant M 2 +V = 0.71 GeV2 is +fixed by the electromagnetic form factor of the nucleon +[66, 67] and the value of gV = 1 by the conserved vector +current (CVC) hypothesis; the value of M 2 +A = 1.09 GeV2 +is determined from neutrino scattering data [68] and that +of gA from neutron decay [69]. +Our IBM-2 results for 76Ge, 82Se, 130Te, and 136Xe are +given in Table III. In the last three columns, we give the +combined NMEs +Mλ = +5 +� +i=1 +CλiMi , Mη = +5 +� +i=1 +CηiMi , +M ′ +η = +7 +� +i=6 +C′ +ηiMi , +(15) +where the values of the coefficients Cλi, Cηi and C′ +ηi are +given in Table II. The values in Table III are calculated +by using unquenched values of gV = 1 and gA = 1.269, +and quenching can be implemented through coefficients +C. +It is worth noting that, in the nuclei considered here, +protons and neutrons occupy the same major shell and +thus the contributions of M6 and M7 vanish, leading to +M ′ +η = 0. One also notices particularly small M1 value +for the case of 82Se originating from the small bosonic +d† +π ˜dν matrix element. +The same also happens for the +first excited 0+ state in 82Se decay, leading to small IBM- +2 82Se(0+ +1 ) → 82Kr(0+ +2 ) 0νββ NME, as can be seen from +Ref. [70]. +Our results can be compared to those of the few ex- +isting studies on this topic. Specifically, the 0+ → 2+ +0νββ decay of 76Ge was studied in Refs. [39] and [38] by +means of projected the Hartree Fock Bogoliubov (PHFB) +method and quasiparticle random-phase approximation +(QRPA), respectively. We observe that the first operator +provides the largest contribution to both Mλ and Mη in +all of these three calculations. Our results, Mλ = 0.035 +and Mη = 0.108, stand in between the PHFB results +(Mλ = 0.002 and Mη = 0.061) and QRPA results +(Mλ = 0.008 − 0.228 and Mη = 0.317 − 0.540). +B. +Leptonic phase-space integrals +The leptonic phase-space integrals, indicated as Gi +(i = 1, 2) in Eq. (4), are given by [28, 39] +Gi = +2 +ln 2 +(GF cos θC)4 +16π5 +m2 +e +4R2 +A +� Q2+ +ββ +m2 +e +m2e +fip1p2E1E2dE1, +(16) +with E2 = Q2+ +ββ + m2 +e − E1. +In the above equation, +GF = 1.1663787(6) · 10−5 GeV−2 is the Fermi cou- +pling constant and θC the Cabibbo angle [69]; Ej and +pj = +� +E2 +j − m2e are the energies and the asymptotic mo- +menta of the electrons, respectively; +f1 = +3 +(meRA)2 +���f −2−1��2 + |f21|2 + +��f −1−2��2 ++ |f12|2� +(17a) +and +f2 = +3 +(meRA)2 +���f −21 +��2 + +��f −12 +��2 + +��f1−2��2 ++ +��f2−1��2� +(17b) +are combinations of electron wave functions as defined in +Ref. [28, Appendix 1]. +In the calculation of G1,2, we have used electron radial +wave functions obtained via a numerical solution of the +Dirac equation with potential [30, 71] +V (r) = +� +−αZF +3−(r/RA)2 +2RA +× ϕ(r) , +r < R , +− αZF +r +× ϕ(r) , +r ≥ R , +(18) + +6 +M1 +M2 +M3 +M4 +M5 +M6 +M7 +Mλ +Mη +M ′ +η +76Ge +0.189 +-0.056 +-0.023 +-0.069 +-0.013 +0 +0 +0.035 +0.108 +0 +82Se +0.003 +0.080 +-0.003 +-0.007 +-0.011 +0 +0 +-0.051 +-0.053 +0 +130Te +0.153 +-0.081 +-0.016 +-0.050 +-0.006 +0 +0 +0.056 +0.112 +0 +136Xe +0.058 +-0.112 +-0.006 +-0.012 +-0.0001 +0 +0 +0.077 +0.092 +0 +Table III: Nuclear matrix elements for 0+ → 2+ 0νββ decay via the 2N-mechanism obtained by using IBM-2. +which includes finite size corrections to the Coulomb po- +tential of the final nucleus with charge ZF and electron +screening, due to the electronic cloud described in the +Thomas-Fermi approximation by the function ϕ(r). The +thus obtained values of the phase-space integrals in Eq. +(16) are given in Table IV. +G1 [10−15 yr−1] +G2 [10−15 yr−1] +Q-value [keV] +76Ge +1.669 +1.157 +1479.9 +82Se +12.357 +9.159 +2221.4 +130Te +18.464 +14.462 +1991.4 +136Xe +8.611 +6.269 +1639.3 +Table IV: Phase-space factors G1 and G2 for 0+ → 2+ 0νββ +decay. The Q-values for the above decays are reported in the +last column. +They are obtained by subtracting the 0+ +1 -2+ +1 +energy splitting in the levels of the daughter nuclei from the +corresponding Q-values of the standard 0+ → 0+ processes +from Refs. [72–77]. +Our results for PSFs are comparable to those of Ref. +[28], where the leptonic phase-space integrals were com- +puted by making use of electron radial wave functions +approximated by their leading terms in a power series ex- +pansion in r. As an example, converted to our notation, +the resulting values for 76Ge read Gpse +1 += 1.865 × 10−15 +yr−1 and Gpse +2 += 1.296 × 10−15 yr−1. It is noteworthy +that the values obtained with approximate wavefunctions +are slightly larger than the values of G1,2 reported in Ta- +ble IV, as was also shown for the decays to 0+ states in +Ref. [30]. +C. +Limits on the ⟨λ and ⟨η⟩ couplings +By combining the calculated values of the leptonic +PSFs in Table IV with our IBM-2 results for the NMEs +in Table III, we can use Eq. (4) to place limits on the +⟨λ⟩ and ⟨η⟩ couplings in L-R models. +The upper limit on the value of the ⟨λ⟩ coupling in L-R +models is obtained by setting ⟨η⟩ to zero and equating Eq. +(4) to the experimental limit on the 0+ → 2+ +1 0νββ half- +life of the mother nucleus. Analogously, by setting ⟨λ⟩ = +0, one can implement the same procedure and obtain the +limit on the value of the ⟨η⟩ coupling. The experimental +results for the 0+ → 2+ +1 0νββ half-lives of 76Ge, 82Se, +130Te and 136Xe are extracted from Refs. +[32, 34–36]. +Our upper limits on the absolute values of the ⟨η⟩ and +⟨λ⟩ L-R model parameters are reported in Table V. The +most stringent limits on these parameters can be set by +making use of the KamLAND-Zen experimental limits +on the 0+ → 2+ +1 0νββ half-life of 136Xe [36], leading to +limits of |⟨λ⟩| < 1.7 · 10−5 and |⟨η⟩| < 1.4 · 10−5. +The dependence of the parameters ⟨η⟩ and ⟨λ⟩ is also +shown in Fig. 1, where limits on the combination of the +⟨λ⟩ and ⟨η⟩ couplings in L-R models are shown. +Figure 1: Limits on the combination of the ⟨λ⟩ and ⟨η⟩ cou- +plings in L-R models in the case of 76Ge (green dotted line), +82Se (continuous blue line), 130Te (red dotted line), and 136Xe +(purple dot-dashed line). The parameter space outside of the +parallel lines, the shaded area, is excluded. +It is also very interesting to compare our results with +those of the IBM-2 study of 0+ → 0+ 0νββ decays. While +one can set even more stringent limits on ⟨η⟩ and ⟨λ⟩ +in the case of 0+ → 0+ 0νββ transitions [42], namely +∼ 10−9 for ⟨η⟩ and ∼ 10−7 for ⟨λ⟩, in this case one can- +not disentangle the ⟨η⟩ and ⟨λ⟩ dependencies from their +dependence on the ⟨mν⟩ parameter. This is one of the +reasons why 0+ → 2+ +1 0νββ decay searches are worth in- +vestigating and why 0+ → 2+ +1 experimental searches are +conducted in parallel with those for standard 0+ → 0+ +transitions. + +0.0003 +0.0002 +130Te +136Xe +0.0001 +0.0000 +-0.0001 +-0.0002 +-0.0003 +-0.0003-0.0002 -0.0001 +0.0000 +0.0001 +0.0002 +0.0003 +<Λ)7 +Decay +Collaboration +τ 0ν,exp +1/2 +[y] +|⟨λ⟩| +|⟨η⟩| +76Ge +Majorana [32] +> 2.1 × 1024 +< 3.0 · 10−4 +< 9.7 · 10−5 +82Se +CUPID-0 [34] +> 3.0 × 1023 +< 2.0 · 10−4 +< 1.9 · 10−4 +130Te +Gran Sasso [35] +> 1.4 × 1023 +< 2.2 · 10−4 +< 1.1 · 10−4 +136Xe +KamLAND-Zen [36] +> 2.6 × 1025 +< 1.7 · 10−5 +< 1.4 · 10−5 +Table V: Calculated limits on the ⟨η⟩ and ⟨λ⟩ L-R model couplings. These limits are estimated by comparing our IBM-2 results +for the 0+ → 2+ +1 half-life, computed by means of Eq. (4), with the experimental limits on the 76Ge, 82Se, 130Te and 136Xe +half-life (90% C.L.). +VI. +CONCLUSIONS +We have computed the neutrinoless double-β (0νββ) +decay nuclear matrix elements between the ground state +and the first excited 2+ state of 76Ge → 76Se, 82Se → +82Kr, 130Te → +130Xe and 136Xe → +136Ba within the +framework of the microscopic Interacting Boson Model +(IBM-2) [29, 46, 47, 55] by considering the exchange +of a Majorana neutrino between two nucleons (2N- +mechanism) [27, 37]. The IBM-2 formalism was widely +used in the past to obtain results for nuclear observ- +ables, including the spectrum, the electromagnetic and +the weak decays, but also to explore the possible emer- +gence of beyond-the-Standard Model effects in the weak +interactions of nuclei; see e.g., Refs. [55, 70, 78–80]. Our +results for the 76Ge(0+ +1 ) → 76Se(2+ +1 ) NMEs stand in be- +tween the PHFB results and QRPA results from the lit- +erature. For other reported cases, to our knowledge, this +is the first calculation. +We have also calculated the relevant leptonic phase- +space integrals numerically by making use of exact Dirac +wave functions with finite nuclear size and electron +screening [30] in order to set some limits on the standard +couplings of L-R models, |⟨λ⟩| and |⟨η⟩|. As in the case +of decays to 0+ states, the PSFs are found to be slightly +smaller than previous values obtained with approximate +wavefunctions. +The most stringent limits on the parameters |⟨λ⟩| and +|⟨η⟩| can be obtained from the 0+ → 2+ +1 0νββ half-life of +136Xe [36], leading to |⟨λ⟩| < 1.7 · 10−5 and |⟨η⟩| < 1.4 · +10−5. While one can set even more stringent limits on ⟨η⟩ +and ⟨λ⟩ in the case of 0+ → 0+ 0νββ transitions, in this +case one cannot disentangle the ⟨η⟩ and ⟨λ⟩ dependencies +from their dependence on the ⟨mν⟩ parameter, making 0+ +→ 2+ +1 0νββ decay searches worth investigating further. +Acknowledgments +This work was supported by the Academy of Finland, +Grant No. 314733, 320062, 345869, and INFN, Italy. +[1] W. Pauli, Phys. Today 31N9, 27 (1978). +[2] E. Fermi, Z. Phys. 88, 161 (1934). +[3] C. L. Cowan, F. Reines, F. B. Harrison, H. W. Kruse and +A. D. McGuire, Science 124, 103 (1956). +[4] C. Giunti and M. Laveder, +Neutrino mixing, +hep- +ph/0310238. +[5] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974) +Erratum: [Phys. Rev. D 11, 703 (1975)]. +[6] R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 +(1975). +[7] G. Senjanovic and R. N. Mohapatra, Phys. Rev. D 12, +1502 (1975). +[8] R. N. Mohapatra, Phys. Rev. D 34, 3457 (1986). +[9] J. D. Vergados, Phys. Lett. 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D 102, 095016 (2020). + diff --git a/RdA0T4oBgHgl3EQfDv9U/content/tmp_files/load_file.txt b/RdA0T4oBgHgl3EQfDv9U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fa143b89fc9a79621d2a3639fae2f1ec5572af0 --- /dev/null +++ b/RdA0T4oBgHgl3EQfDv9U/content/tmp_files/load_file.txt @@ -0,0 +1,1015 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf,len=1014 +page_content='0+ to 2+ neutrinoless double-β decay of 76Ge, 82Se, 130Te and 136Xe in the microscopic interacting boson model J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Ferretti,1, ∗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Magaña Vsevolodovna,2, † J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Kotila,1, 3, 4, ‡ and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Santopinto2, § 1Department of Physics, University of Jyväskylä, PO Box 35, FI-40014, Jyväskylä, Finland 2INFN, Sezione di Genova, via Dodecaneso 33, 16146 Genova (Italy) 3Finnish Institute for Educational Research, University of Jyväskylä, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Box 35, FI-40014 Jyväskylä, Finland 4Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520-8120, USA (Dated: January 6, 2023) Here, we study the neutrinoless double-β (0νββ) decay between the ground state and the first 2+ state of 76Ge → 76Se, 82Se → 82Kr, 130Te → 130Xe and 136Xe → 136Ba systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The relevant nuclear matrix elements (NMEs) involved in the process are calculated within the formalism of the microscopic interacting boson model (IBM-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The IBM-2 has been widely used to obtain predictions for nuclear observables, such as the spectrum, but also to explore the possible emergence of beyond- the-Standard Model effects in the weak interactions of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our calculations are carried out by considering the exchange of a Majorana neutrino between two nucleons (2N-mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In addition to NMEs, we calculate the associated leptonic phase-space factors (PSFs) using electron radial wave functions, which are obtained by solving numerically the Dirac equation of a screened Coulomb potential that takes into account finite nuclear size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' By combining our IBM-2 results for the NMEs with those for the PSFs along with experimental half-life limits, we can set limits on the ⟨λ⟩ and ⟨η⟩ couplings of left-right (L-R) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' INTRODUCTION Neutrinos have a long story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Their existence was pos- tulated by Pauli in 1930 to ensure the conservation of energy and angular momentum in β-decay [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Fermi’s renowned theory of beta decay dates back to 1933 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In 1956, neutrinos were first observed at Los Alamos by Cowan and Reines via the study of inverse beta decay [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Several decades later, neutrinos are still fascinating and mysterious particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Important questions regarding some of their main properties remain unsolved, including the unknown mechanism that generates their masses and a complete understanding of their mixing mechanism and mass hi- erarchy [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Because of the lack in the standard model (SM) of a Yukawa coupling between the Higgs boson and neutrinos, due to the absence of right-handed neutrinos, the SM has to be extended to provide a neutrino mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Extensions of the SM include the L-R symmetric [5–7] and SUSY [8–13] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Some important issues are directly related to the na- ture of neutrinos as Fermi- or Majorana-type particles, a nature which could be directly assessed via the exper- imental observation of neutrinoless double-beta (0νββ) decay process [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' However, despite of the strenuous attempts by many experimental groups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', [15–21], 0νββ-decay has not yet been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Several theoretical investigations on 0νββ decay have ∗Electronic address: jacopo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='ferretti80@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='com †Electronic address: Ruslan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='Magana@ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='it ‡Electronic address: jenni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='kotila@jyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='fi §Electronic address: elena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='santopinto@ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='it been published over the years (for a review see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [22, 23]) in order to guide the experimentalists in their searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Owing to the low-energy character of 0νββ processes, these studies necessarily involve elements of both particle and nuclear physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In particular, nuclear structure models are necessary in order to take care of the nuclear matrix elements (NMEs) [24–26] entering the expression of the 0νββ-decay half-life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Here, we show the results of a calculation of the 0+ to 2+ 0νββ decay of 76Ge, 82Se, 130Te and 136Xe, in which we consider the exchange of a Majorana neutrino be- tween two nucleons, the so-called 2N-mechanism, within an L-R symmetric model [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Specifically, in our study: I) we compute the relevant NMEs within the microscopic interacting boson model (IBM-2) formalism [29] and compare our results with previous calculations for the studied nuclei within different nuclear structure models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' II) we calculate the leptonic phase-space factors (PSFs) by means of electron radial wave functions, ob- tained by solving numerically the Dirac equation of a screened Coulomb potential that takes into account finite nuclear size [30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' and III) by combining the two above ele- ments, namely the results for the NMEs and the leptonic PSFs, with the experimental limits on the half-life, we set limits on the ⟨λ⟩ and ⟨η⟩ couplings of L-R models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Ex- perimental studies on this decay mode can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', in [31–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Previous calculations for the 0+ → 2+ decay rate for the studied nuclei within different nuclear struc- ture models and via the 2N-mechanism can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [27, 28, 37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' This article is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' II the im- portance of 0+ → 2+ 0νββ decay is discussed and some details on the calculation of 0νββ decay rates in L-R symmetric models are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' III the calcula- tion of IBM-2 wave functions is briefly summarized, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='02007v1 [nucl-th] 5 Jan 2023 2 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' IV the decay operators needed for the descrip- tion of 0+ → 2+ 0νββ-decay are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' V the numerical results for the ingredients needed for the calcu- lation of the decay rate are given and discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Finally, the conclusions are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 0+ → 2+ 0νββ DECAY IN THE L-R SYMMETRIC MODEL If we restrict ourselves only to long-range mechanisms for 0νββ decay, the most general effective Lagrangian is the Lorenz-invariant combination of leptonic, jα, and hadronic, Jα, currents with definite tensor structure and chirality [40–42], L = GF cos θc √ 2 � �Uei jµ,i V −AJ† V −A,µ + � α,β ϵβ α,i ji βJ† α + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (1) Here, GF is the Fermi constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' θc is the Cabibbo an- gle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' the hadronic and leptonic currents are defined as J† α = ¯uOαd and ji β = ¯eOβνi, where the index i spans the neutrino mass eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The indices α, β are V ∓ A, S ∓ P, T ∓ T5, where S, P, T and T5 stand for scalar, pseudo-scalar, tensor, and pseudo-tensor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (1) we have isolated the standard model contribution proportional to Uei, where Uei is the PMNS mixing ma- trix element [43, 44], from non-standard contributions, which are those proportional to the couplings ϵβ α,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' By isolating the V ∓ A currents of the L-R models from those allowed in other types of mechanisms, such as SUSY, and performing a non-relativistic reduction of both the leptonic and hadronic currents, one can obtain the expression of the 0νββ decay half-life for a 0+ → 0+ transition in L-R models: � τ 0ν 1/2(0+ → 0+) �−1 = C(0) mm � ⟨mν⟩ me �2 + C(0) λλ ⟨λ⟩2 + C(0) ηη ⟨η⟩2 + 2C(0) mλ ⟨mν⟩ me ⟨λ⟩ + 2C(0) mη ⟨mν⟩ me ⟨η⟩ + 2C(0) λη ⟨λ⟩ ⟨η⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (2) The above equation is a complicated combination of the three parameters ⟨mν⟩, ⟨λ⟩, and ⟨η⟩ and their respective matrix elements and phase-space factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' for details on the combinations C(0) ij , i, j = m, λ, η see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The new physics beyond the standard model is en- closed in the three parameters ⟨mν⟩ = � i U 2 eimi , (3a) ⟨λ⟩ = λ � j UejVej , (3b) and ⟨η⟩ = η � j UejVej , (3c) where ⟨mν⟩ is the average neutrino mass obtained by summing over mass mi of neutrino species i, and ⟨λ⟩ and ⟨η⟩ are the standard couplings of L-R models [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Uej and Vej are the mixing matrix elements of the PMNS matrix [43, 44] for the standard and non-standard (L-R) mechanisms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Thus, the 0+ → 0+ process can occur because either of right- or left-handed leptonic currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' By contrast, if 0νββ-decay to a 2+ state is observed then in addition to proving the Majorana character of neutrinos, the existence of V + A current would also be established, since as a first approximation, this de- cay mode is triggered by right-handed leptonic currents only [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' To be more specific, the combination of the lowest electron partial waves for the 0+ → 2+ transition is the S and P3/2 case since the total angular momen- tum of the two-electron system should be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' However, in order to have a non-zero contribution of the L-L term that is proportional to the average neutrino mass, ⟨mν⟩, the leading term requires the combination of S and D3/2 electron waves, making it negligible compared with the contributions due to L-R terms, which are those propor- tional to ⟨λ⟩ and ⟨η⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' For more details see [27, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Therefore, for the 0+ → 2+ case the half-life can safely be written without the dependence on the average neu- trino mass � τ 0νββ 1/2 (0+ → 2+) �−1 = g4 A � G1 (Mλ⟨λ⟩ − Mη⟨η⟩)2 + G2 � M ′ η⟨η⟩ �2 � , (4) where one can factorize the leptonic phase-space factor (PSF) [28, 30], G, the nuclear matrix elements (NMEs), M, and the axial vector coupling constant, gA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Neu- trinoless double-beta decay to a 2+ state thus provides information that is different from that one could gather from the study of 0+ → 0+ 0νββ processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Moreover, the observation of a 0+ → 2+ 0νββ-decay may also pos- sibly rule out those non-standard mechanisms in which no right-handed gauge bosons or fermions are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' IBM-2 NUCLEAR WAVE FUNCTIONS IBM-2 is a nuclear structure model and was originally introduced as a phenomenological approach to describ- ing collective excitations in nuclei [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Soon afterwards, its relation with the shell model was established [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The starting point of IBM-2 calculations of any nuclear observable, including weak decay rates, is to obtain the nuclear wave functions of the nuclei of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Realistic nuclear wave functions are obtained by fitting the IBM-2 parameters in order to reproduce the experimental en- ergy levels and other nuclear properties, such as elec- tromagnetic transition rates, quadrupole, and magnetic moments etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [29, 49–54], and the relevant two-body op- erators are derived in the IBM-2 formalism [46, 47, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 3 Nucleus ϵdν ϵdπ κ χν χπ ξ1 ξ2 ξ3 c(0) ν c(2) ν c(0) π c(2) π c(4) π 76Ge [49] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='05 76Se [53] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='10 82Se [53] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='10 aGS parameters fitted to reproduce the spectroscopic data of the low-lying energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Table I: Hamiltonian parameters employed in the IBM-2 calculation, along with their references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' All the values are in MeV, with the exception of those of χπ and χν, which are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The IBM-2 parameters not shown here are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The IBM-2 Hamiltonian describing the spectra of even- even nuclei, which is often used in literature, and which is general enough for the phenomenological studies, reads [29, 47] HB = ϵd(ˆndπ + ˆndν) − κ � QB ν · QB π � + 1 2ξ2 �� d† νs† π − d† πs† ν � � ˜dνsπ − ˜dπsν �� + 1 2 � ρ � K=1,3 ξK � d† ν × d† π �(K) · � ˜dπ × ˜dν �(K) + � L=0,2,4 Cρ L � [d† ρd† ρ](L) · [ ˜dρ ˜dρ](L)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (5) In the previous expression, ˆndρ = d† ρdρ and QB ρ = d† ρsρ + s† ρ ˜dρ + χρ[d† ρ × ˜dρ](2) (6) represent the d-boson number operators and the boson quadrupole operators for the proton (ρ = π) and neu- tron (ρ = ν) pairs, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' s† ρ and d† ρ are sρ- and dρ-boson creation operators, and the modified dρ-boson annihilation operator satisfies ˜dρ,m = (−1)mdρ,−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The third term on RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (5) is the so-called Majorana term, which is relevant to the proton-neutron mixed sym- metry, and has been considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', in the context of the isovector collective motion of valence shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The last term on the RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (5) corresponds to the interac- tion between like bosons, and consists of L = 0, 2 and 4 components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' A detailed description of the IBM-2 Hamiltonian is given in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [46] and [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The Hamiltonian parameters are taken from the literature [26, 49–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The values of the Hamiltonian parameters, together with the references from which they are taken, are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 0νββ DECAY OPERATORS FOR 0+ → 2+ TRANSITIONS IN THE IBM-2 In the present study, we focus on the 2N-mechanism discussed by Tomoda [39] in the context of L-R models [5–7, 27, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' To do so, we need to consider a specific set of V (2) i = MiCκi operators, where the index κ refers to the λ, η and η′ contributions of L-R models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The coeffi- cients Cκi and the corresponding two-body operators Mi are enlisted in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The seven operators in this table can be written as a combination of three parts: the relative coor- dinate, Oi(ˆr12)h(r12), the center-of-mass coordinate, Oi(ˆr+12)f(r+12), and the spin part, Oi(σ1, σ2), where the following notation for the coordinates is used: r12 = r2 − r1, r+12 = r2 + r1, ˆr = r/|r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The neutrino potential, which comes from the electron- Majorana neutrino exchange, is given by [57, 58] h(r12) = −r12 ∂ ∂r12 H(r12, A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (7) In the above equation, A = ⟨EN⟩ − E(0+) + me + 1 2Qββ(2+) (8) is the closure energy, where E(0+) is the energy of the initial 0+ state, ⟨EN⟩ the average energy of the interme- diate excited state, and Qββ(2+) is the Q-value of the 0+ → 2+ decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The neutrino propagation func- tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (7), H(r12, A), is given by [57, 58] H(r12, A) = 4π (2π)3 � dp12 exp(ip12 · r12) p12(p12 + A) , (9) where p12 is the conjugate momentum to the r12 coor- dinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' More details on the neutrino potential can be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' On introducing a proton (neutron) creation (annihi- lation) operator π† nljm(˜νnljm) that acts on the single- particle state |nljm⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' the second quantized fermion op- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='erator can be written as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='Mi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='Cλi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='Cηi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='C′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='ηi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='σ1 · σ2[ˆr12 × ˆr12](2)h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='[σ1 × σ2](2) × [ˆr12 ⊗ ˆr12](2)�(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='[ˆr12 × ˆr12](2)h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='(gV/gA)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='−(gV/gA)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='(σ1 + σ2) × [ˆr12 × ˆr12](2)�(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='2(gV/gA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='(σ1 − σ2) × [ˆr12 × ˆr+12](1)�(2) r+12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='r12 h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='2(gV/gA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='(σ1 − σ2) × [ˆr12 × ˆr+12](2)�(2) r+12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='r12 h(r12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='– ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='2(gV/gA) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='Table II: 0+ → 2+ 0νββ decay via the 2N-mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Here, we enlist the two-body operators, Mi, giving the dominant contribution, as well as the coefficients Cλi, Cηi and C′ ηi of the two-body operators [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The coefficients which are not given explicitly are null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' V (λ) i = −1 4 � j1,j2 � j1′,j2′ � J,J′ (−1)J+J′� 1 + (−1)Jδj1,j2 � 1 + (−1)J′δj1′,j2′ ×Gi(j1, j2, J, j1′, j2′, J′, λ)(π† j1 × π† j2)(J)(˜νj′ 1 × ˜νj′ 2)(J′) , (10) with J, J′ = 0, 2 for the current study, and for i = 1 − 5 Gi = � 2 3 � kk′ � k1k2 ik1−k2+λ2 ˆk2 1ˆk2 2 ˆλ2 2 ⟨k10k20λ20⟩ vk1,k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='λ2(r1, r2) ˆkˆk′ˆλ1ˆλ2 � � � � � s1 k1 k s2 k2 k′ λ1 λ2 λ � � � � � ˆJˆλ ˆJ′ � � � � � j1 j2 J j′ 1 j′ 2 J′ k k′ λ � � � � � ׈j1ˆkˆj′ 1 � � � � � 1 2 l1 j1 1 2 l′ 1 j′ 1 s1 k1 k � � � � � ˆj2ˆk′ˆj′ 2 � � � � � 1 2 l2 j2 1 2 l′ 2 j′ 2 s2 k2 k � � � � � ⟨ 1 2∥Σ(s1)∥ 1 2⟩ (−1)k1ˆl1⟨l10k1l′ 10⟩⟨ 1 2∥Σ(s2)∥ 1 2⟩(−1)−k2 ׈l2⟨l20k2l′ 20⟩R(k1k2λ2)(n1, l1, n2, l2, n′ 1, l′ 1, n′ 2, l′ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (11) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (11), Σ(1) = σ, Σ0 = 1, ⟨ 1 2∥Σ(s)∥ 1 2⟩ = � 2(2s + 1), |li − l′ i| ≤ ki ≤ li + l′ i, |j1 − j′ 1| ≤ k ≤ j1 + j′ 1, and |j2 − j′ 2| ≤ k′ ≤ j2 + j′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Also, an additional factor of − √ 3 is needed for G1 and of − � 3/2 for G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' G5 is evaluated in two parts by applying additional factor of 1/3 � 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In our current study the protons and neutrons occupy the same major shell and thus the contributions of G6 and G7 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Regarding the radial integrals, indi- cated as R(k1k2λ2)(n1, l1, n2, l2, n′ 1, l′ 1, n′ 2, l′ 2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (11), their evaluation follows the procedure discussed in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The calculation of the nuclear matrix elements of 0νββ decay would, in principle, proceed by going through all the virtual intermediate states in the odd-odd nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' However, this is a demanding task, which can be greatly simplified by treating the sum over the intermediate states in the closure limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' This is a good approximation in the case of 0νββ decay, as the energy of the virtual neutrino exchange between nucleons is much larger than the typical excitation energy of the intermediate states [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In the closure approximation, one is left with the calculation of two-body NMEs between even-even ini- tial and final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In order to obtain the bosonic image of the fermionic 0νββ decay operator, we proceed in a similar way to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The following step is then to define a map- ping between the IBM-2 JP = 0+ and 2+ boson cre- ation operators, s† and d†, and the shell-model creation operators of collective nucleon pairs, S† = � ρ† j × ρ† j �(0) and D† = � ρ† j × ρ† j′ �(2) M , where the fermion operators ρ† j create nucleons (either neutrons, ρ = ν, or protons, ρ = π) with angular momentum j [46, 47, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' This pro- cedure enables us to find a direct correspondence between the matrix elements between fermionic states in the SD shell-model subspace and the matrix elements in the sd bosonic space of the IBM-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' One has � ρ† j × ρ† j �(0) → Aρ(j) s† ρ (12a) and � ρ† j × ρ† j′ �(2) M → Bρ(j, j′) d† ρ,M , (12b) 5 where the mapping coefficients Aρ(j) and Bρ(j, j′) are obtained by means of the OAI method [47] and depend on the specific normalization of the nuclear structure co- efficients that one considers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our choice is to use the conventions for the Aρ(j) and Bρ(j, j′) coefficients re- ported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [60, 61], which are based on the procedure for diagonalizing the Surface Delta Interaction (SDI) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [62] and the use of the commutator method of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [55, 63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 0+ 1 TO 2+ 1 0νββ DECAY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Nuclear matrix elements Here, we give results for the NMEs relevant to the 0+ 1 to 2+ 1 0νββ decay processes of 76Ge, 82Se, 130Te and 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The nuclear matrix elements of the operators Mi (with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', 7) in Table II are computed in the IBM- 2 formalism [46, 47, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The calculations can be made more realistic by introducing short-range correlation ef- fects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' the two-body operators in Table II need to be multiplied by the short-range correlation function, f(r), squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [55], we make use of the Jastrow function, f(r) = 1 − ce−ar2(1 − br2) , (13) with Argonne parametrization a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='59 fm−2, b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='45 fm−2 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='92 [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The finite size of the nucleon is taken into into account by substituting the coupling constants gA and gV with the form factors, gV(p2 12) = gV � 1 + p2 12 M 2 V �2 (14a) and gA(p2 12) = gA � 1 + p2 12 M 2 A �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (14b) In the above equations, the constant M 2 V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='71 GeV2 is fixed by the electromagnetic form factor of the nucleon [66, 67] and the value of gV = 1 by the conserved vector current (CVC) hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' the value of M 2 A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='09 GeV2 is determined from neutrino scattering data [68] and that of gA from neutron decay [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our IBM-2 results for 76Ge, 82Se, 130Te, and 136Xe are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In the last three columns, we give the combined NMEs Mλ = 5 � i=1 CλiMi , Mη = 5 � i=1 CηiMi , M ′ η = 7 � i=6 C′ ηiMi , (15) where the values of the coefficients Cλi, Cηi and C′ ηi are given in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The values in Table III are calculated by using unquenched values of gV = 1 and gA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='269, and quenching can be implemented through coefficients C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' It is worth noting that, in the nuclei considered here, protons and neutrons occupy the same major shell and thus the contributions of M6 and M7 vanish, leading to M ′ η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' One also notices particularly small M1 value for the case of 82Se originating from the small bosonic d† π ˜dν matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The same also happens for the first excited 0+ state in 82Se decay, leading to small IBM- 2 82Se(0+ 1 ) → 82Kr(0+ 2 ) 0νββ NME, as can be seen from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our results can be compared to those of the few ex- isting studies on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Specifically, the 0+ → 2+ 0νββ decay of 76Ge was studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [39] and [38] by means of projected the Hartree Fock Bogoliubov (PHFB) method and quasiparticle random-phase approximation (QRPA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' We observe that the first operator provides the largest contribution to both Mλ and Mη in all of these three calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our results, Mλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='035 and Mη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='108, stand in between the PHFB results (Mλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='002 and Mη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='061) and QRPA results (Mλ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='008 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='228 and Mη = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='317 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='540).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Leptonic phase-space integrals The leptonic phase-space integrals, indicated as Gi (i = 1, 2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (4), are given by [28, 39] Gi = 2 ln 2 (GF cos θC)4 16π5 m2 e 4R2 A � Q2+ ββ +m2 e m2e fip1p2E1E2dE1, (16) with E2 = Q2+ ββ + m2 e − E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In the above equation, GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1663787(6) · 10−5 GeV−2 is the Fermi cou- pling constant and θC the Cabibbo angle [69];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Ej and pj = � E2 j − m2e are the energies and the asymptotic mo- menta of the electrons, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' f1 = 3 (meRA)2 ���f −2−1��2 + |f21|2 + ��f −1−2��2 + |f12|2� (17a) and f2 = 3 (meRA)2 ���f −21 ��2 + ��f −12 ��2 + ��f1−2��2 + ��f2−1��2� (17b) are combinations of electron wave functions as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [28, Appendix 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' In the calculation of G1,2, we have used electron radial wave functions obtained via a numerical solution of the Dirac equation with potential [30, 71] V (r) = � −αZF 3−(r/RA)2 2RA × ϕ(r) , r < R , − αZF r × ϕ(r) , r ≥ R , (18) 6 M1 M2 M3 M4 M5 M6 M7 Mλ Mη M ′ η 76Ge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='013 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='108 0 82Se 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='011 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='053 0 130Te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='006 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='112 0 136Xe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0001 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='092 0 Table III: Nuclear matrix elements for 0+ → 2+ 0νββ decay via the 2N-mechanism obtained by using IBM-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' which includes finite size corrections to the Coulomb po- tential of the final nucleus with charge ZF and electron screening, due to the electronic cloud described in the Thomas-Fermi approximation by the function ϕ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The thus obtained values of the phase-space integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (16) are given in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' G1 [10−15 yr−1] G2 [10−15 yr−1] Q-value [keV] 76Ge 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='669 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='157 1479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='9 82Se 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='357 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='159 2221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 130Te 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='464 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='462 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 136Xe 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='611 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='269 1639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='3 Table IV: Phase-space factors G1 and G2 for 0+ → 2+ 0νββ decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The Q-values for the above decays are reported in the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' They are obtained by subtracting the 0+ 1 -2+ 1 energy splitting in the levels of the daughter nuclei from the corresponding Q-values of the standard 0+ → 0+ processes from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [72–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our results for PSFs are comparable to those of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [28], where the leptonic phase-space integrals were com- puted by making use of electron radial wave functions approximated by their leading terms in a power series ex- pansion in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' As an example, converted to our notation, the resulting values for 76Ge read Gpse 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='865 × 10−15 yr−1 and Gpse 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='296 × 10−15 yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' It is noteworthy that the values obtained with approximate wavefunctions are slightly larger than the values of G1,2 reported in Ta- ble IV, as was also shown for the decays to 0+ states in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Limits on the ⟨λ and ⟨η⟩ couplings By combining the calculated values of the leptonic PSFs in Table IV with our IBM-2 results for the NMEs in Table III, we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (4) to place limits on the ⟨λ⟩ and ⟨η⟩ couplings in L-R models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The upper limit on the value of the ⟨λ⟩ coupling in L-R models is obtained by setting ⟨η⟩ to zero and equating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (4) to the experimental limit on the 0+ → 2+ 1 0νββ half- life of the mother nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Analogously, by setting ⟨λ⟩ = 0, one can implement the same procedure and obtain the limit on the value of the ⟨η⟩ coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The experimental results for the 0+ → 2+ 1 0νββ half-lives of 76Ge, 82Se, 130Te and 136Xe are extracted from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [32, 34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our upper limits on the absolute values of the ⟨η⟩ and ⟨λ⟩ L-R model parameters are reported in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The most stringent limits on these parameters can be set by making use of the KamLAND-Zen experimental limits on the 0+ → 2+ 1 0νββ half-life of 136Xe [36], leading to limits of |⟨λ⟩| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 · 10−5 and |⟨η⟩| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 · 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The dependence of the parameters ⟨η⟩ and ⟨λ⟩ is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 1, where limits on the combination of the ⟨λ⟩ and ⟨η⟩ couplings in L-R models are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Figure 1: Limits on the combination of the ⟨λ⟩ and ⟨η⟩ cou- plings in L-R models in the case of 76Ge (green dotted line), 82Se (continuous blue line), 130Te (red dotted line), and 136Xe (purple dot-dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The parameter space outside of the parallel lines, the shaded area, is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' It is also very interesting to compare our results with those of the IBM-2 study of 0+ → 0+ 0νββ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' While one can set even more stringent limits on ⟨η⟩ and ⟨λ⟩ in the case of 0+ → 0+ 0νββ transitions [42], namely ∼ 10−9 for ⟨η⟩ and ∼ 10−7 for ⟨λ⟩, in this case one can- not disentangle the ⟨η⟩ and ⟨λ⟩ dependencies from their dependence on the ⟨mν⟩ parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' This is one of the reasons why 0+ → 2+ 1 0νββ decay searches are worth in- vestigating and why 0+ → 2+ 1 experimental searches are conducted in parallel with those for standard 0+ → 0+ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0002 130Te 136Xe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0003-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0002 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0003 <Λ)7 Decay Collaboration τ 0ν,exp 1/2 [y] |⟨λ⟩| |⟨η⟩| 76Ge Majorana [32] > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 × 1024 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0 · 10−4 < 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 · 10−5 82Se CUPID-0 [34] > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0 × 1023 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='0 · 10−4 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='9 · 10−4 130Te Gran Sasso [35] > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 × 1023 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='2 · 10−4 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='1 · 10−4 136Xe KamLAND-Zen [36] > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='6 × 1025 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 · 10−5 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 · 10−5 Table V: Calculated limits on the ⟨η⟩ and ⟨λ⟩ L-R model couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' These limits are estimated by comparing our IBM-2 results for the 0+ → 2+ 1 half-life, computed by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' (4), with the experimental limits on the 76Ge, 82Se, 130Te and 136Xe half-life (90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' CONCLUSIONS We have computed the neutrinoless double-β (0νββ) decay nuclear matrix elements between the ground state and the first excited 2+ state of 76Ge → 76Se, 82Se → 82Kr, 130Te → 130Xe and 136Xe → 136Ba within the framework of the microscopic Interacting Boson Model (IBM-2) [29, 46, 47, 55] by considering the exchange of a Majorana neutrino between two nucleons (2N- mechanism) [27, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The IBM-2 formalism was widely used in the past to obtain results for nuclear observ- ables, including the spectrum, the electromagnetic and the weak decays, but also to explore the possible emer- gence of beyond-the-Standard Model effects in the weak interactions of nuclei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [55, 70, 78–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Our results for the 76Ge(0+ 1 ) → 76Se(2+ 1 ) NMEs stand in be- tween the PHFB results and QRPA results from the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' For other reported cases, to our knowledge, this is the first calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' We have also calculated the relevant leptonic phase- space integrals numerically by making use of exact Dirac wave functions with finite nuclear size and electron screening [30] in order to set some limits on the standard couplings of L-R models, |⟨λ⟩| and |⟨η⟩|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' As in the case of decays to 0+ states, the PSFs are found to be slightly smaller than previous values obtained with approximate wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' The most stringent limits on the parameters |⟨λ⟩| and |⟨η⟩| can be obtained from the 0+ → 2+ 1 0νββ half-life of 136Xe [36], leading to |⟨λ⟩| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='7 · 10−5 and |⟨η⟩| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content='4 · 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' While one can set even more stringent limits on ⟨η⟩ and ⟨λ⟩ in the case of 0+ → 0+ 0νββ transitions, in this case one cannot disentangle the ⟨η⟩ and ⟨λ⟩ dependencies from their dependence on the ⟨mν⟩ parameter, making 0+ → 2+ 1 0νββ decay searches worth investigating further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' Acknowledgments This work was supported by the Academy of Finland, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' 314733, 320062, 345869, and INFN, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf'} +page_content=' [1] W.' metadata={'source': 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(2022) +Preprint 13 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The Core Normal Type Ia Supernova 2019np – An Overall Spherical +Explosion with an Aspherical Surface Layer and an Aspherical 56Ni Core ★ +Peter Hoeflich1,†, Yi Yang (杨轶) 2,3,§,‡, Dietrich Baade4,§, Aleksandar Cikota5,6, Justyn R. Maund7, +Divya Mishra8, Ferdinando Patat4, Kishore C. Patra2,+, Lifan Wang8, J. Craig Wheeler9, +Alexei V. Filippenko2, Avishay Gal-Yam10, Steve Schulze11 +1Department of Physics, Florida State University, Tallahassee, Florida 32306-4350, USA +2Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA +3Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel +4European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Str. 2, 85748 Garching b. München, Germany +5European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile +6 Gemini Observatory/NSF’s NOIRLab, Casilla 603, La Serena, Chile +7Department of Physics and Astronomy, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK +8George P. and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, Texas A.&M. University, 4242 TAMU, College Station, TX 77843, USA +9Department of Astronomy, University of Texas, Austin, TX 78712, USA +10Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel +11The Oskar Klein Centre, Department of Astronomy and Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden +§Bengier-Winslow-Robertson Fellow ++Nagaraj-Noll-Otellini Graduate Fellow +Accepted 1/12/23; Received 12/9/22; in original from 11/09/22 +ABSTRACT +Optical spectropolarimetry of the normal thermonuclear supernova(SN) 2019np from −14.5 to +14.5 days relative to 𝐵-band +maximum detected an intrinsic continuum polarization (𝑝cont) of 0.21%±0.09% at the first epoch. Between days −11.5 and +0.5, +𝑝cont remained ∼0 and by day +14.5 was again significant at 0.19%±0.10%. Not considering the first epoch, the dominant axis +of Si ii𝜆6355 was roughly constant staying close the continuum until both rotated in opposite directions on day +14.5. Detailed +radiation-hydrodynamical simulations produce a very steep density slope in the outermost ejecta so that the low first-epoch +𝑝cont ≈ 0.2% nevertheless suggests a separate structure with an axis ratio ∼2 in the outer carbon-rich (3.5–4)×10−3M⊙. Large- +amplitude fluctuations in the polarization profiles and a flocculent appearance of the polar diagram for the Ca ii near-infrared +triplet (NIR3) may be related by a common origin. The temporal evolution of the polarization spectra agrees with an off-center +delayed detonation. The late-time increase in polarization and the possible change in position angle are also consistent with an +aspherical 56Ni core. The 𝑝cont and the absorptions due to Si ii 𝜆6355 and Ca ii NIR3 form in the same region of the extended +photosphere, with an interplay between line occultation and thermalisation producing 𝑝. Small-scale polarization features may +be due to small-scale structures, but many could be related to atomic patterns of the quasi-continuum; they hardly have an +equivalent in the total-flux spectra. We compare SN 2019np to other SNe and develop future objectives and strategies for SN Ia +spectropolarimetry. +Key words: supernovae: individual (SN 2019np) – polarization +1 INTRODUCTION +Various models of Type Ia supernova (SN) explosions predict pho- +tometric and spectroscopic evolution that reproduce observations +adequately but not uniquely (Alsabti & Murdin 2017), so it is diffi- +★ Based on observations collected at the European Southern Observatory +under ESO program 0102.D-0528. +† E-mail: phoeflich@fsu.edu +‡ E-mail: yiyangtamu@gmail.com +§ E-mail: dbaade@eso.org +cult to judge models merely by their power in matching light curves +and total-flux spectra. However, they predict different explosion ge- +ometries of the progenitor white dwarf (WD), which can be diag- +nosed with polarimetry (Hoeflich et al. 2021). Polarized optical flux +from supernovae (SNe) can be caused by departures from spherical +symmetry of the global ejecta structure or by chemical “clumps” +with different line opacities that block portions of the photosphere +(Wang & Wheeler 2008; Patat 2017). Both schemes can be under- +stood as an incomplete cancellation of the electric vectors integrated +over the photosphere as seen by the observer. Optical polarime- +© 2022 The Authors +arXiv:2301.04721v1 [astro-ph.SR] 11 Jan 2023 + +2 +P. Hoeflich et al. +try probes the geometric properties of the SN explosion and the +structure of the SN ejecta, without spatially resolving the source. +A wavelength-independent continuum polarization would arise from +Thomson scattering of free electrons with a globally aspherical dis- +tribution. In addition or alternatively, it may be caused by energy +input that is spatially offset from the center of mass (Hoeflich et al. +1995; Livne 1999; Kasen et al. 2003; Höflich et al. 2006a). Polar- +ized spectral features can be induced in the SN ejecta by chemically +uneven blocking within the photosphere and by frequency variations +of the associated line opacities in the thermalisation depth. +Any early polarization signal from thermonuclear explosions of- +fers a critical test of the nature of the progenitor systems of Type +Ia SNe. For example, large deviations from global sphericity in the +density distribution and chemical abundances of the ejecta are pre- +dicted for explosions triggered by the dynamical merger of a double +white dwarf (WD) binary (Pakmor et al. 2012; Bulla et al. 2016a). +The resulting polarization is expected to be significant both in the +continuum and across various spectral lines. The continuum po- +larization can be as high as ≳ 0.5–1% at ∼ 1 week after the ex- +plosion if observed out of the orbital plane (Bulla et al. 2016a). +By contrast, an almost spherical density distribution and a mod- +erate degree of chemical inhomogeneity are predicted by delayed- +detonation models (Höflich et al. 2006a; Pakmor et al. 2012, 2013; +Moll et al. 2014; Raskin et al. 2014). A continuum polarization near +zero as well as modest (≲ 1%) signals across major spectral features +were also predicted by specific multidimensional models for both a +selected delayed-detonation and a sub-Chandrasekhar-mass (MCh) +model (Bulla et al. 2016b). +Polarized spectral lines indicate geometric deviations from spheri- +cal symmetry of the associated elements. Chemical inhomogeneities +are imprinted by the propagation of the burning front. Delayed- +detonation models predict an initial subsonic deflagration result- +ing in turbulence and gravitational compression. As the burning +front travels outward, the flame transforms into a supersonic det- +onation because of Rayleigh-Taylor instability at the interface be- +tween unburned and burned material (Khokhlov 1991). Layers of +intermediate-mass elements (IMEs; i.e., from Si to Ca) are then pro- +duced at the front of the detonation wave. At any given epoch, the po- +larization spectrum samples the geometric information of the ejecta +that intersect the photosphere. As the ejecta expand over time, the +electron density decreases and the photosphere recedes into deeper +layers of the ejecta in mass and velocity. Multi-epoch spectropo- +larimetry tomographically maps out the distribution of various ele- +ments. +More recent early-time observations have also found low contin- +uum polarization in other normal Type Ia SNe namely SN 2018gv +(day −13.6; Yang et al. 2020) and SN 2019ein (day −10.9; Patra et al. +2022). SN 2019ein displayed one of the highest expansion velocities +at early phases as inferred from the absorption minimum of the Si ii +𝜆6355 line (∼ 24, 000 km s−1 at 14 days before photometric 𝐵-band +maximum; Pellegrino et al. 2020). The low continuum polarization +on day −10.9 indicates a low degree of asphericity at this phase, +strengthening the existing evidence that the explosions of Type Ia +SNe maintain a high degree of sphericity from their early phases. +The spectropolarimetry of SN 2018gv on day −13.6 was the earliest +such measurement at its time for any Type Ia SN. The 0.2%±0.13% +continuum polarization five days after the explosion (based on phase +estimates from the early light curve) suggests that the photosphere +was moderately aspherical with an axis ratio of 1.1–1.3. 1. However, +1 Throughout the paper, the term equatorial plane is defined by planes through +even at this early phase, the geometry of the outermost ∼ 10−3 to +∼ 10−2 MWD of SN 2018gv still remained observationally uncon- +strained. The polarization is also sensitive to the rapidly-changing +density structure in the outer layers, which intersect the photosphere +in the first few days (Hoeflich et al. 2017). +SN 2019np was discovered at 2019-01-09 15:58 (UT dates are +used throughout this paper) with a 0.5 m telescope at a clear-band +magnitude of 17.8 (Itagaki 2019). Rapid spectroscopic follow-up ob- +servations were carried out as early as ∼ 1 day after the discovery +(Kilpatrick & Foley 2019; Burke et al. 2019; Wu et al. 2019). Spec- +tral cross-correlations with the “Supernova Identification” (SNID; +Blondin & Tonry 2007) and the “Superfit” (Howell et al. 2005) codes +suggest that SN 2019np is a Type Ia SN discovered ∼ 2 weeks before +maximum light. From the photometry by Burke et al. (2022), we +derived that SN 2019np reached its peak 𝐵-band magnitude at MJD +58509.72±0.06±0.51 (see Appendix. A), where the two uncertainties +represent the statistical and the systematic error, respectively. This +estimate is consistent with the respective values of 58510.2±0.8 and +58509.64±0.06 reported by Sai et al. (2022) and Burke et al. (2022). +All phases used throughout the present paper are given relative to +the 𝐵-band maximum light at MJD 58509.72 (2019-01-26.72). A +comprehensive study of the SN by Sai et al. (2022) concluded that +its photometric and spectroscopic properties were similar to those of +other normal Type Ia SNe. +Sai et al. (2022) detected a ≲5% excess in the early bolometric +flux evolution of SN 2019np compared to radiative diffusion mod- +els (Arnett 1982; Chatzopoulos et al. 2012), hinting at additional +energy input compared to the radioactive decay of a Ni core. They +suggested that the blue and relatively fast-rising early light curves +of SN 2019np are best fitted with the mixing of 56Ni from the inner +to the outer layers of the SN ejecta (Piro & Morozova 2016). The +rise time of SN 2019np is not compatible with models that predict +an early interaction between the SN ejecta and any ambient circum- +stellar matter (CSM) or a companion star (Kasen 2010). Moreover, +the colour evolution of SN 2019np is inconsistent with that predicted +for a progenitor WD below 𝑀Ch and surrounded by a thin helium +shell as discussed in Sections 4.4 & 6.1 . In this “double-detonation” +or “He-shell detonation” picture, an initial detonation is triggered in +the surface He shell, sending a shock wave to the inner region of the +C/O WD. The shock generates compression heat and subsequently +triggers the second detonation that ignites the WD (Woosley et al. +1980; Nomoto 1982a,b; Livne 1990; Woosley & Weaver 1994; Hoe- +flich & Khokhlov 1996; Kromer et al. 2010). Burke et al. (2022) +also suggested an excess in the early flux evolution of SN 2019np, +which may have been too weak to have been caused by an interaction +between the ejecta and a companion. Interaction with any CSM is an +additional possibility. +This study presents five epochs of optical spectropolarimetry of +SN 2019np from 𝑡 ≈ −14.5 to +14.5 days and interpretations based +on detailed radiation-hydrodynamic simulations. The paper is or- +ganised as follows. In Section 2 we outline the spectropolarimet- +ric observations and the data-reduction procedure. The polarization +properties of SN 2019np are discussed in Section 3. The analysis of +these properties with hydrodynamic models is carried out in Sec- +tion 4. We summarise our conclusions in Section 5, and develop a +the center, �𝑛�𝑥 = 0, �𝑥 spanning the plane with the symmetry axis of a +rotationally symmetric ellipsoid as the orthogonal vector �𝑛, or �𝑛 being a line +through the center of the WD and the location of an off-center energy source +(Höflich 1995c). The two so defined planes may be different (Section 4). +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +3 +comprehensive appraisal of the potential of spectropolarimetry for +the understanding of Type Ia SNe in Section 6. +2 SPECTROPOLARIMETRY OF SN 2019np +Spectropolarimetry of SN 2019np was conducted with the FO- +cal Reducer and low dispersion Spectrograph 2 (FORS2; Appen- +zeller et al. 1998) on Unit Telescope 1 (UT1, Antu) of the ESO +Very Large Telescope (VLT). The Polarimetric Multi-Object Spec- +troscopy (PMOS) mode was used for all science observations. A +complete set of spectropolarimetry consists of four exposures at +retarder-plate angles of 0, 22.5, 45, and 67.5 degrees. The 300V +grism and a 1′′-wide slit were selected for all observations. The +order-sorting filter GG435 was in place, which has a cut-on at ∼4350 +Å to prevent shorter-wavelength second-order contamination. This +configuration provides a spectral resolving power of 𝑅 ≈ 440 at a +central wavelength of 5849 Å, corresponding to a resolution-element +size of ∼ 13 Å (or ∼ 670 km s−1) according to the VLT FORS2 user +manual (Anderson 2018). +Observations were obtained at five epochs: (in the format +day/UT) −14.5/2019-01-12, −11.4/2019-01-15, −6.4/2019-01-20, ++0.5/2019-01-27, and +14.5/2019-02-10. At the first epoch, the total +4 × 1100 s integration time was split into two sets of exposures to +reduce the impact of cosmic rays. The two loops were carried out at +relatively large and different airmasses, from 1.84 to 1.73 and from +1.73 to 1.70. We conducted a consistency check of the two measure- +ment sets and found that the Stokes parameters derived for the two +loops agree within their 1𝜎 uncertainties over the entire wavelength +range after rebinning the data to larger resolution elements (e.g., +30 Å and 40 Å bin sizes). We thus combined the two datasets by +taking the mean value of the spectra obtained at each retarder-plate +angle. Relative-flux calibration was based on the flux standard star +HD 93621 observed at a half-wave plate angle 0 degrees near epoch +3. The airmass of the flux standard was chosen to be comparable +to that of the spectropolarimetry of SN 2019np. A log of the VLT +spectropolarimety is presented in Table 1. +After bias and flat-field corrections, the ordinary (o) and ex- +traordinary (e) beams in each two-dimensional spectral image were +extracted following standard routines within IRAF2 (Tody 1986, +1993). A typical root-mean-square (RMS) accuracy of ∼ 0.20 Å +was achieved in the wavelength calibrations. Stokes parameters were +then derived using our own routines based on the prescriptions in +Patat & Romaniello (2006) and Maund et al. (2007), which also cor- +rect the bias due to the non-negativity of the polarization degree. The +observed polarization degree and position angle (𝑝obs, PAobs) and +the true values after bias correction (𝑝, PA) can be written as +𝑝obs = +√︃ +𝑄2 + 𝑈2, 𝑝 = +� +𝑝obs − +𝜎2𝑝 +𝑝obs +� +× ℎ(𝑝obs − 𝜎𝑝); +𝑃𝐴obs = 1 +2arctan +�𝑈 +𝑄 +� +, and 𝑃𝐴 = 𝑃𝐴obs. +(1) +Here, 𝑄 and 𝑈 are the intensity (𝐼)-normalised Stokes parameters. +The correction for the polarization bias is based on equations in +Simmons & Stewart (1985) and Wang et al. (1997), where 𝜎𝑝 and ℎ +2 IRAF is distributed by the National Optical Astronomy Observatories, +which are operated by the Association of Universities for Research in Astron- +omy, Inc., under cooperative agreement with the National Science Foundation. +Figure 1. Spectropolarimetry of SN 2019np on day −14.5 (epoch 1) relative +to 𝐵-band maximum light on MJD 58509.7. The five panels (from top to +bottom) display (a) the arbitrarily scaled flux spectrum with major spectral +features identified and the high-velocity component of Ca ii NIR3 labeled +“hv”; (b,c) the normalised Stokes parameters 𝑄 and 𝑈, respectively; (d) the +polarization spectrum (𝑝); and (e) the polarization position angle. Panels (b)– +(e) represent the polarimetry before ISP correction. The grey lines in panels +(b) and (c) show the data with their original sampling while the heavy lines +in panels (b)–(e) use 30 Å bins for clarity. The grey-shaded vertical bands +identify regions of telluric contamination. +denote the 1𝜎 uncertainty in 𝑝obs and the Heaviside step function, +respectively. The brackets are properly set as in Cikota et al. (2019). +A ≲ 0.1% instrumental polarization was also corrected following +the procedure discussed by Cikota et al. (2017). More details of the +reduction of FORS2 spectropolarimetry can be found in the FORS2 +Spectropolarimetry Cookbook and Reflex Tutorial3, as well as in +Cikota et al. (2017) and Yang et al. (2020). +3 POLARIMETRIC PROPERTIES OF SN 2019np +The spectropolarimetry of SN 2019np obtained on days −14.5, +−11.4, −6.4, +0.5, and +14.5 is presented in Figs. 1–5, respectively, +where the data are not corrected for interstellar polarization (ISP). +Polarization spectra are shown together with the associated scaled +total-flux spectra (hereafter referred to as simply “flux spectra”). Both +have been transformed to the rest frame. +3 ftp://ftp.eso.org/pub/dfs/pipelines/instruments/fors/ +fors-pmos-reflex-tutorial-1.3.pdf +MNRAS 000, 1–?? (2022) + +4 +P. Hoeflich et al. +Figure 2. Same as Figure 1, but for day −11.4 (epoch 2). +Figure 3. Same as Figure 1, but for day −6.4 (epoch 3). +Figure 4. Same as Figure 1, but for day +0.5 (epoch 4). The estimated ISP +level is shown by grey-dashed lines in panels (b)–(d). +Figure 5. Same as Figure 1, but for day +14.5 (epoch 5). +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +5 +Figure 6. Intrinsic polarization of SN 2019np from days −14.5 to +14.5 as +labeled from top to bottom in the subpanels. For each epoch, the degree +of polarization is calculated based on ISP-subtracted Stokes 𝑄 and 𝑈, bias- +corrected following Equation 1, and presented (red histograms) with 30 Å +binning, together with the arbitrarily scaled flux spectrum (black lines). Si ii +𝜆6355 and the photospheric component of the Ca ii NIR3 features are marked, +and their velocities (𝑣) are also given. The high-velocity component of the +Ca ii NIR3 feature is labeled “hv.” +3.1 Interstellar Polarization +Removing the polarization imposed by interstellar dust grains in ei- +ther the Milky Way or the host galaxy or both is essential for revealing +the intrinsic polarization of SNe. This ISP is due to dichroic extinc- +tion by nonspherical dust grains aligned by the interstellar magnetic +field. Therefore, the entire observed wavelength range of the spec- +trum is used to determine the overall level of the ISM polarization. As +will be shown in Section 4.3, both the overall level and the continuum +polarization in a narrow wavelength range plus the spectral features +are consistent in our analysis. This provides an argument that the +procedure to find the ISM polarization does not suppress an overall +net-polarization mimicking overall sphericity. The intrinsic contin- +uum polarization of Type Ia SNe around their peak luminosity is +very low (≲ 0.2%; see, e.g., Wang & Wheeler 2008; Patat 2017; Yang +et al. 2020). Therefore, we used the spectrum of SN 2019np from day ++0.5 as an unpolarized standard. We fitted the Stokes 𝑄, 𝑈 param- +eters and the observed degree of polarization, 𝑝, using Serkowski’s +wavelength-dependent law (Serkowski et al. 1975) as well as a mere +constant. In the given low-ISP regime, we found that Serkowski’s +law failed to yield a satisfactory fit, and the ISP can be characterised +by the latter approach, which requires computing the error-weighted +mean values of 𝑄 and 𝑈 over suitably selected spectral regions. +Using the wavelength range 4400–8900 Å but excluding the telluric +features and the strongly polarized Si ii 𝜆6355 line and the Ca ii near- +infrared (NIR) triplet (8500.36 Å, 8544.44 Å, and 8664.52 Å, with a +central wavelength of 𝜆0 ≈ 8570, denoted as Ca ii NIR3 hereafter) +due to the SN, we estimate the ISP as (𝑄ISP, 𝑈ISP) = (0.019±0.121%, +0.322±0.072%), and 𝑝ISP = 0.343 ± 0.075%. These values are well +consistent with the ISP derived over the wavelength ranges which are +considered to be depolarized due to blanketing by numerous iron ab- +sorption lines (see, e.g., Howell et al. 2001; Höflich et al. 2006b; Patat +et al. 2008; Maund et al. 2013; Patat et al. 2015; Yang et al. 2020). +Adopting the Galactic and the host-galaxy reddening of SN 2019np +of 𝐸(𝐵−𝑉)Gal = 0.018 mag and 𝐸(𝐵−𝑉)host = 0.10±0.03 mag (Sai +et al. 2022), we find the estimated ISP consistent with the empirical +upper limit caused by dichroic extinction and established for dust in +the Galaxy, 𝑝ISP < 9%× 𝐸(𝐵`𝑉), following Serkowski et al. (1975). +3.2 Intrinsic Continuum Polarization +After subtracting the ISP, we determined the continuum polariza- +tion of SN 2019np at all epochs from the Stokes parameters over the +wavelength range 6400–7000 Å, which is considered to be free of +significant polarized spectral features (Patat et al. 2009). The error- +weighted mean Stokes parameters within this region are given in Ta- +ble 2. The uncertainty was estimated by adding the statistical errors +and the standard deviation computed from the 30 Å-binned spectra +within the chosen wavelength range in quadrature. The continuum +polarization within this wavelength interval is consistent with that +computed over the entire observed wavelength range after exclusion +of the broad, polarized Si ii 𝜆6355 and Ca ii NIR3 lines. +The intrinsic continuum polarization of SN 2019np on day −14.5 +was 0.21%±0.09%. After only three days, it had dropped to ∼ 0 by +day −11.4 and remained low until the SN reached its peak lumi- +nosity. By day +14.5, the continuum polarization had increased to +0.19%±0.10%.4 From the power-law fit of the earliest light curves +of SN 2019np, we place the time of first light at 𝑡0 = −17.92 ± 0.06 +day, where 0.06 day is only the statistical error (see Appendix A). An +additional systematic error of ±0.51 day results from the determina- +tion of the time of the peak luminosity. The times of the five epochs +of VLT spectropolarimetry relative to this time of the SN explosion +are 3.5, 6.5, 11.5, 18.5, and 32.4 days, respectively. The first epoch +is the earliest such measurement for any Type Ia SN to date. +3.3 The Dominant Axes in the Q–U Plane +For each epoch of our ISP-corrected spectropolarimetry, we ex- +amine in the Stokes 𝑄–𝑈 plane the axial symmetry of the ejecta of +SN 2019np as they enter the extended photosphere. We do this sep- +arately for suitable wavelength ranges covering the continuum and +the Si ii 𝜆6355 and Ca ii NIR3 lines. This method was introduced +by Wang et al. (2001); a different graphical rendering of the same +data will be discussed in Section 3.4. Purely axially symmetric ejecta +imprint a linear structure on the 𝑄–𝑈 plane, since the orientation of +the structure is defined by a single polarization position angle, while +varying scattering and polarization efficiencies lead to deviations +from a straight line. By projecting the data onto the best-fitting axis +and measuring the scatter about this so-called dominant axis, +𝑈 = 𝛼 + 𝛽𝑄 , +(2) +one may characterise the degree of axial symmetry of the SN ejecta +(Wang et al. 2003; Maund et al. 2010a). +4 Note that the time variation in 𝑝cont is at a 2 𝜎 level. However, the signif- +icance of the variation is strongly supported by the change in the dominant +axes in the 𝑄–𝑈 plane. Moreover, the estimate of the uncertainty in 𝑝cont +includes real spectral variations in 𝑝 caused by spectral features (see Sections +3.2, 3.3, and 4.3). +MNRAS 000, 1–?? (2022) + +6 +P. Hoeflich et al. +Figure 7 displays the ISP-corrected Stokes parameters on the +𝑄–𝑈 plane between days −14.5 and +14.5. The dominant axis of +SN 2019np as determined from its polarization projected on the 𝑄– +𝑈 plane was derived by performing an error-weighted linear least- +squares fit to the entire observed wavelength range (4350 ≤ 𝜆 ≤ +9100 Å) with the prominent and polarized Si ii 𝜆6355 and Ca ii NIR3 +features excluded. Data points covering the Si ii𝜆6355 and Ca ii NIR3 +profiles were omitted in the top row, where the dominant axis appears +as the black long-dashed line. +To examine the difference between the fits including and excluding +the Si ii 𝜆6355 and Ca ii NIR3 lines, we list the dominant axis and +the corresponding position angles for both cases in Table 2. The +dominant axes of SN 2019np fitted for both cases are consistent with +each other within their 1𝜎 uncertainties except for epochs 3 and 4, +when SN 2019np reached its peak luminosity and the discrepancy +between the two fits amounts to ∼ 2𝜎. We consider the fits with both +broad and polarized individual features excluded a more reasonable +characterisation of the orientation of the SN ejecta since these Si and +Ca features generally exhibit significant deviations from the rest of +the wavelength range (Leonard et al. 2005). +In the middle and bottom rows of Figure 7, the directions of the +symmetry axes of the Si ii 𝜆6355 and Ca ii NIR3 features are shown +by the green and purple dot-dashed lines, respectively. The fitting +procedures were the same as for the continuum but over the velocity +ranges from 24,000 to 0 km s−1 for Si ii 𝜆6355 and from 28,000 to +0 km s−1 for the Ca ii NIR3 complex. The derived parameters are also +listed in Table 2. On day −14.5, the spectropolarimetry over the op- +tical range is poorly represented by a dominant axis. The Ca ii NIR3 +feature is barely described by the linear fits. Additionally, as shown +by the 𝑄–𝑈 diagrams for day −14.5, data points across Si ii 𝜆6355 +deviate from the clustering in the continuum, indicating a conspicu- +ous polarization across the line. However, owing to the relatively low +signal-to-noise ratio (SNR) and the moderate level of polarization, +it is hard to quantitatively determine whether Si ii 𝜆6355 and the +ejecta of SN 2019np determined from the optical continuum (as far +as recorded by FORS2) follow different geometric configurations. +Starting from day −11.4, the ejecta of SN 2019np have developed +a more discernible symmetry axis compared to day −14.5. This is +indicated by the significantly reduced uncertainties in the linear least- +squares fits to the polarimetry on the 𝑄–𝑈 plane (see the 𝛼∗, 𝛽∗, and +𝜃∗ +𝑑 values in Table 2). The dominant axis of SN 2019np shows little +temporal evolution between days −11.4 and +0.5 and rotates by ∼15◦ +from days +0.5 to +14.5. Polar diagrams for Ca ii NIR3 appeared +very flocculent, and somewhat misaligned with the dominant axes of +Si ii 𝜆6355 and 𝑝cont. Qualitatively, the temporal evolution of Si ii +𝜆6355 and Ca ii NIR3 features are similar. This can be seen from +the middle and bottom rows of Figure 7. Not considering the first +epoch, the dominant axis of Si ii 𝜆6355 was roughly constant and +stayed close to that of the continuum until both rotated in opposite +directions on day +14.5. Not considering day −14.5, we suggest that +SN 2019np belongs to the spectropolarimetric type D1 (Wang & +Wheeler 2008), in which a dominant axis can be determined while +the scatter of the data points about the dominant axis is conspicuous. +At the earliest epoch, a dominant axis cannot be clearly identified, and +the continuum polarization measurements cluster around a location +offset from the origin. +Apart from Si ii 𝜆6355 and Ca ii NIR3, there are numerous minor +peaks scattered all over the polarization spectra. Nominally, these fea- +tures are significant at ∼ 2𝜎 and occasionally at 3𝜎. Careful quality +control of the data and our reduction procedures have not identified +them as artifacts, although some of them will undoubtedly be spu- +rious. Most of them are volatile and, in consecutive observations, +do not appear at the same location. This can be expected because +the spectral features form in layers with different abundances (see +Section 4.3). In our analysis in Section 4, we will refer to them as +“wiggles”. +3.4 Line Polarization in Polar Coordinates +To further visualise the geometric distribution of the Si ii and Ca ii +opacities in the ejecta of SN 2019np, we cast the line polarization into +the format of polar plots where the radial axis indicates the velocity +across the spectral profile and the angle from the reference direction +represents the polarization position angles on the plane of the sky at +the corresponding wavelength (introduced by Maund et al. (2009), +and see, e.g., Reilly et al. 2016; Hoeflich 2017; Stevance et al. 2019). +Figure 8 presents the polar plots for the Si ii 𝜆6355 and Ca ii NIR3 +lines from days −14.5 to +14.5. +On day −14.5, relatively highly polarized Si ii is present mostly +above the photospheric velocity. The orientation of the Si-rich ma- +terial appears to be different from the direction of the dominant axis +as determined in Section 3.3 and indicated by the grey sector in the +left panel of Figure 8. Note that the angular size of the fan-shaped +sector represents the 1𝜎 uncertainty of the position angle. Unlike +the Si-rich material that is confined in a relatively narrow range in +position angle, the Ca-rich component exhibits a more diverse radial +profile. The Ca-rich material below the high-velocity (hv) compo- +nent at ∼ 20, 000 km s−1 shows a range in position angle that is +consistent with (i) the dominant axes plotted as black dashed lines +in the left panels of Figure 7, and (ii) the grey fan-shaped sector in +the left panel of Figure 8. However, the component above the high- +velocity threshold exhibits a range in position angle that is distinct +from the dominant axis but has a similar orientation as the Si-rich +material above the photosphere. Therefore, the high-velocity Si-rich +and Ca-rich components seen on day −14.5 are likely to share a +similar geometric distribution that differs from that of the optical +continuum. +On day −11.4, the dominant axis has rotated relative to day −14.5, +as indicated by the position angle of the grey fan-shaped sector in +the second polar plot of Figure 8. Additionally, based on its reduced +angular extent, we deduce that the symmetry axis of the SN ejecta +becomes more prominent and well-defined as the photosphere pro- +gressively recedes. Most of the Si- and the Ca-rich material gets +almost aligned with the optical dominant axis, with larger offsets +seen in the radial profile of the Ca-rich component. This alignment +suggests that a similar axial symmetry is shared by the total ejecta +and the line opacities. An overall similar geometry of SN 2019np +can be derived from the polar plots for days −6.4 and +0.5 (third +and fourth panels in Figure 8), which indicate no significant evo- +lution since day −11.4. From day −11.4 to +0.5, the orientation of +the dominant axis persists. The widths in velocity of the fan-shaped +sectors display an overall decreasing trend for both the Si-rich and +the Ca-rich components. Since the line velocities decrease and the +high-velocity components diminish with time, the polarization sig- +nal measured at the high-velocity end decreases and becomes less +significant as indicated by the large uncertainties. +By day +14.5, the dominant axis has rotated compared to that +measured during the rising phase of SN 2019np. The scatter has +increased again in radial profiles of the Ca-rich material, suggesting a +more complex structure of the line-forming regions in the more inner +layers of the SN ejecta. The high-velocity component has become +indiscernible in the flux spectrum (Figures 5 and 6, and Sai et al. +2022). +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +7 +Figure 7. Intrinsic polarization of SN 2019np displayed on the Stokes 𝑄–𝑈 plane. The top row shows the data over the wavelength range 4250–9100 Å. The +wavelength of each 30 Å bin is indicated by the colour bar on the right side. The middle and the bottom rows display the polarization for the Si ii 𝜆6355 and +Ca ii NIR3 features over the velocity ranges of 24,000–4000 km s−1 and 28,000–2000 km s−1, respectively. The velocities are also indicated by the corresponding +colour bars. The epochs of the observations are labeled with their phases at the top of each column. In each panel, the black long-dashed line shows the dominant +axis calculated over the wavelength range 4250–9100 Å with the Si ii 𝜆6355 and Ca ii NIR3 features excluded (the values of the fitting parameters 𝛼 and 𝛽 in +eq. 2 are listed in Table 2). In the middle and the bottom rows, the green and purple dot-dashed lines in each subpanel represent linear fits to the displayed data +points that cover the Si ii 𝜆6355 and the Ca ii NIR3 features, respectively. +Figure 8. Polar plots of the polarization of SN 2019np across the Si ii 𝜆6355 and Ca ii NIR3 lines at all five epochs. In each panel, the radial distance and the +angle represent the expansion velocity and the polarization position angle, respectively. The center of each fan-shaped bin gives the average position angle +calculated over the velocity range covered by the radial extent of the bin. The angular widths of the fan-shaped bins represent the 1𝜎 uncertainty on the position +angle rather than the underlying physical dimensions. The velocity is labeled in km s−1, and the celestial position angles are measured in degrees from North to +East. The blue and the orange colour bars indicate the ISP-corrected polarization degree across the Si ii 𝜆6355 and Ca ii NIR3 profiles, respectively. The data +have been rebinned to 30 Å for better visualisation. The direction of the dominant axis is shown by grey-shaded regions with their angular width representing +the 1𝜎 uncertainty (i.e., 𝜃∗ +𝑑 in Table 2). The blue and red semicircles mark the estimated photospheric velocity and the high-velocity component as measured +from the absorption minima of the Si ii 𝜆6355 and Ca ii NIR3 lines, respectively. +An overall property of the polar diagrams is their patchy appear- +ance, especially in Ca ii NIR3 (Fig. 6). These “flocculent” structures +tend to become gradually less conspicuous with time, and increase +again at day +14.5. +4 NUMERICAL MODELLING +This section conducts a quantitative study of the degree of as- +phericity of SN 2019np inferred from the observations described in +Section 3. We also investigate their temporal evolution and interpret +the nature of the polarization variations on small wavelength scales. +As a baseline, we will use an off-center delayed-detonation model +MNRAS 000, 1–?? (2022) + +8 +P. Hoeflich et al. +Figure 9. Temporal evolution of the intrinsic polarization of SN 2019np from +days −14.5 to +14.5. The 𝐵 and 𝑉 light curves (from Burke et al. 2022) are +displayed in the top panel. The middle panel gives the continuum polarization +calculated from the error-weighted mean values of the Stokes parameters in +the range 6400–7000 Å. The bottom panel presents the peak polarization +measured across the Si ii 𝜆6355 line. Values measured with both 30 Å and +20 Å bin sizes are plotted as labeled. Second-order polynomial fits to either +bin size are indicated by the solid black and dotted grey curves, respectively. +(Khokhlov 1991), namely the explosion of an 𝑀Ch WD in which a +deflagration front starts in the center and transitions to a detonation +for reasons described in Section 4.1. +A low level of polarization along the continuum spectrum of a SN +is most likely generated by spherically symmetric ejecta leading to +complete cancellation of the electric vectors. However, an aspherical +but rotationally symmetric object may also be viewed along its sym- +metry axis, which has the same effect. To distinguish these two pos- +sibilities, we will use both the polarization over the quasi-continuum +and the modulation of the polarization across major spectral features +in order to separate the intrinsic asphericity and the polarization +actually observed from a certain direction. In our analysis, we will +employ an approach of minimum complexity rather than fine tuning +the parameterised geometry to optimise the fitting. The modeling will +address whether the 0.1%–0.2% polarization variations with wave- +length in the quasi-continuum seen at all epochs can be understood +in terms of opacity variations. Furthermore, we will discuss whether +the temporal and spectral resolution of our VLT spectropolarimetry +is sufficient to detect and probe any small-scale structures in density +and/or abundances. +The VLT spectropolarimetry of SN 2019np between days −14.5 +and +14.5 was analysed through simulations employing modules of +the HYDrodynamical RAdiation (HYDRA) code5 (Höflich 1995a, +2003; Penney & Hoeflich 2014; Hristov et al. 2021; Hoeflich et al. +2021). HYDRA solves the time-dependent radiation transport equa- +5 Many of the HYDRA modules are regularly used by other groups and are +available on request to P.H. +tion (RTE) including the rate equations that calculate the nuclear +reactions based on a network with 211 isotopes and statistical equa- +tions for the atomic level populations, the equation of state, the matter +opacities, and the hydrodynamic evolution. The resulting polariza- +tion is obtained by post-processing the given level populations and +the density and abundance structure through a Monte Carlo (MC) +approach (Khokhlov 1991; Höflich 1995a, 2003; Penney & Hoeflich +2014; Hristov et al. 2021; Hoeflich et al. 2021). Atomic models were +considered for the ionisation stages I–III of C, N, O, Ne, Mg, Na, +Ca, Si, S, Ar, V, Ti, Cr, Fe, Co, and Ni, but without forbidden tran- +sitions. For the luminosity evolution of the multidimensional model +as a function of time, a spherical reference model with 911 depth +points was adopted, which is adequate considering the small devi- +ation from spherical symmetry. Moreover, the timescales are dom- +inated by the inner layers which are almost spherical in off-center +delayed-detonations whereas the spectra are formed in the photo- +sphere. This allows us to compare the observations with snapshots of +the multidimensional model, neglecting time derivatives in the rate +and radiation transport equations. +For the polarization spectra, we use ∼ 700 frequency counters +between 2800 and 10,200 Å. The resulting spatial discretisation cor- +responds to a formal spectral resolving power of 𝑅 ≈ 500, which +matches that of the observations (𝑅 ≈ 440, Section 2). However, in a +rapidly expanding atmosphere with gradients, the spatial resolution +degrades 𝑅 to ∼ 150 since the solution of the radiation transport +equation depends on the spatial gradients of the physical quantities. +Simulating a large number of configurations by multidimensional +models is prohibitively expensive. Therefore, we employ a scattering +approach with a thermalisation depth to find and discuss estimates +for the degree of asphericity in the surface as well as deeper layers +(Höflich 1991). +The continuum polarization may be caused by an aspherical +electron-scattering photosphere or an off-center energy input or both +(Höflich 1995c; Kasen 2006; Bulla et al. 2016a) (see Fig. 10). In the +spectra of a Type Ia SN, opacities from bound-bound transitions form +a wavelength-dependent quasi-continuum and also produce individ- +ual spectral features. The quasi-continuum may exhibit polarization +signals when the sizes of any opacity clumps are comparable to the +free mean path of Thomson scattering. One should keep in mind that, +in the high Thomson optical-depth regime (𝜏 ≳ 3–4), the continuum +polarization in the quasi-continuum will be lower compared to that +at 𝜏 ≈ 1 and reach an asymptotic limit for large optical depths since +any information about asphericity will be blurred by multiple scat- +tering (see, e.g., Figures 1 and 5 of Höflich 1991). If the opacity of +the quasi-continuum becomes much larger than the optical depth of +the Thomson scattering, the degree of polarization 𝑝 ∝ 𝜏sc, where +𝜏sc denotes the electron-scattering optical depth of layers at which +photons thermalise. +4.1 The Reference Model +As the spherically symmetric reference, we adopt the delayed- +detonation Model 25 for a normal-bright Type Ia SN from Hoeflich +(2017) because it shows light-curve properties very similar to those +of SN 2019np. The explosion disrupts a WD with mass close to 𝑀Ch. +Burning starts as a deflagration front near the center and transitions to +a detonation by the mixing of unburned fuel and hot ashes (Khokhlov +1991). +The explosion originates from a C/O WD with a main-sequence +mass of 5 M⊙ as the progenitor star, solar metallicity, and a central +density 𝜌𝑐 = 2 × 109 g cm−3. The deflagration–detonation transi- +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +9 +Figure 10. The symmetry axis �𝑛 (black arrow) is defined by the minor axis of a rotationally symmetric ellipsoid (i.e., an oblate spheroid, left plot) or by the +vector (right plot) through the center (gray dot) and the location of the DDT (green dot). The equatorial plane E (red) is given by �𝑛 �𝑥= 0 with �𝑥 spanning E. The +viewing angle 𝜃 is the angle between the plane E and the direction to the observer (blue line). 𝜃 = +90◦, −90◦, and 0◦ correspond to the north pole, south pole, +and the equator, respectively. As common, 𝜃 is measured counter-clockwise. +Figure 11. Left: The mass above the photosphere as seen by photons in the 𝑈, 𝐵, and 𝑉 bands as a function of time for the normal Type Ia SN calculated in the +off-center angle-averaged version of the delayed-detonation Model 25 (Hoeflich 2017). The exponential index (𝑛) of the radial density distribution at the position +of the photosphere as a function of time is also shown by the red triple-dot-dashed line. The five epochs of VLT spectropolarimetry are marked by grey vertical +dashed lines. Middle: Angle-averaged abundance structure as a function of expansion velocity, also calculated using Model 25. Vertical grey-dashed lines indicate +the location of the scattering photosphere — that is, 𝜏sc = 1 at the times when the VLT spectropolarimetry was obtained. The region with electron-capture +elements is indicated by EC. Right: The 56Ni distribution as seen above the photosphere on day +14.5 based on the hydrodynamical simulation of the off-center +detonation. The mass fraction of off-center 56Ni above the photospheric radius (dark-red) is ∼ 6%. At this phase, the radius of the photosphere is close to +the location (black dot) where the deflagration-to-detonation transition takes place, and it expands with ∼ 7000 km s−1. The mass fraction is colour-coded in a +domain size of ±23,500 km s−1. +tion was triggered when the density at the front had dropped below +2.5 × 107 g cm−3 when ∼ 0.24 M⊙ of the material had been burned +by the deflagration front. For the construction of the off-center de- +layed detonation transition (DDT), we follow the description of Livne +(1999) that has been previously employed (Höflich et al. 2006b; Fe- +sen et al. 2015; Hoeflich et al. 2021). To terminate the deflagration +phase, the delayed-detonation transition is triggered with the mass- +coordinate 𝑀DDT as an additional free parameter. The time series +of the flux and the polarization spectra were generated without fur- +ther tuning of the model parameters. The photometric properties +predicted by the spherical model are similar to those measured for +SN 2019np, namely Δ𝑚15(𝑉/𝐵) = 1.14/0.68 mag (Model 25) and +1.04/0.67 mag (Sai et al. 2022). +According to the above prescription, the axial symmetry of the +SN model is defined by the location(s) of the point(s) where the +deflagration-to-detonation transition took place. The asphericity in +the density distribution near the surface layers was characterised by +introducing an additional free parameter when modeling the contin- +uum polarization at the earliest phase. For the actual implementation +see the last paragraph of this subsection. The symmetry axis that +determines the geometric properties of the outermost layers and that +which is defined by the location of the deflagration-to-detonation +transition in the inner regions are not correlated with each other, +since the latter is stochastic and expected to take place deeper in +the WD. As the DDT is turbulently driven in the regime of dis- +tributed burning, its location depends on the ignition process of the +thermonuclear runaway, namely multispot or off-center ignition, and +initial magnetic fields. In contrast to the inner symmetry axis, that of +the surface layers is likely determined by the direction of the angular +momentum of the progenitor system, i.e., the equatorial plane of a +companion or the plane of an accretion disc. Since the luminosity +originates from the energy source that is well below the photosphere +in the first few days after the SN explosion, and the outermost layers +do not affect the emission at later phases, our simulations treat these +two symmetry axes as independent parameters. +In Figure 11, we present the mass above the photosphere as a func- +tion of time (left panel) and the radial distribution of the chemical +abundances as a function of expansion velocity (middle panel). Over- +all, the exploding envelope has the familiar onion-shell-like structure. +However, the onion is no longer spherical but elongated as a result of +MNRAS 000, 1–?? (2022) + +n +DDT +0 +0 +Equatorial planeC +0 +44Ti +0.8 +66Ni +S +Ne +EC +Mg +0.6 +0.4 +X +0.2 +0 +5 +10 +15 +20 +25 +0 +v [1000 km/sec]1 +X, +56 +Ni +.5 +0. +-20 +-10 +0 +10 +20 +3 km/sec] +Vx[10310 +P. Hoeflich et al. +Figure 12. Left: Continuum polarization as a function of asphericity for an oblate ellipsoidal scattering-dominated photosphere viewed equator-on. A steep +density gradient can be expected at early phases (see Figure 11), when the polarization approaches the limit of large optical depth. The exponential index of the +radial density distribution on day −14.5 can be best represented by configurations with 𝑛 ≈ 13–14 (left plot of Fig. 11). The horizontal line and the cyan-shaded +area indicate the level and the associated uncertainty of the continuum polarization (respectively) measured on day −14.5. Therefore, an axis ratio between +∼ 1.25 and 1.4 is expected. To aid the discussion in the text, the inset panel shows the continuum polarization as a function of thermalisation optical depth for +an oblate ellipsoid with an axis ratio of 2 and 𝜌 ∝ 𝑟−2 (from Höflich 1991). Right: The degree of continuum polarization produced by a spherical photosphere +plus an off-center energy source as a function of viewing angle. For the illustration of the effect of an off-center energy source, a radial density distribution index +𝑛 = 3 and an optical depth 𝜏 = 1 are chosen to represent the SN photosphere around two weeks after maximum brightness (see Figure 11). The horizontal line +and the pink-shaded area mark the level and the error of the continuum polarization, respectively, on day +14.5. +the off-center DDT (see also Fig. 2 in Hoeflich et al. 2021). In the left +panel, we also mark the times of the VLT spectropolarimetry with +respect to both the estimated time of the explosion and the 𝐵-band +light-curve peak. The earliest spectropolarimetry to date of any Type +Ia SN on day −14.5 probes the outermost ≲ (2.5–3) ×10−3 MWD layer +of the exploding WD, corresponding to a mass of ≲ 4×10−3 M⊙. As +deduced from the red triple-dot-dashed curve, at such an early phase, +the exponential index of the radial density distribution is 𝑛 ≈ 13–14. +In the middle panel of Figure 11, we mark the locations of the photo- +sphere at each epoch of the VLT spectropolarimetry in velocity space. +Note that the absorption minimum of, for example, Si ii 𝜆6355 does +not measure the expansion velocity at the photosphere but the aver- +age projected velocity toward an observer. The difference compared +to the expansion velocity at the photosphere is particularly large in +zones with steep density profiles. Since the photosphere recedes over +time, multi-epoch spectropolarimetry can tomographically map out +the degree of asphericity at different chemical layers. As indicated by +the middle panel of Figure 11, the cadence of the VLT observations +of SN 2019np only provides a resolution in expansion velocity of +∼ 6000 km s−1, at which a discrimination of any structures smaller +than several thousand km s−1 in the radial direction is not possible. +For a detailed discussion, see Section 6.1. +We employ a delayed-detonation model considering the fact that +C ii was seen in the first epoch on day −14.5 (Figure 1), correspond- +ing to the very outer layers of ≲ 4 × 10−3 M⊙, making a sub-𝑀Ch +explosion an unlikely candidate even for the case of C/He mixtures +(Shen & Moore 2014). Note that 𝑀Ch explosions may have a thin +H/He-rich surface layer as a result of the accretion phase but at a +significantly smaller mass, (1–5) ×10−4 M⊙ (Hoeflich et al. 2019), +an amount below our numerical resolution. Therefore, we neglect the +H/He layer in our simulation. +In Model 25, the early-time spectra originate from the region +with incomplete explosive carbon burning and an inward-increasing +contribution by explosive oxygen burning (Fig. 11). By the time of +maximum light, the photosphere is formed in layers of complete oxy- +gen burning and partial silicon burning as indicated by the presence +of Ar and Ca. The emergence of Ar lines in the mid-infrared was +predicted by our models. In SN 2014J, they were detected by Telesco +et al. (2015). At ∼ 2 weeks after peak luminosity, the spectrum on +day +14.5 obtained by our last epoch of VLT observations is formed +at the interface between partial, distributed silicon burning and with +burning to nuclear statistical equilibrium (NSE). The position of this +layer coincides with the location where the DDT has been triggered. +Note that in our simulation the point of the DDT does not lead to +a strong refraction wave (Gamezo et al. 2005) as in all spherical +delayed-detonation models (Khokhlov 1991). The innermost layers +undergo weak reactions under NSE conditions, resulting in the pro- +duction of electron-capture (EC) elements. +An asphericity in the outermost layers as indicated by the positive +detection of the continuum polarization at the first epoch (see Sec- +tion 3.2) is not produced by our hydrodynamical reference model. To +estimate the degree of asphericity at that early epoch, we describe +the density structure of SN 2019np by stretching along the radial +direction using an oblate ellipsoid with the axis ratio as a free pa- +rameter. The density and abundance structure are directly taken from +our reference model, transforming the distance from the center of +an element as 𝑟(𝑚) ⇒ 𝑟(𝑚, 𝜃) (Höflich 1991). In the toy models +for the continuum developed in Section 4.2, we treat the orienta- +tion as a free parameter. For reasons of computational feasibility of +the full model, we assume that the symmetry axes of the density +and abundances are aligned. When the deflagration front has burned +∼ 0.25 M⊙, we trigger the detonation by mixing burned and un- +MNRAS 000, 1–?? (2022) + +1.0 +0.6 +Density Structure pαr-n +Spherical + Off-Centered Source +% +0.5 +at T=1 and pαr-3 +P [%] +0.8 +d +3.0 +n=3 +Polarization [ +Polarization +with 10% off-center +n=5 +2.0 +0.4 +with 5% off-center +n=7 +0.6 +n=10 +n=15 +0.3 +0.4 +Day +14.5, +Jo +Jo +0.2 +0.19%± 0.10% +ree +Day -14.5 +0. +.2 +0.21%±0.09% +p +0.1 +9 +0.0 +0.0 +1.1 +1.2 +1.3 +1.0 +1.4 +-50 +50 +0 +Axis Ratio +Inclination +[Degree]Geometry of Type Ia SN 2019np +11 +burned fuel at 𝑀DDT = 0.5 M⊙6, the so-called Zel’dovich reactivity +gradient mechanism (Zel’Dovich et al. 1970). +4.2 Continuum Polarization +On days −14.5 (the first epoch) and +14.5 (the fifth and last +epoch), the level of the continuum polarization has been measured as +0.21%±0.09% and 0.19%±0.10%, respectively, both at a ∼ 2𝜎 level +(see footnote in Section 3.2). The former corresponds to the very +outer layers and the latter probes the inner layers near the position +where the deflagration-to-detonation transition takes place. Between +day −11.4 (second epoch) and day +0.5 (fourth epoch), the contin- +uum polarization was consistent with zero within the uncertainties +(see Section 3.2). +At early times, the thermalisation depth of the photons emitted by +a SN is large (i.e., 𝜏 ≳ 3), and the polarization degree reaches its +asymptotic value (see Figs. 1 and 11 of Hoeflich et al. (1993) and +Höflich (1995c), respectively, and the inset in Figure 12). The max- +imum polarization degree is expected when 𝜏 ≈ 1 (Höflich 1991). +Linear polarization produced by aspherical density structures follows +the relation 𝑝 ∝ 𝑠𝑖𝑛2𝜃, where 𝜃 is the angle between the polar direc- +tion and the observer. As the radial density exponent 𝑛 is high at the +first epoch (𝑛 ≈ 13–14; see left panel of Figure 11), a minimum axis +ratio of 1.25–1.4 can be inferred from the continuum polarization +of 0.21%±0.09% on day −14.5 (see left panel of Figure 12). For an +equator-on perspective (𝜃 = 0o), the high axis ratio implies aspheric- +ity in excess of 30% in the 4 × 10−3 M⊙ of the carbon-rich layers in +the outermost part of the exploding WD (see Figure 11). +Only three days later, on day −11.4, the continuum polarization +had dropped rapidly to a level consistent with zero. By contrast, +for a constant global asphericity, the degree of polarization would +increase with time because (i) the density slope becomes flatter (see +the left panel of Figure 11), and (ii) the thermalisation optical depth +decreases to ∼ 1 as the SN reaches maximum light, when the quasi- +continuum opacity in the iron-rich region becomes comparable to, +or larger than, the Thomson opacity (see, e.g., Figure 2 in Höflich +et al. 1993). +Therefore, the rapid decrease in continuum polarization observed +in SN 2019np suggests that the large-scale asphericity in the density +structures seen at the earliest phase is limited to the very outer layers. +We find that an additional structural component is only required at +the first epoch. For all deeper layers, we do not have to impose any +asphericity on the density distribution. The difference in polarization +position angle between the surface and the deeper layers may be +attributed to the additional structural component. +On day +14.5, the continuum polarization of SN 2019np exhibited +an increase to 0.19%±0.10%, although the scattering optical depth +had decreased significantly well below 𝜏sc = 1, where 𝑝 ∝ 𝜏sc, +and, hence, a decrease in 𝑝 may be expected. As will be discussed +in Section 4.3, the continuum polarization can be understood as a +consequence of our off-center DDT Model 25, which produces an +aspherical distribution of 56Ni, and thus an inhomogeneous central +energy source. +6 Using the amplitude of the Si ii 𝜆6355 polarization as the criterion, we +chose this mass fraction from a set of intermediate models for levels of +0.1, 0.3, 0.5, and 0.9 M⊙. For the Monte Carlo post-processing to obtain 𝑝 +presented in this paper, a number of particles per resolution element has been +used to obtain a statistical absolute error of ≲ 0.015%. +An off-center energy source is needed because the quasi- +continuum dominates the Thomson scattering causing thermalisa- +tion at low 𝜏sc and, thus, only photons with grazing incidence on the +outer photosphere get polarized. The reason is that photons scattered +into the direction of their travel, the Poynting vector, are unpolar- +ized whereas light scattered orthogonally to the Poynting vector is +100% polarized. Radially traveling photons are most likely to escape +when they are scattered along the Poynting vector, whereas grazing- +incidence photons can escape most easily when they are radially +scattered by 90◦. +A similar increase of the continuum polarization after maximum +light was reported for SN 2019ein, namely 0.28%±0.10% on day ++10 and 1.31%±0.32% on day +21 (Maund et al. 2021; Patra et al. +2022). This rise of the broad-band polarization at late phases was +also attributed to the emergence of an aspherical central energy input +as the photosphere reaches the Si/Fe interface (Patra et al. 2022). +As a first step, we quantify the level of the asphericity in the 56Ni +distribution required for a toy model that does not depend on details +of the explosion process. We estimate the amount of off-center 56Ni +at the photosphere relative to the main, symmetric component of +the 56Ni distribution based on previous simulations (Höflich 1995b). +Motivated by the low continuum polarization between days −11.4 +and +0.5 (see Section 3.2), we assume7 that the low continuum po- +larization by any global asphericity in the density at the photosphere +can be neglected and, based on Model 25, that the off-center source +is at about the photosphere (see right panel of Figure 11). For the +toy model, a point-like off-center source at 𝜏𝑠𝑐 = 1 in a spherical +envelope is assumed to obtain a first-order estimate. +The relative contribution by the off-center component at day +14.5 +to the total energy input at the photospheric level is found to be +between 5% and 10% (see the right panel of Figure 12). Using as +reference the axis defined by the center and the location the DDT, a +tangential energy source causes a flip in the polarization angle or, in +the 𝑄–𝑈 diagram, the polarization axis should rotate by 90◦(Höflich +1995c). However, only ∼ 70◦ are observed relative to the layers seen +at day −14.5 . Thus, our toy model predicts a difference between +the symmetry axes of the outer structural component and the inner +layers which causes a change in PA in the 𝑄–𝑈 diagram of about +20◦ compared to day +0.5 (Fig. 7). This estimate is obtained by the +vectorial addition of the polarization contributions by the off-center +source and the spherical source. +Although a change in position angle from day −14.5 to −11.5 +to +14.5 is hard to measure in SN 2019np owing to the very low +intrinsic continuum polarization, we suggest that the rotation of the +dominant axis fitted to the same optical wavelength range as above +(see Figure 7) is compatible with the prediction of an off-center +energy source beginning to be exposed to the observer at this phase. +One of the major effects of the very steep density slope in the +outer layers is that even the small ∼ 0.2% continuum polarization +∼ 3.5 days after the SN explosion implies a significant aspherical +density distribution. The polarization of SN 2019np at the earliest +epoch is higher than that measured in other Type Ia SNe at later +phases, which are closer to epoch 2 on day −11.4 and epoch 3 on +day −6.4 of SN 2019np. For example, 0.10%±0.07% was observed +in SN 2019ein (Patra et al. 2022) on day −10.9, and 0.06%±0.12% +in SN 2012fr (Maund et al. 2013) on day −11. +However, the 0.21%±0.09% continuum polarization of SN 2019np +on day −14.5 (∼ 3.5 days after the explosion) is comparable to +7 As shown in Section 4.3, we cannot allow a large-scale density asymmetry +in Model 25. +MNRAS 000, 1–?? (2022) + +12 +P. Hoeflich et al. +Figure 13. Estimating the viewing angle 𝜃 from polarization spectra (see +text). As examples, the full polarization spectrum of the off-center DDT +Model 25 at day −6.4 is shown for inclinations 𝜃 of about 0◦ (blue, equator- +on), 45◦ (red), and 60◦ (cyan), and, as reference, the intrinsic polarization of +SN 2019np at a corresponding resolution (gray). The estimate of 𝜃 is based +on all epochs, and the error in 𝜃 is constrained by the uncertainty of the +observations. For ±90o the polarization is close to zero. Note the sensitivity +of the ratio between line and continuum polarization to 𝜃. +the marginal detection of a 0.20%±0.13% continuum polarization +in SN 2018gv on day −13.5, ∼ 5 days after the explosion, (Yang +et al. 2020). At that moment, the density exponent in SN 2018gv had +dropped to ∼ 9–10 (see the left panel of Figure 12 and Figure 21 +of Yang et al. 2020). This leads to a ∼ 10%—35% deviation from +spherical symmetry within the outermost ∼ (0.5–2) ×10−2 MWD for +an equator-on configuration. The cases of SNe 2019np and 2018gv +may provide a hint that any asphericity in the outer layers of normal- +bright Type Ia SNe becomes apparent in polarization only during the +very earliest phases and thereafter quickly almost vanishes. There- +fore, given the high density gradient near the surface layers of the +ejecta of Type Ia SNe, a low but nonzero continuum polarization +measured in the first few days after the explosion does not necessar- +ily imply a low deviation from sphericity in their outermost layers. +4.3 Polarization Spectra +The spectral evolution of SN 2019np is similar to that of other +normal-bright Type Ia SNe (Sai et al. 2022), enabling us to compare +the polarization spectra of SN 2019np with the models for normal +Type Ia SNe discussed by Höflich (1995a). The inclination was ob- +tained by comparing the direction-dependent synthetic polarization +spectra to those of SN 2019np at all epochs and minimising the 𝜒2 +averaged over 100 Å-wide bins. To fit the evolution of the polar- +ization spectra (see Section 3.4 and Figs. 6 and 9), we find that a +viewing angle of 𝜃 = +45◦ ±10◦ is the most plausible approximation +of the actual case8. In Figure 16, the off-center delayed-detonation +model viewed at this angle is in good overall agreement with the +observations of SN 2019np. +In Model 25, the polarization across Si ii 𝜆6355 is formed within +an extended geometrical structure between 9000 and 27,000 km s−1 +which undergoes complete and incomplete oxygen burning in veloc- +ity (middle panel of Fig. 11, where the velocity of the photosphere at +the time of the observations is indicated). In the outermost region of +8 For example, at maximum light, compared to 𝜃 = +45◦, the polarization in +Si ii 𝜆6355 is larger by ∼ +50% at 𝜃 ≈ +35◦, vanishes at +90◦, and becomes +small for negative angles depending on the phase, whereas the overall level +of 𝑝 peaks at 𝜃 ≈ +10◦. +partial explosive oxygen and carbon burning, the polarization of Si ii +𝜆6355 is weaker since its abundance diminishes with increasing ve- +locity. At early times, this line forms close to the region with 𝜏sc = 1. +Because the polarization by electron scattering is mostly formed in +the range 0.1 < 𝜏sc < 1, the polarization across Si ii lines is generally +low. The Si polarization increases as the photosphere continuously +recedes and, without the structural component, reaches its peak when +the photosphere enters the layers with quasi-equilibrium conditions +around the Si group. Thus, the polarization in Si ii 𝜆6355 increases +with growing distance between the optical depth at a given wave- +length and the layer with 𝜏sc ≈ 1, which is always more internal. Any +aspherical distribution is expected to be most prominent around this +phase, when the photosphere passes the inner boundary of the ex- +plosive C- and O-burning, and the QSE(Si)/NSE interface becomes +exposed (Höflich et al. 2006a). After peak luminosity, the polariza- +tion of Si ii 𝜆6355 decreases because the quasi-continuum opacities +increasingly dominate the electron scattering. +Apart from Si ii 𝜆6355, the polarization over a quasi-continuum +wavelength range also increases in the same region that forms various +other spectral features, which are resolved (see Figures 14, 15, and +16). This concerns the entire wavelength range occupied by blends +of the Fe group, Si ii, S ii, and O i. Depending on time, features in +the polarization spectra appear around (for example) 4400, 4800, +5400, 5800, 6800, 7200, 7500, 8300, and 9000 Å (Figure 16). Over- +all, thousands of overlapping lines are involved (see, e.g., Figures 1 +and 2 of Hoeflich et al. 1993). The variations in this quasi-continuum +depend on the velocity gradients, the abundances, and the ionisation +level. The polarization is very sensitive to this pattern as it influences +the thermalisation optical depth by individual components because +spectral lines mostly depolarize. In flux spectra, these variations are +mostly blurred because photons are absorbed and emitted, but they +are visible in the line-formation radii traced by spectropolarimetry +(Fig.15). As a result, spectropolarimetry is effectively much more +sensitive to spectral lines than flux spectroscopy because its observ- +able signatures are much less volatile. Nevertheless, some individ- +ual patterns can be identified by comparing the line identifications +(Figures 1 to 5 and 15) and, from the models, by variations in the +wavelength-dependent radii of line formation as presented in Fig- +ure 14. For instance, the rather persistent feature at 9000 Å can be +attributed to a strong Fe ii + Co ii blend which becomes obvious as +a change in the thermalisation radius and appears in both observed +and synthetic spectra (Figure 16). +In the models, a maximum or minimum in polarization is produced +if the thermalisation optical depth is above or below 𝜏sc ≈ 1, respec- +tively. Owing to the sensitivity of the polarization, maxima in the +observations can be minima in the synthetic spectra for moderately +strong blends which appear in the optical depth (Figure 14). One +example is the Fe/Co blend at ∼ 9000 Å. This feature toggles with +time between maxima and minima in both theory and observations. +At most epochs, the changes in model and observations are synchro- +nised, except for days −11.4 and +0.5. Another example is the S/Fe +blend at ∼ 5800 Å, for which the simulations mostly reproduce the +observations but on day +0.5. Both features can be identified as an +elevation in the radius of photon decoupling as shown by Figure 14. +For these examples, the time to toggle from high to low polarization +can be estimated from the rate with which the photosphere recedes +over the abundance gradients. The gradients typically extend over +∼ 500–1000 km s−1 (Figure 11) and so correspond to a timescale of +∼ 1 day. From spectral analysis, a similar timescale of a few days +is well established for changes in ionisation stages (Branch et al. +1981; Mazzali et al. 1993; Höflich 1995c; Lentz et al. 2001; Baron +et al. 2006; Dessart et al. 2014). This is faster than our observing +MNRAS 000, 1–?? (2022) + +Model 25 at day -6.4 seen from different directions +0.6 +00 ++45° ++60° +0.4 +d +0.2 +0 +5000 +6000 +7000 +8000 +9000 +Rest-frame Wavelength.Geometry of Type Ia SN 2019np +13 +Figure 14. Flux and polarization spectra observed in SN 2019np compared to the spectral formation radius computed with Model 25. Top panels: Scaled flux +spectrum (black curve) and degree of polarization (red histogram) observed on days −6.4 (left), and +0.5 (right). Bottom panels: Photon-decoupling radius 𝑅 +for 𝜏 = 0.1 and 1 computed with delayed-detonation Model 25 at similar phases (left panel for day −7 and right panel for day +0). Major spectral lines are +labeled. Note that the polarization is mostly produced by Thomson scattering between 𝜏 = 0.1 and 1. Even strong features such as Si ii 𝜆6355, the Ca ii NIR3 +complex, and various blended features below 5500 Å are formed in the same region, setting a qualitative limit to the picture of line polarization being produced +by chemically selective blocking of an underlying scattering-dominated photosphere. +Figure 15. Spectral formation as a function of radius at maximum light. +We compare the observation with the spectrum of Model 25 (upper panel) +and the radii corresponding to optical depths of 0.1 and 1, between which the +absorption features and the polarization are formed (Figure 12). The observed +and synthetic spectra generally agree without fine tuning. For strong lines like +Si ii 𝜆6355, the flux minima and the blue wings typically correspond to an +optical depth of 1 and 0.1, respectively. This is also true for other strong +features such as Ca ii NIR3 at ∼ 8200 Å and many of the line blends below +5500 Å. Strong spectral features form in the same region where the continuum +polarization is also produced. The high-frequency structure in the radius of +formation of Si ii 𝜆6355 is due to Fe ii transitions which do not appear in the +flux spectra, but show up in the line profiles as discussed in Figure 16. Note +the reduced effect of the quasi-continuum, justifying the wavelength range +used for the determination of 𝑝cont in Section 3.2. +frequency of SN 2019np and may reflect an insignificant phase shift +in the evolution of the models relative to the observations. This phase +shift may also point toward small-scale structures such as Rayleigh- +Taylor fingers or Kelvin-Helmholtz instabilities not included in our +models, which would reveal themselves in short-term variations in +the polarization spectra, but are not resolved in our dataset. The +numerous wiggles (Section 3.3 and Figure 16) are not resolved in the +current observations. They may possibly be understood in the same +way as resolved features: namely in terms of atomic physics. Many +coincide with features produced by the model. Some of them may +indicate genuine small-scale structures, and others may be just noise +in the data or ejecta. Their true nature cannot be determined with +the current observations. This ambiguity points toward a need for +high-cadence observations to separate small scale instabilities from +imprints governed by atomic physics. +The change of the polarization profiles of (for example) Si ii 𝜆6355 +can also be understood within the same framework, leading to a new +diagnostic (Figures 14 and 16) of substructures in lines, although the +spectral resolution of our polarimetry may not be sufficient to fully +reveal the underlying velocity structure. Overall, the location of the +peak (its Doppler shift) agrees between observations and synthetic +profiles including the evolution of the line width. This supports the +interpretation by large-scale asphericity in the abundance distribu- +tions. From the models, this evolution can be understood even though, +at higher granularity, some discrepancies need to be discussed. (a) +Binning of the data may introduce artifacts, in particular at very early +times when the SNR is low in the current data as on day −14.59 or +on day +0.5, when the polarization peak in Si ii 𝜆6355 occupies just +one wavelength bin whereas the associated change in the position +angle takes place over three bins (Figure 4). (b) Some discrepancies +between observations and model profiles may also hint at the model +9 On day −14.5, Si ii 𝜆6355 shows multiple components at 25 Å binning. +MNRAS 000, 1–?? (2022) + +14 +P. Hoeflich et al. +Figure 16. Polarization calculated with off-center DDT Model 25 (blue +histograms) compared to the intrinsic polarization of SN 2019np (red his- +tograms) from days −14.5 to +14.5. The grey dotted curve in each panel +shows the observed scaled flux spectrum at the given epoch. In the model +calculations, an inclination of ∼ 45◦ was adopted. +Fe opacities being too weak between 6000 and 6500 Å, possibly ow- +ing to a lack of Rayleigh-Taylor mixing, too low excitation of the +atomic levels, or slightly too low a metallicity in the progenitor. As +discussed above, even the strong lines are blended with many weak +lines, which do not appear in the flux but in the polarization. +In the simulations, the strength of the polarization depends on +the thermalisation depth in the atmosphere and the density profile +(see Section 4.2). If at some wavelength the thermalisation depth is +close to the Thomson optical depth of 1, the polarization peaks at +that wavelength. It becomes smaller with decreasing thermalisation +depth, and reaches the asymptotic value for large depths. As a result, +the line profile is broader at early times when, owing to steep density +profiles combined with decreasing abundances in the region of in- +complete oxygen burning, namely around day −11.4, the radii of the +photon decoupling regions are similar and, thus, the resulting profiles +are broad. With time, the density slope flattens and, to first order, the +profile becomes narrower. Note that, in an expanding atmosphere, +the absorption is determined by the Sobolev optical depth which is +not inherently spherically symmetric in wavelength (see Figure 14). +By days −6.4 and +0.5, the Si profile is formed in the QSE region and +a flat density gradient leads to an increasing blueshift of the peak. +On day −11.4, Si ii 𝜆6355 is blended with Fe ii 𝜆𝜆6293, 6358, 6497 +and weaker Fe ii and Fe iii transitions from excited levels, leading to +a more complicated profile. In the models, the iron blends seem to be +weaker than in the observations. Because line absorption depolarizes, +this can explain the lack of depolarization in the model profile in +both the blue and red. For the same reason, at days −6.4 and +0.5, +the observed profile has a steep decline whereas the synthetic profile +shows a long red tail. The imprint of Fe ii 𝜆6497 may be seen in the +observations. A similar shape of the Si ii 𝜆6355 polarization profile +has also been seen in SN 2018gv around peak luminosity (Yang et al. +2020). +In the models and the observations after peak luminosity of +SN 2019np, Si ii 𝜆6355 becomes progressively blended with sev- +eral strong Fe ii lines. The Si line has vanished by day +14.5 as the +photosphere recedes into the NSE region, which displays strong Fe- +group elements that form both a quasi-continuum and discrete lines +in the spectrum. The feature at ∼ 6200 Å, which is conventionally +attributed to Si ii 𝜆6355, becomes increasingly dominated by Fe ii +lines and, as a consequence, the corresponding polarization across +this wavelength range also disappears. Overall, the quasi-continuum +on day +14.5 is produced by numerous overlapping Fe-group lines +from Fe, Co, Ni, etc. +Without fine-tuning of our model, the polarization spectra across +several major spectral features can also be reproduced and generally +agree with the observed polarization spectra (see Figure 16). For the +calculation of the continuum polarization, we use the wavelength +range 6400–7000 Å applied in Section 3.2. The global asphericity +in the electron density distribution on day −14.5 is not accounted +for in the hydro simulation. Therefore, we imposed an overall ellip- +tical distribution with an axis ratio of 2 to obtain the overall level of +the polarization over the entire wavelength range observed (see Sec- +tion 4.1). The choice of the axis ratio is motivated by Figure 12 and +results quantitatively from the most likely viewing angle, 𝜃 ≈ 45◦ +(Section 4.3) and the relation 𝑝 ∝ sin2𝜃 (Section 4.2). +At all later epochs, the continuum polarization is calculated di- +rectly without modifying the hydro model (Section 4.1). The con- +tinuum polarization produced by the detailed off-center model (Fig- +ure 16) is within the 1𝜎 error range of the observed values (Table 2). +On day −11.4, the asphericity in the electron density is caused by the +aspherical abundance distributions.10 Its value is ill-defined because +of steep changes of the synthetic polarization at the edges of the +6400–7000 Å wavelength range (see second panel from top of Figure +16), although we used the same range as in Section 3.2 to minimise +the effect of lines. Therefore, the value of 0.16%±0.04% returned by +the models is somewhat larger than the observed level of 0.099%± +0.080%, but well within the error range. The synthetic continuum po- +larization on days −6.4, +0.5, and +14.5 are 0.05%(0.075±0.080%), +0.06% (0.110±0.100%), and 0.22% (0.186±0.104%), respectively, +with the observed values given in brackets11. +Most of the discrete polarization features are at the level of ≲ 0.2– +0.4%. In the models, they are produced by depolarization or the fre- +quency variation in the thermalisation optical depth. Whether they +appear as local maxima or minima in the polarization spectrum de- +pends on the scattering optical depth of the corresponding region +of formation. Many of these wiggles in the observed polarization +10 In SNe Ia and unlike SNe II, the resulting asphericity in the electron dis- +tribution remains rather small, 5–10%, because the free electrons per nucleon +are about equal for Si/S II and Fe/Co II-III. For example, in the hydrogen- +rich envelope of SNe IIP, the opacity drops by 4 orders of magnitudes over +the recombination front of hydrogen, causing highly aspherical Thomson- +scattering dominated photospheres even in case of slightly aspherical 56Ni +distributions or rotation (Höflich et al. 2001; Leonard & Filippenko 2005). +11 Note that the variations in the observed continuum polarization are on a 2𝜎 +level (Sect. 3.1). However, they also coincide with a change in the dominant +axis in the 𝑄–𝑈 diagram (Fig. 7), and 𝜎 includes spectral variations by lines. +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +15 +spectra (Section 3.3) coincide with features in the synthetic polar- +ization spectra. Discrepancies may be due to small-scale structures +that the observations of SN 2019np do not resolve in time and wave- +length. This may hint at the possibility of significant detection of +numerous weakly polarized lines in future higher SNR observations +with FORS2 at ESO’s VLT. Since some patterns do not have mutual +counterparts, such observations should also aim for higher spectral +resolution. Simulations with matching resolution are feasible with +moderate additional effort (see Section 6.1). +Although our models reproduce many (even relatively minor) as- +pects of the observations, some limitations are also apparent. For +instance, on day −14.5, the synthetic polarization spectra exhibit +fairly similar overall patterns, but the models do not show the large- +amplitude fluctuations with wavelength observed in the polarization +spectra of the strongest resonance lines, namely in Ca ii NIR3 (see +Figure 8 and Section 3.4) and possibly in Si ii 𝜆6355.12 The involve- +ment of Ni, Co, or Fe seems to be ruled out because, in the spectral +region strongly affected by iron-group elements (𝜆 ≲ 4700 Å and +𝜆 ≳ 8500 Å), similar patterns do not exist and observations and +synthetic spectra agree well for our model with solar abundances in +the outer C/O layers and Fe, Co, and Ni mass fractions of 0.002, +0.00005, and 0.0001, respectively (Anders & Grevesse 1989). As +discussed in Section 6.1, an amount of 𝑀(56Ni) ≈ 0.02–0.03 M⊙ +in the outer 0.2 M⊙ (∼ 0.1 in mass fraction; alternatively produced +in a sub-𝑀Ch explosion) is needed to explain the early bumps in +light curves of SNe 2017cbv and 2018oh. The associated spectra are +dominated by Fe, Co, and Ni lines (Höflich 1998; Magee & Maguire +2022), traces of which are likely just barely seen in all spectra of +SN 2019np (Figure 16). +Another example for the shortcomings of our current model is +the increased polarization in Ca ii NIR3 around and after maximum +light, which is not reproduced by our models. This resonance line has +by far the largest cross-section and is optically thick even in regions +with solar abundances, so that very minor inhomogeneities can have +a big impact on the polarization. Extended, inhomogeneous radial +components in the Ca distribution may be expected from Rayleigh- +Taylor instabilities, interactions with a companion star, and/or sheet- +like/caustic structures, which may develop within 5–10 days after the +explosion as the result of mixing of radioactive 56Ni and electron- +capture elements (Marietta et al. 2000; Fesen et al. 2007; Hoeflich +2017; Maeda et al. 2018). Additionally, the late rise of the Ca ii NIR3 +polarization may also be caused by the alignment of calcium atoms in +the presence of a magnetic field as recently suggested by Yang et al. +(2022). If combined with near- and mid-infrared nebular spectra, +later-epoch polarimetry of the Ca ii NIR3 feature will allow us to +discriminate between various possibilities concerning the nature of +the progenitor and the explosion mechanism as discussed by Höflich +et al. (2004), Telesco et al. (2015), Hoeflich et al. (2021), and Ashall +et al. (2021), since the spatial distribution of radioactive Co and +stable Fe, Ni, and Co can be probed independently. +4.4 SN 2019np in Polarimetric Context with other Type Ia SNe +The polarization properties of some Type Ia SNe are remarkably +different from those that are typical for normally bright thermonu- +clear SNe (Cikota et al. 2019; Patra et al. 2022) and SN 2019np. An +12 On day −14.5, the feature at ∼ 6000 Å (within the Si ii 𝜆6355 profile, +Figure 16), which occupies a single bin with 30 Å binning, breaks up into two +components with peaks at 0.48% and 0.39% if 40 Å bins are used. +example is SN 2004dt, which exhibited exceptionally high polariza- +tion in some spectral lines. For instance, the peak polarization across +Si ii 𝜆6355 reached ∼ 2.4% and ∼ 3% after binning to 50 Å and +25 Å, respectively (Wang et al. 2006; Cikota et al. 2019). Although +the continuum polarization was as low as ≲ 0.2%–0.3% around peak +brightness (Leonard et al. 2005; Wang et al. 2006), many features +of Si, S, and Mg in the synthetic polarization spectrum13 had their +equivalent in the observations (Höflich et al. 2006b) as commonly +found in Type Ia SNe. These findings might be accounted for by +either a violent merger of two C-O WDs (Bulla et al. 2016a) or an +off-center delayed detonation model within a continuum of param- +eters consisting of especially the position of the delayed-detonation +transition, the amount of burning during the deflagration phase, and +the viewing angle of the observer (Höflich et al. 2006a). +For normally bright Type Ia SNe, the polarization of Si ii 𝜆6355 +five days before 𝐵-band maximum light (𝑝−5 d +Si ii ), which is repre- +sentative of the maximum value (𝑝max +Si ii ), correlates with the light- +curve stretch parameter measured as the decline in magnitude +within 15 days after 𝐵 maximum (Δm15(𝐵); Wang et al. 2007; +Cikota et al. 2019). For SN 2019np, Sai et al. (2022) measured +Δm15(𝐵) = 1.04 ± 0.04 mag, and 𝑝max +Si ii amounted to 0.62±0.03% +on day +0.5 (with 30 Å binning; Figure 9). Therefore, we conclude +that SN 2019np is consistent with the 𝑝max +Si ii –Δm15(𝐵) relation. +In a follow-up study, Maund et al. (2010b) investigated Si ii 𝜆6355 +observations of a sample of nine normal Type Ia SNe and found that +𝑝−5 d +Si ii is also correlated with the temporal velocity gradient �𝑣Si ii. +Accordingly, the deceleration of the SN expansion is also correlated +with the degree of chemical asphericity. The interpolated velocity +gradient of SN 2019np was 21±5 km s−1 day−1 on day +10 (Sai et al. +2022). By interpolating the Si ii 𝜆6355 velocity evolution estimated +from our VLT observations, we estimated velocity gradients of 53±19 +and 22±9 km s−1 day−1 on days +0 and +10, respectively. This means +that SN 2019np was also consistent with the 𝑝−5 d +Si ii –�𝑣Si ii relation. +Subluminous Type Ia SNe exhibit substantially different polariza- +tion properties than discussed above. For instance, a polarization of +∼ 0.7% in the optical continuum but only ∼ 0.3% across Si ii 𝜆6355 +were observed in SNe 1999by (Howell et al. 2001) and 2005ke (Patat +et al. 2012) at ∼ 0 and −7 days relative to maximum light, respec- +tively. The high degree of continuum polarization can be explained +by a global asphericity of as much as 15% (Patat et al. 2012). These +two events are outliers from the correlation proposed by Wang et al. +(2007) between the Si ii 𝜆6355 polarization five days before 𝐵-band +maximum and Δm15, nor do they match the relation between the +velocity gradient of Si ii 𝜆6355 and the associated peak polarization +(Maund et al. 2010b). The mismatch may be due to SNe 1999by and +2005ke perhaps being typical representatives of underluminous Type +Ia SNe. Their spectroscopic and polarimetric properties can be un- +derstood within the frameworks of delayed detonations originating +from a rapidly rotating WD or WD-WD mergers (Patat et al. 2012). +5 CONCLUSIONS +At five epochs between days −14.5 and +14.5 from maximum light, +we have obtained high-quality optical VLT spectropolarimetry of the +normal Type Ia SN 2019np. The first epoch of our observation is the +earliest such measurement carried out to date for any Type Ia SN. The +data have been analysed with detailed radiation-hydrodynamic non- +LTE simulations in the framework of an off-center delayed detonation +13 The continuum polarization in the off-center DDT is ∼ 0.1%–0.2%. +MNRAS 000, 1–?? (2022) + +16 +P. Hoeflich et al. +which produces aspherical distributions in the burning products and, +in particular, an aspherical 56Ni core. The observations are also +compatible with the presence of a central energy source that deviates +from spherical symmetry. The understanding of SN 2019np that we +have achieved with our simulations can be summarised as follows. +(1) A viewing angle of ∼ 45◦ provides the best fit to the amplitude +and temporal evolution of the polarization spectra including Si ii +𝜆6355 and the continuum (Section 4). As discussed in point (3) +below, the continuum polarization at the first epoch on day −14.5 +requires a separate component. +(2) The five epochs roughly cover the time interval in which the +photosphere receded through the layers of incomplete carbon burn- +ing, complete carbon burning, incomplete and complete (QSE14) ex- +plosive oxygen burning, and incomplete Si burning at the interface to +NSE. The cadence of the observations corresponds to a resolution of +∼ 6000 km s−1 in expansion velocity. Higher-cadence observations +than obtained for SN 2019np are required to resolve any structures in +the ejecta at smaller scales. For instance, considering the stratifica- +tion in expansion velocity of the abundance layers and the recession +speed of the photosphere, a cadence of ∼ 1 day would be essential to +map out the interfaces between different chemical layers. +(3) The outermost ≲ 4 × 10−3 M⊙ region seen during the first +∼ 3.5 days after the SN explosion is consistent with a C/O-rich layer. +The change of the polarization position angle (Figure 7) and the ro- +tation of the dominant axis (Figure 8) from the first to later epochs +suggest a different orientation of the outermost layer compared to +the inner regions. To account for the outer asymmetry, a separate +structure had to be added to the models. This renders it relatively +unlikely that SN 2019np originated from a sub-𝑀Ch double detona- +tion induced by a helium shell at the surface of the WD because the +initial shape of the core of the WD can be expected to be symmetric, +being governed by gravity. A deformation of the core may occur in +a rapidly, differentially rotating WD with extreme specific angular +momentum close to the center (Eriguchi & Mueller 1993), but is not +likely. +(4) Although the continuum polarization on day −14.5 was only +0.21%±0.09%, the hydrodynamic modeling of the polarization with +an evolving density profile suggests the presence of a remarkably +aspherical outermost layer of the SN, comprising a fraction of ∼ +(2–3) ×10−3 of the WD-progenitor mass. In conjunction with the +inclination of ∼ 45◦, the axis ratio of the electron density distribution +predicted by the models amounts to ∼ 2–3. That is, in the presence +of a steep density gradient in the outermost layers, a low but nonzero +continuum polarization cannot be taken as an indicator of a high level +of sphericity. +(5) The rise of the continuum polarization to 0.19%±0.07% about +two weeks after peak luminosity is consistent with an aspherical 56Ni +distribution in the core as predicted by off-center delayed-detonation +models. From the models, a change of the polarization position angle +by ∼ 20◦ can be expected. The time-evolving distribution of the +polarimetric signals in the polar plots (Figure 8) and the rotating +dominant axis on the 𝑄–𝑈 plane (Figure 7 and Section 3.3) are +consistent with this prediction. Around maximum light, the off-center +contribution causes a continuum polarization of ≲ 0.1%. +(6) Small 𝑝cont seems to be a characteristic of off-center DDT +models. This justifies the assumption of negligible intrinsic polariza- +tion made for the determination of the ISP (Section 3.1). This method +differs from those commonly applied to core-collapse SNe (Section +3.1). The difference between the continuum polarization 𝑝cont and +14 QSE: Quasi-Statistical -Equilibrium +the average polarization over the entire spectrum with many line- +dominated spectral regions remains small, ∼ 0.1% from −6.4 to ++0.5 days for our normally bright off-center DDT model. The differ- +ence is consistent with the observations of SN 2019np. To understand +the physical reason for being small, see Section 4.3. Despite much +stronger line polarization, a similar small value of 𝑝cont and a small +difference in the continuum is found for the Type Ia SN 2004dt when +analysed within the framework of off-center DDT models (Section +4.4). +(7) The polarization observed across Si ii 𝜆6355 on day −14.5 is +higher than in our model (Figure 16). The dominant axes fitted to this +line and the optical continuum are both not well defined at this phase +(Sect. 3.3). From our full-star models, it is hard to deduce whether +the asphericity of these two components in the outermost layers has a +common origin — for instance, owing to interaction with a low-mass +accretion disc, which is a free parameter and remains unconstrained +in the current modeling process of the full star. However, the toy +model for the continuum polarization (Section 4.2) demands different +symmetry axes for the density and the abundances. +(8) High asphericity is found to be confined to the very outer +layers. It emerges from the fitting when introduced as an additional +free parameter not included in the hydrodynamical model. Possible +physical causes may include a short-lived interaction between the +SN ejecta with a low-mass accretion disc (Gerardy et al. 2007) or a +companion star (Marietta et al. 2000), or surface burning as found in +sub-𝑀Ch explosions (Shen et al. 2012) but see item (3), or the imprint +of the burning of H/He-rich material originating from the surface +(Hoeflich et al. 2019). Interaction with a donor star seems less likely +because it would affect not only the outermost layers. As shown in +Figures 1 to 3 and discussed in Section 4, the sub-𝑀Ch model is in +tension with the observation of the Ca ii NIR3 feature. A similar early- +time polarization has been observed in the normal-bright SN 2018gv +(Yang et al. 2020). However, in the underluminous SN 2005ke, a +significant polarization was observed in the outer ∼ 0.2 M⊙, hinting +toward rapid rotation or a dynamical merger (Patat et al. 2012). +(9) The continuum polarization of SN 2019np vanished by day +−11.5 and remained consistent with zero within one 𝜎 until the SN +reached its peak luminosity, indicating a high degree of spherical +symmetry between the outermost 0.02–0.03 MWD and ∼ 0.5 MWD. +In the case of a highly aspherical configuration extending into deeper +layers, the continuum polarization should increase with time since +the density distribution flattens significantly and the scattering opti- +cal depth decreases from the optically thick regime. However, such an +increase in continuum polarization was not observed in SN 2019np. +This partly invalidates the conventional assumption that a low con- +tinuum polarization provides evidence of low asphericity. In the +presence of a steep density gradient as in the outermost layers of +SN 2019np, major deviations from spherical symmetry are well pos- +sible. High-cadence observations are needed to distinguish between +these alternatives. +(10) The increased continuum polarization on day +14.5 can be +explained by abundance asphericities in an off-center delayed det- +onation. The direction of the dominant axis of SN 2019np has also +changed between days +0 and +14.5 (Figure 7), which may indicate a +small, off-center distribution of the central energy source, 5%–10% +of the total amount of 56Ni. The model also predicts a change of the +polarization position angle of the quasi-continuum. Although quali- +tatively indicated by the observations, the change in the polarization +position angle of the continuum is hard to quantify from the available +observations because the intrinsic continuum polarization is very low +(Section 3), and the numerous small wiggles at this low level may +also cause a problem (Figures 14–16). +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +17 +(11) None of the possible mechanisms causing the rise of the +Ca ii NIR3 polarization on day +14.5 has been included in our sim- +ulations. Its origin remains uncertain, as discussed in Section 4.3. +(12) In the optical domain, spectral line formation and polar- +ization by electron scattering take place in the same region of the +expanding atmosphere (Section 4.3 and Figure 15). By contrast, +the canonical polarization-by-obscuration picture (Wang & Wheeler +2008) requires that spectral lines are formed mainly above the last +continuum-scattering surface. To first order, this approximation can +be used to place an upper limit on the peak polarization from large- +scale asymmetries, which are produced by the large Sobolev optical +depth over the entire photosphere, if the formation of the quasi- +continuum is dominated by Thomson scattering. In SNe Ia, the +Si ii 𝜆𝜆6348, 6373 doublet provides an example. It originates from +a low-excitation state, and Si accounts for ∼60% of the total mass +fraction corresponding to up to ∼ 2, 500 times the solar value found +in the H-rich envelopes of CC-SNe. The expansion velocities of Si- +rich layers range from ∼ 9000 to more than 22,000 km s−1. In the +presence of large-scale asphericity in Si, this line will be significantly +polarized unless the region of asymmetric density or Si abundance +is hidden behind an extended photospheric region. For small-scale +or multiple structures, the resulting polarization depends sensitively +on 𝜏sc at their location. +(13) Overall, the polarization spectra and their temporal evolution +can be understood as a variable thermalisation optical depth and +partial blocking of the photosphere at a given geometric depth. +(14) There are strong high-amplitude fluctuations in the polariza- +tion spectra on day −14.5. They can be attributed to the strongest +S/Si and the Ca ii lines, in particular Ca ii NIR3. The lack of similar +patterns and the agreement between the observations and synthetic +spectra in the spectral region dominated by iron-group elements, +namely the 𝑈 and 𝐵 bands and longward of 8500 Å, seem to rule out +Ni, Co, or Fe as being affected by the same mechanism(s) responsi- +ble for the fluctuations just mentioned.. For details and their possible +origin see Section 4.3. +(15) At the same time, there are also flocculent structures in the +polar diagram for Ca ii NIR3 (Section 3.4, Figure 16) with a broader +distribution in position angle and a lower polarization degree (see +Figure 8), suggesting a more complex structure of the outermost +layers than adopted in the models. +(16) Many relatively weak polarization features within the wave- +length range ∼ 4500–6000 Å and beyond 6800 Å can be interpreted +as signatures of spectral line blends and unresolved wiggles. The +general agreement between data and model suggests that many of +these weak polarization features are real. Some discrepancies do not +necessarily invalidate this conclusion, but likely point toward small- +scale structures (Section 4.3). However, the noise in the data for +SN 2019np sets a limit to probing those scales in this event. +(17) Polarization is able to pick up spectral signatures not visible +in the flux spectra because lines depolarize but both absorb and emit +photons (see Section 4.3). Spectropolarimetry of sufficient spectral +resolution can reveal spectral lines that are undetectable in flux spec- +tra, and, at proper cadence, their depth of formation can be inferred. +This added diagnostic power is independent of any asphericity and +can be important to discriminate explosion scenarios. +(18) The asphericity in the 56Ni distribution is significant (Fig- +ure 11) although the continuum polarization is relatively low. As +discussed in Section 4.2, a low thermalisation depth results in van- +ishingly low continuum polarization, except for photons that graze +the photosphere at low optical Thomson-scattering depth (see Sec- +tion 4.3). As a corollary, high polarization can be expected if the +off-center component dominates, but polarization may fail to detect +even significant asphericity in the density produced by a Fe/Co/Ni +core. +(19) The polarization profiles of strong isolated features provide +new diagnostics to probe for mixing on small and medium scales, +and to explore the chemical stratification (Section 4.3). +6 FUTURE OPPORTUNITIES +6.1 The Diagnostic Power of Spectropolarimetry of Type Ia SNe +Asphericity holds a key to the understanding of the nature of ther- +monuclear SNe. It involves three main components: (1) the exploding +WD, (2) all other matter bound in the progenitor system, which may +include a companion star and any bound CSM such as a common en- +velope or an accretion disc, and the inner parts of several winds (e.g., +from the WD, a companion star or its Roche lobe, and an accretion +disc), and (3) the unbound CSM consisting of the outer faster and +less dense parts of the winds and ultimately the interstellar medium. +A detailed discussion of the effects of these three constituents +on the geometry is beyond the scope of this paper. A broad intro- +duction to the geometrical signatures in polarization and nebular +spectra expected from all commonly considered explosion models +and progenitor channels was recently given by Hoeflich et al. (2021). +Reviews on various topics can be found among the articles collected +in Alsabti & Murdin (2017). +a) To properly plan observing sequences, timescales are critical. +For all explosion paths, the initial phase of the explosion takes a few +seconds to a minute, and the main spatial dimension is given by the +exploding WD or the two merging WDs and ranges from 1.5×108 to +109 cm. The hydrodynamical interaction of the explosively expand- +ing envelope with a companion star, any accretion disc, and the inner +parts of the wind(s) takes place within ∼ 1010–1013 cm. Considering +the velocities and masses in the outer layers of the explosion, this +corresponds to timescales of minutes to about an hour for interaction +with the bound matter. Interaction with the wind(s) and the ISM can +extend over days to many years, and is only limited by the transition +to the supernova-remnant phase. +b) The impact of the various components on the geometrical struc- +ture and the associated polarization depends on the mass of the com- +panion and any bound CSM relative to that of the ejecta which differs +strongly between the explosion processes (∼ 0.6–2 M⊙). +c) The mass-loss rate from the system may range from ∼ 10−6 +to 10−4 M⊙ yr−1 and can be due to any model-dependent combina- +tion of the wind from the WD, the wind from a companion, super- +Eddington accretion onto the WD (e.g., Nomoto et al. 1976), the +Roche lobe in a single-degenerate system, or the high-velocity wind +from accretion discs in cataclysmic variables, or it may be produced +during the final phase of dynamical mergers. Upper limits to the total +mass content of these winds integrated over the time considered for +early polarimetry are ∼ 10−4–10−7 M⊙ (Dragulin & Hoeflich 2016; +Chevalier & Fransson 2017). Even at the earliest times when sig- +nificant polarization can still be expected, the dynamical effects of +the impact of the ejecta on the unbound CSM are likely to be small +(Figure 11). Nevertheless, in early-time photometry, some small ad- +ditional blue flux may appear owing to energy released during the +interaction. For very high wind densities, the hard radiation at the +shock discontinuity and the reverse shock may lead to enhanced ioni- +sation in the photosphere of the SN. In SN 2019np, we see no obvious +evidence of such effects (Section 4.3). +MNRAS 000, 1–?? (2022) + +18 +P. Hoeflich et al. +d) The CSM bound in the system may include matter in a Roche +lobe and/or an accretion disc. Observational evidence has been re- +ported by (for example) Aldering et al. (2006), and, from high- +velocity Ca ii absorptions in early-phase spectra, its mass has been +estimated to be of order 10−2–10−3 M⊙ (Gerardy et al. 2007). This +mass is comparable to that in the layers probed by our early polarime- +try of SN 2019np, and it is compatible with the large asymmetries +proposed (Figure 11). The small-scale structures depend on the scale +height of the material (e.g., the Roche lobe or the disc) and the +sound-crossing time during the hydrodynamical phase of the inter- +action. However, since the mass of the bound CSM is much larger +than that contained in the wind, the structure produced by the inter- +action of the ejecta with this matter can be expected to be conserved +in the subsequent possible interaction with the outer winds and the +ISM, although the temperatures and sound speed are likely to be high +(Margutti et al. 2014; Hsiao et al. 2020). Therefore, polarimetry will +provide a unique way to explore the bound CSM. As discussed above +(Sections 3.3 & 4.3), some of the wiggles in polarization spectra +and flocculent structures in polar plots seen in the spectropolarimetry +of SN 2019np may have their origin in Rayleigh-Taylor or Kelvin- +Helmholtz instabilities and crossing shock waves produced during +the injection with an orientation imprinted by the bound CSM. Po- +larization measurements with a latency between hours and one day +are needed (Figure 17) to learn whether the polarization position +angle persists, which would suggest a large-scale structure, domi- +nated by instabilities imprinting their characteristic size as wiggles +and large-amplitude fluctuations in the polarization spectra ans well +as flocculent structures in the polar diagrams, or a combination of +large and small scales. These structures become most prominent in +Ca ii NIR3, which is an excellent tracer of structure owing to its large +atomic cross section. +e) Alternatively, the early wiggles and the flocculent structures +in SN 2019np may be related to the explosion mechanism, in which +case they would have an origin internal to the SN proper. Poten- +tial sources are explosive surface He-burning in sub-𝑀Ch explosions +(e.g., Shen & Moore 2014) or in NSE-rich material rising from the +central region in gravitationally confined thermonuclear explosions +(Kasen & Plewa 2005). In both cases, some 0.02–0.1 M⊙ of 56Ni +may reach near-surface regions. This is well within the mass range +that can be probed by polarization as in SN 2019np (Figure 11) be- +cause the characteristics and especially the structures associated with +these processes are different. A central distinguishing criterion is the +presence of products from low-density burning in sub-𝑀Ch explo- +sions and of NSE-dominated material from high-density burning and +a mixture of NSE and QSE in gravitationally confined detonations. +Prompted by detections of early excess luminosity in optical light +curves, Piro & Kollmeier (2018) suggested 56Ni as a possible energy +source. At a first glance, early photometry of iPTF16abc (Miller et al. +2018) and SN 2019np (Sai et al. 2022) makes it plausible that both +light curves can be well explained by 56Ni mixed into the outer layers +of the ejecta. A more detailed study has been presented by Magee +& Maguire (2022). The authors proposed that the bump in the light +curve can be understood by a Ni shell of 0.02–0.03 M⊙ in the outer +0.2 M⊙ but also noted that the colours are too blue, and the spectra +would be dominated by Fe/Co/Ni. Since both of these side effects are +not supported by the observations, the authors suggested as a remedy +that the inferred 10% of 56Ni is concentrated in a small clump. +However, the observed flux and polarization spectra safely rule +out 56Ni-powered early-time light curves (Section 4.3). Moreover, +the proposed small clump is unlikely to be the correct explanation; +the luminosity would be dominated by a plume resulting in a flip +in polarization position angle (Figure 12) because even as little as +0.02 M⊙ of 56Ni would dominate the energy input. In both the sub- +𝑀Ch and the gravitationally confined detonation, the hypothetical +56Ni would be in the surface layer. It would contribute 50% of the +total heating and be equivalent to the energy output from 0.1 M⊙ of +56Co, and may not solve the blue-colour problem. The latter may +be an opacity effect in a rapidly expanding atmosphere instead of a +heat indicator (Ashall et al. 2022). The presence of burned material +from a He-triggered sub-𝑀Ch explosion can be clearly demonstrable +with spectropolarimetry because it can detect iron-group elements +down to solar metallicity (Section 4.3), which is not possible from +total-flux spectra alone. Furthermore, polarimetry is the ideal tool +to distinguish explosions without surface burning and He-triggers in +sub-𝑀Ch WDs. +f) Interaction of the SN ejecta with a companion star is a conse- +quence of most explosion models except for the violent, dynamical, +and secular mergers. The early bump identified in the photometry +of SN 2017cbv appears blue and has been modeled by interactions +with a subgiant star at a distance of 56 solar radii from the exploding +WD (Hosseinzadeh et al. 2017). However, the model also predicts a +stronger ultraviolet flux than was observed. All SN mechanisms with +an internally triggered explosion seem to have a problem explaining +the early blue excess flux and for the explanation to resort to inter- +action with some CSM. An interaction with a relatively high-mass +object can be expected for many progenitor systems, and it would +leave its imprint not only in the surface layer but all the way down +to the central region. So large a structure is likely to become visible +in polarization. But it may also produce small-scale structures in the +abundances and, depending on the donor star, the density (Marietta +et al. 2000). Interaction with a companion causes a tight connection +between outer and inner regions of the expanding envelope, including +a common and persistent symmetry axis and, initially, a cone with +an opening angle of ∼ 30◦. Although our modeling of SN 2019np +did not include a companion, the observed change in polarization +angle (Figures 1–4) probably disfavours a dominant effect of an +ejecta/companion interaction on the polarization (Figure 16). Any +such effect depends on the size of, and distance to, the companion +star, and the large-scale asymmetry imposed by a small companion +can be expected to be largest in deep layers (Marietta et al. 2000). +Furthermore, in contrast to an off-center DDT, a small-scale structure +produced by Rayleigh-Taylor instabilities will occupy a cone instead +of a spherical layer. Unfortunately, in our study of SN 2019np, we +lack the cadence to resolve small scales and their distribution from +the outer to the inner regions. +g) For deflagration fronts, we must expect small-scale Rayleigh- +Taylor instabilities and the imprint of the thermonuclear runaway, +the caustic distribution of the burning processes, and magnetic fields +as discussed throughout this paper (especially Sections 4.3 and 5) +and by Hristov et al. (2021). Polarimetry of Ca ii NIR3 is a highly +sensitive tool to probe for the associated structures. The polarization +of this triplet in the central region is strong evidence for mixing from +the outside because burning to hot NSE destroys Ca (Figure 11). +However, the huge atomic cross section of Ca ii NIR3 desensitises +it to abundance effects (Section 4.3). When observations of this +feature and the continuum polarization with similar characteristics +as ours of SN2019 np but with higher cadence are combined with +high-resolution nebular spectra in the near- and mid-infrared at later +epochs, a fairly complete three-dimensional model of the structure +of thermonuclear SNe can be assembled (Kotak et al. 2004; Telesco +et al. 2015; Hoeflich et al. 2021). +h) Off-center DDT explosions of 𝑀Ch WDs can generally be thor- +oughly investigated by spectropolarimetry. For sub-𝑀Ch explosions, +the location of the secondary ignition of the C/O core can only be ex- +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +19 +Figure 17. +Rate of recession of the photosphere in the spherical high- +resolution Model 25 as a function of time. The recession rate 𝑑𝑣phot/𝑑𝑡 for +Model 25 (see Section 4) was calculated using the Rosseland mean opacity +in the optical and a range of ±2 days to determine 𝑣phot. The exact values +depend on the explosion model used (see Figure 9 of Quimby et al. 2007). +pected to show an imprint on the polarization spectrum if it happens +in the regime of distributed burning. +i) For SN 2019np, we found evidence of an off-center component +in the 56Ni distribution, which disfavours sub-𝑀Ch explosions. How- +ever, we also showed that large-scale asymmetries in the extended +central Fe/Co/Ni region cannot be excluded (Section 4.2). Contrary +to normal-bright SNe and SN 2019np in particular, polarimetry of +underluminous Type Ia SNe provided strong evidence for signifi- +cant asphericity even in deeper layers. This may favour fast rotating +WDs or dynamical mergers (Section 5). A possible discriminator +may be the presence of Rayleigh-Taylor instabilities in models with +deflagration burning, and their absence in pure detonation models +like dynamical mergers (García-Berro & Lorén-Aguilar 2017). As +explained above, time-resolved polarimetry can establish the actual +facts. +j) The Introduction has discussed polarization of Type Ia SNe as +a diagnostic tool, also in connection with minority events like vio- +lent mergers (Pakmor 2017; Kushnir et al. 2013). Such events should +exhibit prominent polarization signatures because the off-center com- +ponent would dominate (Figure 12). Therefore, the amplitude in the +continuum polarization with time would be larger by a factor of 5–6, +and large polarization can also be expected in spectral lines. +6.2 Implications for the Design of Spectropolarimetric +Observing Sequences and Simulations +Spectropolarimetry with adequate cadence from the earliest possible +moment and (to potentially expose the EC layers) up to ∼ 3 weeks +after maximum brightness is necessary to get a comprehensive pic- +ture of normal Type Ia SNe on all scales and to discriminate between +competing interpretations. The SNR must be high, and the spectral +resolution should be better than that of our SN 2019np data. In our +observations of SN 2019np, the combination of a spectral resolving +power of 𝑅 ≈ 440 and a ∼ 4–6 day cadence only achieved a reso- +lution of ∼ 6000 km s−1 in comoving-frame velocity between mass +elements. This limited our analysis of any smaller-scale structure in +the SN ejecta and the surface layer. Observations with low cadence +disqualify for several quantitative comparisons with models, most +importantly the identification of individual clumps (predicted by +Rayleigh-Taylor instabilities), layers of specific chemical elements, +the interface between the Ca-rich and the inner NSE-region (Fig- +ure 11), and the origin of the asphericity in the outermost layers. +To devise a more optimised observing strategy, the evolution of the +recession velocity of the photosphere needs to be considered, which +Figure 17 illustrates for Model 25. Starting ∼ 1 week after explo- +sion, during the rising and photospheric phases of Type Ia SNe, this +velocity declines from ∼ 1000 to ∼ 200 km s−1 d−1. Accordingly, a +2–4 day observing cadence suffices to locate the interfaces between +various chemical constituents relevant for the internal structure of the +explosion (items (e)–(j) above and Fig. 11). With one-day cadence +and a matching spectral resolving power of 1000 (corresponding to +a transversal resolution of 300 km s−1), the morphology and origin +of the individual structures discussed in items (e)–(j) can also be +explored, namely the properties of the burning front including sec- +ondary detonations or a DDT, shear instabilities due to interactions +with a companion star, the imprint of the thermonuclear runaway, the +consequence of mixing of radioactive and EC elements, and possibly +magnetic fields. +To also determine the shape of the outermost layers of the ex- +ploding WD calls for rapid-response and high-cadence observations +because the radial density index governing this shape changes early +and rapidly. Some 3.5 days after the explosion, a one-day cadence +corresponds to a radial resolution of ∼ 3500 km s−1 and 10−4 M⊙ +in mass (Figure 11). Therefore, such observations resolve both the +bound CSM (item (d) above) and the He layer and 56Ni mass (item +(e)) to be expected for sub-𝑀Ch or confined detonation models. At an +expansion velocity of 24,000 km s−1 (a typical value of high-velocity +components in Ca ii NIR3 and the cutoff velocity of the blue wings +in Si ii 𝜆6355; Quimby et al. 2007; Gerardy et al. 2007), the transver- +sal resolution (which is set by the spectral resolution of 300 km s−1) +exceeds the radial resolution by a factor ∼ 10. These resolutions de- +scribe cones with approximate opening angles of 20◦ and 2◦ for the +radial and transversal directions (respectively), and are respectively +governed by the cadence and the spectral resolution.15 The apices +of the cones are at the center of the WD and the numerical grid. +The opening angles are comparable to the angle suspended by (for +example) the region affected by an interaction with a companion star, +or about 20% and 2% of the photospheric radius. +With such data, the interaction of the ejecta with an accretion +disc, a Roche lobe (item (d) above) and a progenitor star (item (f)) +can be significantly distinguished. They may also be sufficient to +characterise the structures and underlying physical mechanism(s) +which produce the large-amplitude fluctuations in the polarization +profile and flocculent features in polar diagrams of strong spectral +lines. +The numerical models applied in this study have an effective spa- +tial resolution of ∼ 600 km s−1 for clumps.16 This is just sufficient +to resolve the largest Rayleigh-Taylor instabilities, for example, but +already appears to be higher than all SN spectropolarimetry obtained +15 The opening angles are given by the arcsin of the structure size divided +by the photospheric radius in velocity space. +16 The effective resolution is estimated from the grid resolution, i.e., the +domain size (2 × 25, 000 km s−1) divided by the number of grid points (330) +in a differential scheme (contributing a factor of ∼ 2). For spherical clumps +as the simplest structures, it is multiplied by another factor 2. This results in +75 km s−1 × 23. +MNRAS 000, 1–?? (2022) + +Model 25 +4 +3 +2 +d +0 +0 +10 +20 +30 +Days Since Explosion [days]20 +P. Hoeflich et al. +to date. Higher-resolution models to resolve small-scale structures +can be achieved with sub-star instead of full-star simulations or, bet- +ter, a space-domain implementation17 for our Variable Eddington +Tensor solver in HYDRA as discussed by Hristov et al. (2021, and +references therein). Investigation of the effects caused by details of +the thermonuclear runaway in a 𝑀Ch explosion or any secondary +C/O ignition would require spectropolarimetry at late times in com- +bination with late-time near- and mid-infrared flux spectra (Hoeflich +et al. 2021). Spectropolarimetry will provide a tomographic sam- +pling of the geometric properties of the layer near the Si/Fe interface +in the inner regions of the WD remnant, while nebular flux profiles +at sufficient spectral resolution will probe the physical conditions +and kinematics of this region through unblended line profiles, which +are also sensitive to the aspect angle of the observer (Hoeflich et al. +2021). The combination of these two datasets will subject models +to critical consistency checks because they investigate the same re- +gion from different perspectives. Analogous simulations should be +employed to test other models and make use of hydro-dynamical +simulations to investigate points (e)–(j) in the list above. A detailed +discussion of many other scenarios is beyond the scope of this paper. +Obviously, similar simulations can be applied to a wide variety of +explosion scenarios and other transients (e.g., Leonard et al. 2012; +Maund et al. 2019; Dessart et al. 2021; Buckley et al. 2021), and +those will be performed in the future. +By using details previously not considered in combination with +extensive modelling, we obtained new insights into the formation of +𝑝 and obtained many results for SN 2019np in Section 5. However, we +pushed the analysis to the limit of current data and modelling efforts. +This study should be regarded as a pathfinder for a new approach to +the analysis of SNe Ia data, to evaluate the limits and to identify the +potential and shortcomings of current observations and theoretical +models, to evaluate methods to correct for the ISM, and to develop +future polarization programs (Section 6). +Acknowledgements: We are grateful to the European Organisation for +Astronomical Research in the Southern Hemisphere (ESO) for the +generous allocation of observing time. We especially thank the staff +at Paranal for their proficient and highly motivated support of this +project in service mode. P.H. acknowledges support by the National +Science Foundation (NSF) through grant AST-1715133. A.V.F.’s +supernova group at U.C. Berkeley is grateful for financial assistance +from the Christopher R. Redlich Fund and many individual donors, +including Gary and Cynthia Bengier, Clark and Sharon Winslow, +Sanford Robertson, Sunil Nagaraj, Landon Noll, and Sandy Ottelini. +The research of Y.Y. has also been supported through a Benoziyo +Prize Postdoctoral Fellowship. +Facilities: The observations were obtained with FORS2 and the Very +Large Telescope at the European Southern Observatory’s La Silla +Paranal Observatory in Chile. The simulations have been performed +on the computer cluster of the astro-group at Florida State University. +Software: IRAF is distributed by the National Optical Astronomy +Observatories, which are operated by the Association of Universities +for Research in Astronomy, Inc., under cooperative agreement with +the National Science Foundation. 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The ear- +liest photometric dataset of SN 2019np consists of the SDSS 𝑔- and +𝑟-band light curves generated by the Zwicky Transient Facility (ZTF) +alert packets (Patterson et al. 2019). Owing to the lack of 𝑔-band pho- +tometry at the earliest phases of SN 2019np, we fitted the power law +to the 𝑟-band flux, for which we considered two subsets, namely ob- +servations before days ∼ −14 and ∼ −12, respectively. The fitting is +also constrained by the last nondetection on day −18.2. As shown +by the filled black and open orange circles in the upper-left panel +of Figure A1, after arbitrarily scaling the 𝑔 and 𝑟 light curves of +SN 2019np, the evolution of the flux in both bandpasses is consistent +between days −15.7 and −12.7, when photometry is available for +both filters. The best fit to the observations before day −14 gives +𝑚 = 1.20 ± 0.04 and a rise time 𝑡0 = −17.92 ± 0.06 days. The sta- +tistical error of 0.06 day in 𝑡0 is much smaller than the systematic +uncertainty of the time of the 𝐵-band light-curve maximum, which +amounts to ±0.51 day in our analysis and has to be added. +Figure A1 also compares the flux evolution of SN 2019np to its +fit with Equation A1 and includes data for selected other SNe with +well-sampled photometry at similarly early phases. The 𝑔-band light +curve of the normal Type Ia SN 2011fe is well approximated by an +expanding fireball model (i.e., 𝑚 ≈ 2; Nugent et al. 2011). Two cases +with an early flux excess, namely SN 2017cbv (Hosseinzadeh et al. +2017) and iPTF16abc (Miller et al. 2018), exhibit a fast rise within +the first ∼ 5 days after the explosion and favour a power-law index +around unity. At 𝑚 = 1.20 ± 0.04, SN 2019np is similar to both of +them. +MNRAS 000, 1–?? (2022) + +Geometry of Type Ia SN 2019np +23 +Figure A1. Best-fit 𝑓 ∝ (𝑡 − 𝑡0)𝑚 model to describe the early flux evolution of SN 2019np compared to that of selected other SNe with well-sampled early +photometry. All flux distributions are normalised to the peak magnitude measured for each SN in the given bandpasses. In the upper-left panel, the black and the +brown solid lines fit the 𝑟-band flux (filled-black circles) of SN 2019np before −14 and −12 days relative to the 𝐵-band maximum on MJD 58509.7, respectively. +In the inset, the inner to outer contours represent the 1𝜎, 2𝜎, and 3𝜎 confidence levels of the power-law parameters. The open orange circles mark the 𝑔-band +photometry of SN 2019np. The residuals of the fits to the 𝑟-band light curves before −14 and −12 days are shown by the black and brown dots, respectively, in +the bottom-left panel. The upper-right panel compares the fit of SN 2019np to the 𝑔 light curves of SNe 2017cbv, 2011fe, and iPTF16abc within the first four +days after explosion. SN 2019np exhibits a similar power-law index as SN 2017cbv and iPTF16abc, for which a blue excess has been identified within the first +∼ 5 days. The residuals are shown in the bottom-right panel. +MNRAS 000, 1–?? (2022) + diff --git a/RdE3T4oBgHgl3EQfygvM/content/tmp_files/load_file.txt b/RdE3T4oBgHgl3EQfygvM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d4256e36008c5689fdb4767c2379bcd71c1a63d3 --- /dev/null +++ b/RdE3T4oBgHgl3EQfygvM/content/tmp_files/load_file.txt @@ -0,0 +1,2323 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf,len=2322 +page_content='MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Preprint 13 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 The Core Normal Type Ia Supernova 2019np – An Overall Spherical Explosion with an Aspherical Surface Layer and an Aspherical 56Ni Core ★ Peter Hoeflich1,†, Yi Yang (杨轶) 2,3,§,‡, Dietrich Baade4,§, Aleksandar Cikota5,6, Justyn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Maund7, Divya Mishra8, Ferdinando Patat4, Kishore C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patra2,+, Lifan Wang8, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Craig Wheeler9, Alexei V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Filippenko2, Avishay Gal-Yam10, Steve Schulze11 1Department of Physics, Florida State University, Tallahassee, Florida 32306-4350, USA 2Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA 3Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel 4European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2, 85748 Garching b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' München, Germany 5European Organisation for Astronomical Research in the Southern Hemisphere (ESO), Alonso de Cordova 3107, Vitacura, Casilla 19001, Santiago de Chile, Chile 6 Gemini Observatory/NSF’s NOIRLab, Casilla 603, La Serena, Chile 7Department of Physics and Astronomy, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK 8George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics & Astronomy, Texas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='&M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' University, 4242 TAMU, College Station, TX 77843, USA 9Department of Astronomy, University of Texas, Austin, TX 78712, USA 10Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel 11The Oskar Klein Centre, Department of Astronomy and Department of Physics, Stockholm University, AlbaNova, SE-106 91 Stockholm, Sweden §Bengier-Winslow-Robertson Fellow +Nagaraj-Noll-Otellini Graduate Fellow Accepted 1/12/23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Received 12/9/22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' in original from 11/09/22 ABSTRACT Optical spectropolarimetry of the normal thermonuclear supernova(SN) 2019np from −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days relative to 𝐵-band maximum detected an intrinsic continuum polarization (𝑝cont) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09% at the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Between days −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, 𝑝cont remained ∼0 and by day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 was again significant at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Not considering the first epoch, the dominant axis of Si ii𝜆6355 was roughly constant staying close the continuum until both rotated in opposite directions on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Detailed radiation-hydrodynamical simulations produce a very steep density slope in the outermost ejecta so that the low first-epoch 𝑝cont ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2% nevertheless suggests a separate structure with an axis ratio ∼2 in the outer carbon-rich (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5–4)×10−3M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Large- amplitude fluctuations in the polarization profiles and a flocculent appearance of the polar diagram for the Ca ii near-infrared triplet (NIR3) may be related by a common origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The temporal evolution of the polarization spectra agrees with an off-center delayed detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The late-time increase in polarization and the possible change in position angle are also consistent with an aspherical 56Ni core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The 𝑝cont and the absorptions due to Si ii 𝜆6355 and Ca ii NIR3 form in the same region of the extended photosphere, with an interplay between line occultation and thermalisation producing 𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Small-scale polarization features may be due to small-scale structures, but many could be related to atomic patterns of the quasi-continuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' they hardly have an equivalent in the total-flux spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We compare SN 2019np to other SNe and develop future objectives and strategies for SN Ia spectropolarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Key words: supernovae: individual (SN 2019np) – polarization 1 INTRODUCTION Various models of Type Ia supernova (SN) explosions predict pho- tometric and spectroscopic evolution that reproduce observations adequately but not uniquely (Alsabti & Murdin 2017), so it is diffi- ★ Based on observations collected at the European Southern Observatory under ESO program 0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='D-0528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' † E-mail: phoeflich@fsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='edu ‡ E-mail: yiyangtamu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='com § E-mail: dbaade@eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='org cult to judge models merely by their power in matching light curves and total-flux spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, they predict different explosion ge- ometries of the progenitor white dwarf (WD), which can be diag- nosed with polarimetry (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polarized optical flux from supernovae (SNe) can be caused by departures from spherical symmetry of the global ejecta structure or by chemical “clumps” with different line opacities that block portions of the photosphere (Wang & Wheeler 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patat 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Both schemes can be under- stood as an incomplete cancellation of the electric vectors integrated over the photosphere as seen by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Optical polarime- © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04721v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='SR] 11 Jan 2023 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' try probes the geometric properties of the SN explosion and the structure of the SN ejecta, without spatially resolving the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A wavelength-independent continuum polarization would arise from Thomson scattering of free electrons with a globally aspherical dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In addition or alternatively, it may be caused by energy input that is spatially offset from the center of mass (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Livne 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Kasen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polar- ized spectral features can be induced in the SN ejecta by chemically uneven blocking within the photosphere and by frequency variations of the associated line opacities in the thermalisation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Any early polarization signal from thermonuclear explosions of- fers a critical test of the nature of the progenitor systems of Type Ia SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For example, large deviations from global sphericity in the density distribution and chemical abundances of the ejecta are pre- dicted for explosions triggered by the dynamical merger of a double white dwarf (WD) binary (Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The resulting polarization is expected to be significant both in the continuum and across various spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The continuum po- larization can be as high as ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5–1% at ∼ 1 week after the ex- plosion if observed out of the orbital plane (Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By contrast, an almost spherical density distribution and a mod- erate degree of chemical inhomogeneity are predicted by delayed- detonation models (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Pakmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Moll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Raskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A continuum polarization near zero as well as modest (≲ 1%) signals across major spectral features were also predicted by specific multidimensional models for both a selected delayed-detonation and a sub-Chandrasekhar-mass (MCh) model (Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polarized spectral lines indicate geometric deviations from spheri- cal symmetry of the associated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Chemical inhomogeneities are imprinted by the propagation of the burning front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Delayed- detonation models predict an initial subsonic deflagration result- ing in turbulence and gravitational compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As the burning front travels outward, the flame transforms into a supersonic det- onation because of Rayleigh-Taylor instability at the interface be- tween unburned and burned material (Khokhlov 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Layers of intermediate-mass elements (IMEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', from Si to Ca) are then pro- duced at the front of the detonation wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At any given epoch, the po- larization spectrum samples the geometric information of the ejecta that intersect the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As the ejecta expand over time, the electron density decreases and the photosphere recedes into deeper layers of the ejecta in mass and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Multi-epoch spectropo- larimetry tomographically maps out the distribution of various ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' More recent early-time observations have also found low contin- uum polarization in other normal Type Ia SNe namely SN 2018gv (day −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020) and SN 2019ein (day −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' SN 2019ein displayed one of the highest expansion velocities at early phases as inferred from the absorption minimum of the Si ii 𝜆6355 line (∼ 24, 000 km s−1 at 14 days before photometric 𝐵-band maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Pellegrino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The low continuum polarization on day −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='9 indicates a low degree of asphericity at this phase, strengthening the existing evidence that the explosions of Type Ia SNe maintain a high degree of sphericity from their early phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The spectropolarimetry of SN 2018gv on day −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6 was the earliest such measurement at its time for any Type Ia SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='13% continuum polarization five days after the explosion (based on phase estimates from the early light curve) suggests that the photosphere was moderately aspherical with an axis ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, 1 Throughout the paper, the term equatorial plane is defined by planes through even at this early phase, the geometry of the outermost ∼ 10−3 to ∼ 10−2 MWD of SN 2018gv still remained observationally uncon- strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The polarization is also sensitive to the rapidly-changing density structure in the outer layers, which intersect the photosphere in the first few days (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' SN 2019np was discovered at 2019-01-09 15:58 (UT dates are used throughout this paper) with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 m telescope at a clear-band magnitude of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='8 (Itagaki 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Rapid spectroscopic follow-up ob- servations were carried out as early as ∼ 1 day after the discovery (Kilpatrick & Foley 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Spec- tral cross-correlations with the “Supernova Identification” (SNID;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Blondin & Tonry 2007) and the “Superfit” (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2005) codes suggest that SN 2019np is a Type Ia SN discovered ∼ 2 weeks before maximum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From the photometry by Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022), we derived that SN 2019np reached its peak 𝐵-band magnitude at MJD 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='51 (see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A), where the two uncertainties represent the statistical and the systematic error, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This estimate is consistent with the respective values of 58510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='8 and 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06 reported by Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) and Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' All phases used throughout the present paper are given relative to the 𝐵-band maximum light at MJD 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='72 (2019-01-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A comprehensive study of the SN by Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) concluded that its photometric and spectroscopic properties were similar to those of other normal Type Ia SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) detected a ≲5% excess in the early bolometric flux evolution of SN 2019np compared to radiative diffusion mod- els (Arnett 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Chatzopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012), hinting at additional energy input compared to the radioactive decay of a Ni core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' They suggested that the blue and relatively fast-rising early light curves of SN 2019np are best fitted with the mixing of 56Ni from the inner to the outer layers of the SN ejecta (Piro & Morozova 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The rise time of SN 2019np is not compatible with models that predict an early interaction between the SN ejecta and any ambient circum- stellar matter (CSM) or a companion star (Kasen 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Moreover, the colour evolution of SN 2019np is inconsistent with that predicted for a progenitor WD below 𝑀Ch and surrounded by a thin helium shell as discussed in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 & 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In this “double-detonation” or “He-shell detonation” picture, an initial detonation is triggered in the surface He shell, sending a shock wave to the inner region of the C/O WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The shock generates compression heat and subsequently triggers the second detonation that ignites the WD (Woosley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Nomoto 1982a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Livne 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Woosley & Weaver 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoe- flich & Khokhlov 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Kromer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) also suggested an excess in the early flux evolution of SN 2019np, which may have been too weak to have been caused by an interaction between the ejecta and a companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Interaction with any CSM is an additional possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This study presents five epochs of optical spectropolarimetry of SN 2019np from 𝑡 ≈ −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days and interpretations based on detailed radiation-hydrodynamic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The paper is or- ganised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In Section 2 we outline the spectropolarimet- ric observations and the data-reduction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The polarization properties of SN 2019np are discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The analysis of these properties with hydrodynamic models is carried out in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We summarise our conclusions in Section 5, and develop a the center, �𝑛�𝑥 = 0, �𝑥 spanning the plane with the symmetry axis of a rotationally symmetric ellipsoid as the orthogonal vector �𝑛, or �𝑛 being a line through the center of the WD and the location of an off-center energy source (Höflich 1995c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The two so defined planes may be different (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 3 comprehensive appraisal of the potential of spectropolarimetry for the understanding of Type Ia SNe in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2 SPECTROPOLARIMETRY OF SN 2019np Spectropolarimetry of SN 2019np was conducted with the FO- cal Reducer and low dispersion Spectrograph 2 (FORS2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Appen- zeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1998) on Unit Telescope 1 (UT1, Antu) of the ESO Very Large Telescope (VLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The Polarimetric Multi-Object Spec- troscopy (PMOS) mode was used for all science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A complete set of spectropolarimetry consists of four exposures at retarder-plate angles of 0, 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, 45, and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The 300V grism and a 1′′-wide slit were selected for all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The order-sorting filter GG435 was in place, which has a cut-on at ∼4350 Å to prevent shorter-wavelength second-order contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This configuration provides a spectral resolving power of 𝑅 ≈ 440 at a central wavelength of 5849 Å, corresponding to a resolution-element size of ∼ 13 Å (or ∼ 670 km s−1) according to the VLT FORS2 user manual (Anderson 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Observations were obtained at five epochs: (in the format day/UT) −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5/2019-01-12, −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4/2019-01-15, −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4/2019-01-20, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5/2019-01-27, and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5/2019-02-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At the first epoch, the total 4 × 1100 s integration time was split into two sets of exposures to reduce the impact of cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The two loops were carried out at relatively large and different airmasses, from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='84 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='73 and from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='73 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We conducted a consistency check of the two measure- ment sets and found that the Stokes parameters derived for the two loops agree within their 1𝜎 uncertainties over the entire wavelength range after rebinning the data to larger resolution elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 30 Å and 40 Å bin sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We thus combined the two datasets by taking the mean value of the spectra obtained at each retarder-plate angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Relative-flux calibration was based on the flux standard star HD 93621 observed at a half-wave plate angle 0 degrees near epoch 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The airmass of the flux standard was chosen to be comparable to that of the spectropolarimetry of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A log of the VLT spectropolarimety is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' After bias and flat-field corrections, the ordinary (o) and ex- traordinary (e) beams in each two-dimensional spectral image were extracted following standard routines within IRAF2 (Tody 1986, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A typical root-mean-square (RMS) accuracy of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='20 Å was achieved in the wavelength calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Stokes parameters were then derived using our own routines based on the prescriptions in Patat & Romaniello (2006) and Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2007), which also cor- rect the bias due to the non-negativity of the polarization degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The observed polarization degree and position angle (𝑝obs, PAobs) and the true values after bias correction (𝑝, PA) can be written as 𝑝obs = √︃ 𝑄2 + 𝑈2, 𝑝 = � 𝑝obs − 𝜎2𝑝 𝑝obs � × ℎ(𝑝obs − 𝜎𝑝);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 𝑃𝐴obs = 1 2arctan �𝑈 𝑄 � , and 𝑃𝐴 = 𝑃𝐴obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (1) Here, 𝑄 and 𝑈 are the intensity (𝐼)-normalised Stokes parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The correction for the polarization bias is based on equations in Simmons & Stewart (1985) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (1997), where 𝜎𝑝 and ℎ 2 IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astron- omy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', under cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Spectropolarimetry of SN 2019np on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (epoch 1) relative to 𝐵-band maximum light on MJD 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The five panels (from top to bottom) display (a) the arbitrarily scaled flux spectrum with major spectral features identified and the high-velocity component of Ca ii NIR3 labeled “hv”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (b,c) the normalised Stokes parameters 𝑄 and 𝑈, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (d) the polarization spectrum (𝑝);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' and (e) the polarization position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Panels (b)– (e) represent the polarimetry before ISP correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The grey lines in panels (b) and (c) show the data with their original sampling while the heavy lines in panels (b)–(e) use 30 Å bins for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The grey-shaded vertical bands identify regions of telluric contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' denote the 1𝜎 uncertainty in 𝑝obs and the Heaviside step function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The brackets are properly set as in Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1% instrumental polarization was also corrected following the procedure discussed by Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' More details of the reduction of FORS2 spectropolarimetry can be found in the FORS2 Spectropolarimetry Cookbook and Reflex Tutorial3, as well as in Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2017) and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3 POLARIMETRIC PROPERTIES OF SN 2019np The spectropolarimetry of SN 2019np obtained on days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 is presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1–5, respectively, where the data are not corrected for interstellar polarization (ISP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polarization spectra are shown together with the associated scaled total-flux spectra (hereafter referred to as simply “flux spectra”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Both have been transformed to the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3 ftp://ftp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='org/pub/dfs/pipelines/instruments/fors/ fors-pmos-reflex-tutorial-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='pdf MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Same as Figure 1, but for day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 (epoch 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Same as Figure 1, but for day −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 (epoch 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Same as Figure 1, but for day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (epoch 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The estimated ISP level is shown by grey-dashed lines in panels (b)–(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Same as Figure 1, but for day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (epoch 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 5 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Intrinsic polarization of SN 2019np from days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 as labeled from top to bottom in the subpanels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For each epoch, the degree of polarization is calculated based on ISP-subtracted Stokes 𝑄 and 𝑈, bias- corrected following Equation 1, and presented (red histograms) with 30 Å binning, together with the arbitrarily scaled flux spectrum (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Si ii 𝜆6355 and the photospheric component of the Ca ii NIR3 features are marked, and their velocities (𝑣) are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The high-velocity component of the Ca ii NIR3 feature is labeled “hv.” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 Interstellar Polarization Removing the polarization imposed by interstellar dust grains in ei- ther the Milky Way or the host galaxy or both is essential for revealing the intrinsic polarization of SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This ISP is due to dichroic extinc- tion by nonspherical dust grains aligned by the interstellar magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, the entire observed wavelength range of the spec- trum is used to determine the overall level of the ISM polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As will be shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3, both the overall level and the continuum polarization in a narrow wavelength range plus the spectral features are consistent in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This provides an argument that the procedure to find the ISM polarization does not suppress an overall net-polarization mimicking overall sphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The intrinsic contin- uum polarization of Type Ia SNe around their peak luminosity is very low (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Wang & Wheeler 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patat 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, we used the spectrum of SN 2019np from day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 as an unpolarized standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We fitted the Stokes 𝑄, 𝑈 param- eters and the observed degree of polarization, 𝑝, using Serkowski’s wavelength-dependent law (Serkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1975) as well as a mere constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the given low-ISP regime, we found that Serkowski’s law failed to yield a satisfactory fit, and the ISP can be characterised by the latter approach, which requires computing the error-weighted mean values of 𝑄 and 𝑈 over suitably selected spectral regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Using the wavelength range 4400–8900 Å but excluding the telluric features and the strongly polarized Si ii 𝜆6355 line and the Ca ii near- infrared (NIR) triplet (8500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='36 Å, 8544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='44 Å, and 8664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='52 Å, with a central wavelength of 𝜆0 ≈ 8570, denoted as Ca ii NIR3 hereafter) due to the SN, we estimate the ISP as (𝑄ISP, 𝑈ISP) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='019±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='121%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='322±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='072%), and 𝑝ISP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='343 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='075%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These values are well consistent with the ISP derived over the wavelength ranges which are considered to be depolarized due to blanketing by numerous iron ab- sorption lines (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Adopting the Galactic and the host-galaxy reddening of SN 2019np of 𝐸(𝐵−𝑉)Gal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='018 mag and 𝐸(𝐵−𝑉)host = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='03 mag (Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022), we find the estimated ISP consistent with the empirical upper limit caused by dichroic extinction and established for dust in the Galaxy, 𝑝ISP < 9%× 𝐸(𝐵`𝑉), following Serkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 Intrinsic Continuum Polarization After subtracting the ISP, we determined the continuum polariza- tion of SN 2019np at all epochs from the Stokes parameters over the wavelength range 6400–7000 Å, which is considered to be free of significant polarized spectral features (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The error- weighted mean Stokes parameters within this region are given in Ta- ble 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The uncertainty was estimated by adding the statistical errors and the standard deviation computed from the 30 Å-binned spectra within the chosen wavelength range in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The continuum polarization within this wavelength interval is consistent with that computed over the entire observed wavelength range after exclusion of the broad, polarized Si ii 𝜆6355 and Ca ii NIR3 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The intrinsic continuum polarization of SN 2019np on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' After only three days, it had dropped to ∼ 0 by day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and remained low until the SN reached its peak lumi- nosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the continuum polarization had increased to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 From the power-law fit of the earliest light curves of SN 2019np, we place the time of first light at 𝑡0 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06 day, where 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06 day is only the statistical error (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An additional systematic error of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='51 day results from the determina- tion of the time of the peak luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The times of the five epochs of VLT spectropolarimetry relative to this time of the SN explosion are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 days, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The first epoch is the earliest such measurement for any Type Ia SN to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 The Dominant Axes in the Q–U Plane For each epoch of our ISP-corrected spectropolarimetry, we ex- amine in the Stokes 𝑄–𝑈 plane the axial symmetry of the ejecta of SN 2019np as they enter the extended photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We do this sep- arately for suitable wavelength ranges covering the continuum and the Si ii 𝜆6355 and Ca ii NIR3 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This method was introduced by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' a different graphical rendering of the same data will be discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Purely axially symmetric ejecta imprint a linear structure on the 𝑄–𝑈 plane, since the orientation of the structure is defined by a single polarization position angle, while varying scattering and polarization efficiencies lead to deviations from a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By projecting the data onto the best-fitting axis and measuring the scatter about this so-called dominant axis, 𝑈 = 𝛼 + 𝛽𝑄 , (2) one may characterise the degree of axial symmetry of the SN ejecta (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4 Note that the time variation in 𝑝cont is at a 2 𝜎 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the signif- icance of the variation is strongly supported by the change in the dominant axes in the 𝑄–𝑈 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Moreover, the estimate of the uncertainty in 𝑝cont includes real spectral variations in 𝑝 caused by spectral features (see Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 7 displays the ISP-corrected Stokes parameters on the 𝑄–𝑈 plane between days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The dominant axis of SN 2019np as determined from its polarization projected on the 𝑄– 𝑈 plane was derived by performing an error-weighted linear least- squares fit to the entire observed wavelength range (4350 ≤ 𝜆 ≤ 9100 Å) with the prominent and polarized Si ii 𝜆6355 and Ca ii NIR3 features excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Data points covering the Si ii𝜆6355 and Ca ii NIR3 profiles were omitted in the top row, where the dominant axis appears as the black long-dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To examine the difference between the fits including and excluding the Si ii 𝜆6355 and Ca ii NIR3 lines, we list the dominant axis and the corresponding position angles for both cases in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The dominant axes of SN 2019np fitted for both cases are consistent with each other within their 1𝜎 uncertainties except for epochs 3 and 4, when SN 2019np reached its peak luminosity and the discrepancy between the two fits amounts to ∼ 2𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We consider the fits with both broad and polarized individual features excluded a more reasonable characterisation of the orientation of the SN ejecta since these Si and Ca features generally exhibit significant deviations from the rest of the wavelength range (Leonard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the middle and bottom rows of Figure 7, the directions of the symmetry axes of the Si ii 𝜆6355 and Ca ii NIR3 features are shown by the green and purple dot-dashed lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The fitting procedures were the same as for the continuum but over the velocity ranges from 24,000 to 0 km s−1 for Si ii 𝜆6355 and from 28,000 to 0 km s−1 for the Ca ii NIR3 complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The derived parameters are also listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the spectropolarimetry over the op- tical range is poorly represented by a dominant axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The Ca ii NIR3 feature is barely described by the linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Additionally, as shown by the 𝑄–𝑈 diagrams for day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, data points across Si ii 𝜆6355 deviate from the clustering in the continuum, indicating a conspicu- ous polarization across the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, owing to the relatively low signal-to-noise ratio (SNR) and the moderate level of polarization, it is hard to quantitatively determine whether Si ii 𝜆6355 and the ejecta of SN 2019np determined from the optical continuum (as far as recorded by FORS2) follow different geometric configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Starting from day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, the ejecta of SN 2019np have developed a more discernible symmetry axis compared to day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This is indicated by the significantly reduced uncertainties in the linear least- squares fits to the polarimetry on the 𝑄–𝑈 plane (see the 𝛼∗, 𝛽∗, and 𝜃∗ 𝑑 values in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The dominant axis of SN 2019np shows little temporal evolution between days −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and rotates by ∼15◦ from days +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polar diagrams for Ca ii NIR3 appeared very flocculent, and somewhat misaligned with the dominant axes of Si ii 𝜆6355 and 𝑝cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Qualitatively, the temporal evolution of Si ii 𝜆6355 and Ca ii NIR3 features are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This can be seen from the middle and bottom rows of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Not considering the first epoch, the dominant axis of Si ii 𝜆6355 was roughly constant and stayed close to that of the continuum until both rotated in opposite directions on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Not considering day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, we suggest that SN 2019np belongs to the spectropolarimetric type D1 (Wang & Wheeler 2008), in which a dominant axis can be determined while the scatter of the data points about the dominant axis is conspicuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At the earliest epoch, a dominant axis cannot be clearly identified, and the continuum polarization measurements cluster around a location offset from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Apart from Si ii 𝜆6355 and Ca ii NIR3, there are numerous minor peaks scattered all over the polarization spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Nominally, these fea- tures are significant at ∼ 2𝜎 and occasionally at 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Careful quality control of the data and our reduction procedures have not identified them as artifacts, although some of them will undoubtedly be spu- rious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Most of them are volatile and, in consecutive observations, do not appear at the same location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This can be expected because the spectral features form in layers with different abundances (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In our analysis in Section 4, we will refer to them as “wiggles”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 Line Polarization in Polar Coordinates To further visualise the geometric distribution of the Si ii and Ca ii opacities in the ejecta of SN 2019np, we cast the line polarization into the format of polar plots where the radial axis indicates the velocity across the spectral profile and the angle from the reference direction represents the polarization position angles on the plane of the sky at the corresponding wavelength (introduced by Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2009), and see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Reilly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Stevance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 8 presents the polar plots for the Si ii 𝜆6355 and Ca ii NIR3 lines from days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, relatively highly polarized Si ii is present mostly above the photospheric velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The orientation of the Si-rich ma- terial appears to be different from the direction of the dominant axis as determined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 and indicated by the grey sector in the left panel of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that the angular size of the fan-shaped sector represents the 1𝜎 uncertainty of the position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Unlike the Si-rich material that is confined in a relatively narrow range in position angle, the Ca-rich component exhibits a more diverse radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The Ca-rich material below the high-velocity (hv) compo- nent at ∼ 20, 000 km s−1 shows a range in position angle that is consistent with (i) the dominant axes plotted as black dashed lines in the left panels of Figure 7, and (ii) the grey fan-shaped sector in the left panel of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the component above the high- velocity threshold exhibits a range in position angle that is distinct from the dominant axis but has a similar orientation as the Si-rich material above the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, the high-velocity Si-rich and Ca-rich components seen on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 are likely to share a similar geometric distribution that differs from that of the optical continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, the dominant axis has rotated relative to day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, as indicated by the position angle of the grey fan-shaped sector in the second polar plot of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Additionally, based on its reduced angular extent, we deduce that the symmetry axis of the SN ejecta becomes more prominent and well-defined as the photosphere pro- gressively recedes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Most of the Si- and the Ca-rich material gets almost aligned with the optical dominant axis, with larger offsets seen in the radial profile of the Ca-rich component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This alignment suggests that a similar axial symmetry is shared by the total ejecta and the line opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An overall similar geometry of SN 2019np can be derived from the polar plots for days −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (third and fourth panels in Figure 8), which indicate no significant evo- lution since day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the orientation of the dominant axis persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The widths in velocity of the fan-shaped sectors display an overall decreasing trend for both the Si-rich and the Ca-rich components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Since the line velocities decrease and the high-velocity components diminish with time, the polarization sig- nal measured at the high-velocity end decreases and becomes less significant as indicated by the large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the dominant axis has rotated compared to that measured during the rising phase of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The scatter has increased again in radial profiles of the Ca-rich material, suggesting a more complex structure of the line-forming regions in the more inner layers of the SN ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The high-velocity component has become indiscernible in the flux spectrum (Figures 5 and 6, and Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 7 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Intrinsic polarization of SN 2019np displayed on the Stokes 𝑄–𝑈 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The top row shows the data over the wavelength range 4250–9100 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The wavelength of each 30 Å bin is indicated by the colour bar on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The middle and the bottom rows display the polarization for the Si ii 𝜆6355 and Ca ii NIR3 features over the velocity ranges of 24,000–4000 km s−1 and 28,000–2000 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The velocities are also indicated by the corresponding colour bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The epochs of the observations are labeled with their phases at the top of each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In each panel, the black long-dashed line shows the dominant axis calculated over the wavelength range 4250–9100 Å with the Si ii 𝜆6355 and Ca ii NIR3 features excluded (the values of the fitting parameters 𝛼 and 𝛽 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2 are listed in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the middle and the bottom rows, the green and purple dot-dashed lines in each subpanel represent linear fits to the displayed data points that cover the Si ii 𝜆6355 and the Ca ii NIR3 features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polar plots of the polarization of SN 2019np across the Si ii 𝜆6355 and Ca ii NIR3 lines at all five epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In each panel, the radial distance and the angle represent the expansion velocity and the polarization position angle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The center of each fan-shaped bin gives the average position angle calculated over the velocity range covered by the radial extent of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The angular widths of the fan-shaped bins represent the 1𝜎 uncertainty on the position angle rather than the underlying physical dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The velocity is labeled in km s−1, and the celestial position angles are measured in degrees from North to East.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The blue and the orange colour bars indicate the ISP-corrected polarization degree across the Si ii 𝜆6355 and Ca ii NIR3 profiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The data have been rebinned to 30 Å for better visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The direction of the dominant axis is shown by grey-shaded regions with their angular width representing the 1𝜎 uncertainty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 𝜃∗ 𝑑 in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The blue and red semicircles mark the estimated photospheric velocity and the high-velocity component as measured from the absorption minima of the Si ii 𝜆6355 and Ca ii NIR3 lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An overall property of the polar diagrams is their patchy appear- ance, especially in Ca ii NIR3 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These “flocculent” structures tend to become gradually less conspicuous with time, and increase again at day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4 NUMERICAL MODELLING This section conducts a quantitative study of the degree of as- phericity of SN 2019np inferred from the observations described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We also investigate their temporal evolution and interpret the nature of the polarization variations on small wavelength scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As a baseline, we will use an off-center delayed-detonation model MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Temporal evolution of the intrinsic polarization of SN 2019np from days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The 𝐵 and 𝑉 light curves (from Burke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022) are displayed in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The middle panel gives the continuum polarization calculated from the error-weighted mean values of the Stokes parameters in the range 6400–7000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The bottom panel presents the peak polarization measured across the Si ii 𝜆6355 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Values measured with both 30 Å and 20 Å bin sizes are plotted as labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Second-order polynomial fits to either bin size are indicated by the solid black and dotted grey curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (Khokhlov 1991), namely the explosion of an 𝑀Ch WD in which a deflagration front starts in the center and transitions to a detonation for reasons described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A low level of polarization along the continuum spectrum of a SN is most likely generated by spherically symmetric ejecta leading to complete cancellation of the electric vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, an aspherical but rotationally symmetric object may also be viewed along its sym- metry axis, which has the same effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To distinguish these two pos- sibilities, we will use both the polarization over the quasi-continuum and the modulation of the polarization across major spectral features in order to separate the intrinsic asphericity and the polarization actually observed from a certain direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In our analysis, we will employ an approach of minimum complexity rather than fine tuning the parameterised geometry to optimise the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The modeling will address whether the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1%–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2% polarization variations with wave- length in the quasi-continuum seen at all epochs can be understood in terms of opacity variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Furthermore, we will discuss whether the temporal and spectral resolution of our VLT spectropolarimetry is sufficient to detect and probe any small-scale structures in density and/or abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The VLT spectropolarimetry of SN 2019np between days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 was analysed through simulations employing modules of the HYDrodynamical RAdiation (HYDRA) code5 (Höflich 1995a, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Penney & Hoeflich 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' HYDRA solves the time-dependent radiation transport equa- 5 Many of the HYDRA modules are regularly used by other groups and are available on request to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' tion (RTE) including the rate equations that calculate the nuclear reactions based on a network with 211 isotopes and statistical equa- tions for the atomic level populations, the equation of state, the matter opacities, and the hydrodynamic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The resulting polariza- tion is obtained by post-processing the given level populations and the density and abundance structure through a Monte Carlo (MC) approach (Khokhlov 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Höflich 1995a, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Penney & Hoeflich 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Atomic models were considered for the ionisation stages I–III of C, N, O, Ne, Mg, Na, Ca, Si, S, Ar, V, Ti, Cr, Fe, Co, and Ni, but without forbidden tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the luminosity evolution of the multidimensional model as a function of time, a spherical reference model with 911 depth points was adopted, which is adequate considering the small devi- ation from spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Moreover, the timescales are dom- inated by the inner layers which are almost spherical in off-center delayed-detonations whereas the spectra are formed in the photo- sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This allows us to compare the observations with snapshots of the multidimensional model, neglecting time derivatives in the rate and radiation transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the polarization spectra, we use ∼ 700 frequency counters between 2800 and 10,200 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The resulting spatial discretisation cor- responds to a formal spectral resolving power of 𝑅 ≈ 500, which matches that of the observations (𝑅 ≈ 440, Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, in a rapidly expanding atmosphere with gradients, the spatial resolution degrades 𝑅 to ∼ 150 since the solution of the radiation transport equation depends on the spatial gradients of the physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Simulating a large number of configurations by multidimensional models is prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, we employ a scattering approach with a thermalisation depth to find and discuss estimates for the degree of asphericity in the surface as well as deeper layers (Höflich 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The continuum polarization may be caused by an aspherical electron-scattering photosphere or an off-center energy input or both (Höflich 1995c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Kasen 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016a) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the spectra of a Type Ia SN, opacities from bound-bound transitions form a wavelength-dependent quasi-continuum and also produce individ- ual spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The quasi-continuum may exhibit polarization signals when the sizes of any opacity clumps are comparable to the free mean path of Thomson scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' One should keep in mind that, in the high Thomson optical-depth regime (𝜏 ≳ 3–4), the continuum polarization in the quasi-continuum will be lower compared to that at 𝜏 ≈ 1 and reach an asymptotic limit for large optical depths since any information about asphericity will be blurred by multiple scat- tering (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Figures 1 and 5 of Höflich 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' If the opacity of the quasi-continuum becomes much larger than the optical depth of the Thomson scattering, the degree of polarization 𝑝 ∝ 𝜏sc, where 𝜏sc denotes the electron-scattering optical depth of layers at which photons thermalise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 The Reference Model As the spherically symmetric reference, we adopt the delayed- detonation Model 25 for a normal-bright Type Ia SN from Hoeflich (2017) because it shows light-curve properties very similar to those of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The explosion disrupts a WD with mass close to 𝑀Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Burning starts as a deflagration front near the center and transitions to a detonation by the mixing of unburned fuel and hot ashes (Khokhlov 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The explosion originates from a C/O WD with a main-sequence mass of 5 M⊙ as the progenitor star, solar metallicity, and a central density 𝜌𝑐 = 2 × 109 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The deflagration–detonation transi- MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 9 Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The symmetry axis �𝑛 (black arrow) is defined by the minor axis of a rotationally symmetric ellipsoid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', an oblate spheroid, left plot) or by the vector (right plot) through the center (gray dot) and the location of the DDT (green dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The equatorial plane E (red) is given by �𝑛 �𝑥= 0 with �𝑥 spanning E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The viewing angle 𝜃 is the angle between the plane E and the direction to the observer (blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 𝜃 = +90◦, −90◦, and 0◦ correspond to the north pole, south pole, and the equator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As common, 𝜃 is measured counter-clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Left: The mass above the photosphere as seen by photons in the 𝑈, 𝐵, and 𝑉 bands as a function of time for the normal Type Ia SN calculated in the off-center angle-averaged version of the delayed-detonation Model 25 (Hoeflich 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The exponential index (𝑛) of the radial density distribution at the position of the photosphere as a function of time is also shown by the red triple-dot-dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The five epochs of VLT spectropolarimetry are marked by grey vertical dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Middle: Angle-averaged abundance structure as a function of expansion velocity, also calculated using Model 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Vertical grey-dashed lines indicate the location of the scattering photosphere — that is, 𝜏sc = 1 at the times when the VLT spectropolarimetry was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The region with electron-capture elements is indicated by EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Right: The 56Ni distribution as seen above the photosphere on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 based on the hydrodynamical simulation of the off-center detonation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The mass fraction of off-center 56Ni above the photospheric radius (dark-red) is ∼ 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At this phase, the radius of the photosphere is close to the location (black dot) where the deflagration-to-detonation transition takes place, and it expands with ∼ 7000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The mass fraction is colour-coded in a domain size of ±23,500 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' tion was triggered when the density at the front had dropped below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 × 107 g cm−3 when ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='24 M⊙ of the material had been burned by the deflagration front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the construction of the off-center de- layed detonation transition (DDT), we follow the description of Livne (1999) that has been previously employed (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Fe- sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To terminate the deflagration phase, the delayed-detonation transition is triggered with the mass- coordinate 𝑀DDT as an additional free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The time series of the flux and the polarization spectra were generated without fur- ther tuning of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The photometric properties predicted by the spherical model are similar to those measured for SN 2019np, namely Δ𝑚15(𝑉/𝐵) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='14/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='68 mag (Model 25) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='67 mag (Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' According to the above prescription, the axial symmetry of the SN model is defined by the location(s) of the point(s) where the deflagration-to-detonation transition took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The asphericity in the density distribution near the surface layers was characterised by introducing an additional free parameter when modeling the contin- uum polarization at the earliest phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the actual implementation see the last paragraph of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The symmetry axis that determines the geometric properties of the outermost layers and that which is defined by the location of the deflagration-to-detonation transition in the inner regions are not correlated with each other, since the latter is stochastic and expected to take place deeper in the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As the DDT is turbulently driven in the regime of dis- tributed burning, its location depends on the ignition process of the thermonuclear runaway, namely multispot or off-center ignition, and initial magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In contrast to the inner symmetry axis, that of the surface layers is likely determined by the direction of the angular momentum of the progenitor system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', the equatorial plane of a companion or the plane of an accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Since the luminosity originates from the energy source that is well below the photosphere in the first few days after the SN explosion, and the outermost layers do not affect the emission at later phases, our simulations treat these two symmetry axes as independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In Figure 11, we present the mass above the photosphere as a func- tion of time (left panel) and the radial distribution of the chemical abundances as a function of expansion velocity (middle panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Over- all, the exploding envelope has the familiar onion-shell-like structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the onion is no longer spherical but elongated as a result of MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) n DDT 0 0 Equatorial planeC 0 44Ti 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='8 66Ni S Ne EC Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 0 5 10 15 20 25 0 v [1000 km/sec]1 X, 56 Ni .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 20 10 0 10 20 3 km/sec] Vx[10310 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Left: Continuum polarization as a function of asphericity for an oblate ellipsoidal scattering-dominated photosphere viewed equator-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A steep density gradient can be expected at early phases (see Figure 11), when the polarization approaches the limit of large optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The exponential index of the radial density distribution on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 can be best represented by configurations with 𝑛 ≈ 13–14 (left plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The horizontal line and the cyan-shaded area indicate the level and the associated uncertainty of the continuum polarization (respectively) measured on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, an axis ratio between ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To aid the discussion in the text, the inset panel shows the continuum polarization as a function of thermalisation optical depth for an oblate ellipsoid with an axis ratio of 2 and 𝜌 ∝ 𝑟−2 (from Höflich 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Right: The degree of continuum polarization produced by a spherical photosphere plus an off-center energy source as a function of viewing angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the illustration of the effect of an off-center energy source, a radial density distribution index 𝑛 = 3 and an optical depth 𝜏 = 1 are chosen to represent the SN photosphere around two weeks after maximum brightness (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The horizontal line and the pink-shaded area mark the level and the error of the continuum polarization, respectively, on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the off-center DDT (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2 in Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the left panel, we also mark the times of the VLT spectropolarimetry with respect to both the estimated time of the explosion and the 𝐵-band light-curve peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The earliest spectropolarimetry to date of any Type Ia SN on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 probes the outermost ≲ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5–3) ×10−3 MWD layer of the exploding WD, corresponding to a mass of ≲ 4×10−3 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As deduced from the red triple-dot-dashed curve, at such an early phase, the exponential index of the radial density distribution is 𝑛 ≈ 13–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the middle panel of Figure 11, we mark the locations of the photo- sphere at each epoch of the VLT spectropolarimetry in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that the absorption minimum of, for example, Si ii 𝜆6355 does not measure the expansion velocity at the photosphere but the aver- age projected velocity toward an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The difference compared to the expansion velocity at the photosphere is particularly large in zones with steep density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Since the photosphere recedes over time, multi-epoch spectropolarimetry can tomographically map out the degree of asphericity at different chemical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As indicated by the middle panel of Figure 11, the cadence of the VLT observations of SN 2019np only provides a resolution in expansion velocity of ∼ 6000 km s−1, at which a discrimination of any structures smaller than several thousand km s−1 in the radial direction is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For a detailed discussion, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We employ a delayed-detonation model considering the fact that C ii was seen in the first epoch on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (Figure 1), correspond- ing to the very outer layers of ≲ 4 × 10−3 M⊙, making a sub-𝑀Ch explosion an unlikely candidate even for the case of C/He mixtures (Shen & Moore 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that 𝑀Ch explosions may have a thin H/He-rich surface layer as a result of the accretion phase but at a significantly smaller mass, (1–5) ×10−4 M⊙ (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019), an amount below our numerical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, we neglect the H/He layer in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In Model 25, the early-time spectra originate from the region with incomplete explosive carbon burning and an inward-increasing contribution by explosive oxygen burning (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By the time of maximum light, the photosphere is formed in layers of complete oxy- gen burning and partial silicon burning as indicated by the presence of Ar and Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The emergence of Ar lines in the mid-infrared was predicted by our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In SN 2014J, they were detected by Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At ∼ 2 weeks after peak luminosity, the spectrum on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 obtained by our last epoch of VLT observations is formed at the interface between partial, distributed silicon burning and with burning to nuclear statistical equilibrium (NSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The position of this layer coincides with the location where the DDT has been triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that in our simulation the point of the DDT does not lead to a strong refraction wave (Gamezo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2005) as in all spherical delayed-detonation models (Khokhlov 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The innermost layers undergo weak reactions under NSE conditions, resulting in the pro- duction of electron-capture (EC) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An asphericity in the outermost layers as indicated by the positive detection of the continuum polarization at the first epoch (see Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2) is not produced by our hydrodynamical reference model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To estimate the degree of asphericity at that early epoch, we describe the density structure of SN 2019np by stretching along the radial direction using an oblate ellipsoid with the axis ratio as a free pa- rameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The density and abundance structure are directly taken from our reference model, transforming the distance from the center of an element as 𝑟(𝑚) ⇒ 𝑟(𝑚, 𝜃) (Höflich 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the toy models for the continuum developed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2, we treat the orienta- tion as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For reasons of computational feasibility of the full model, we assume that the symmetry axes of the density and abundances are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' When the deflagration front has burned ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='25 M⊙, we trigger the detonation by mixing burned and un- MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6 Density Structure pαr-n Spherical + Off-Centered Source % 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 at T=1 and pαr-3 P [%] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='8 d 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 n=3 Polarization [ Polarization with 10% off-center n=5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 with 5% off-center n=7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6 n=10 n=15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 Day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, Jo Jo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10% ree Day -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09% p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 50 50 0 Axis Ratio Inclination [Degree]Geometry of Type Ia SN 2019np 11 burned fuel at 𝑀DDT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 M⊙6, the so-called Zel’dovich reactivity gradient mechanism (Zel’Dovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 Continuum Polarization On days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (the first epoch) and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (the fifth and last epoch), the level of the continuum polarization has been measured as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10%, respectively, both at a ∼ 2𝜎 level (see footnote in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The former corresponds to the very outer layers and the latter probes the inner layers near the position where the deflagration-to-detonation transition takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Between day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 (second epoch) and day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (fourth epoch), the contin- uum polarization was consistent with zero within the uncertainties (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At early times, the thermalisation depth of the photons emitted by a SN is large (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 𝜏 ≳ 3), and the polarization degree reaches its asymptotic value (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1 and 11 of Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (1993) and Höflich (1995c), respectively, and the inset in Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The max- imum polarization degree is expected when 𝜏 ≈ 1 (Höflich 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Linear polarization produced by aspherical density structures follows the relation 𝑝 ∝ 𝑠𝑖𝑛2𝜃, where 𝜃 is the angle between the polar direc- tion and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As the radial density exponent 𝑛 is high at the first epoch (𝑛 ≈ 13–14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' see left panel of Figure 11), a minimum axis ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='25–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 can be inferred from the continuum polarization of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09% on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (see left panel of Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For an equator-on perspective (𝜃 = 0o), the high axis ratio implies aspheric- ity in excess of 30% in the 4 × 10−3 M⊙ of the carbon-rich layers in the outermost part of the exploding WD (see Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Only three days later, on day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, the continuum polarization had dropped rapidly to a level consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By contrast, for a constant global asphericity, the degree of polarization would increase with time because (i) the density slope becomes flatter (see the left panel of Figure 11), and (ii) the thermalisation optical depth decreases to ∼ 1 as the SN reaches maximum light, when the quasi- continuum opacity in the iron-rich region becomes comparable to, or larger than, the Thomson opacity (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Figure 2 in Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, the rapid decrease in continuum polarization observed in SN 2019np suggests that the large-scale asphericity in the density structures seen at the earliest phase is limited to the very outer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We find that an additional structural component is only required at the first epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For all deeper layers, we do not have to impose any asphericity on the density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The difference in polarization position angle between the surface and the deeper layers may be attributed to the additional structural component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the continuum polarization of SN 2019np exhibited an increase to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10%, although the scattering optical depth had decreased significantly well below 𝜏sc = 1, where 𝑝 ∝ 𝜏sc, and, hence, a decrease in 𝑝 may be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As will be discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3, the continuum polarization can be understood as a consequence of our off-center DDT Model 25, which produces an aspherical distribution of 56Ni, and thus an inhomogeneous central energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 6 Using the amplitude of the Si ii 𝜆6355 polarization as the criterion, we chose this mass fraction from a set of intermediate models for levels of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='9 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the Monte Carlo post-processing to obtain 𝑝 presented in this paper, a number of particles per resolution element has been used to obtain a statistical absolute error of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='015%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An off-center energy source is needed because the quasi- continuum dominates the Thomson scattering causing thermalisa- tion at low 𝜏sc and, thus, only photons with grazing incidence on the outer photosphere get polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The reason is that photons scattered into the direction of their travel, the Poynting vector, are unpolar- ized whereas light scattered orthogonally to the Poynting vector is 100% polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Radially traveling photons are most likely to escape when they are scattered along the Poynting vector, whereas grazing- incidence photons can escape most easily when they are radially scattered by 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A similar increase of the continuum polarization after maximum light was reported for SN 2019ein, namely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='28%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10% on day +10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='31%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='32% on day +21 (Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This rise of the broad-band polarization at late phases was also attributed to the emergence of an aspherical central energy input as the photosphere reaches the Si/Fe interface (Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As a first step, we quantify the level of the asphericity in the 56Ni distribution required for a toy model that does not depend on details of the explosion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We estimate the amount of off-center 56Ni at the photosphere relative to the main, symmetric component of the 56Ni distribution based on previous simulations (Höflich 1995b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Motivated by the low continuum polarization between days −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2), we assume7 that the low continuum po- larization by any global asphericity in the density at the photosphere can be neglected and, based on Model 25, that the off-center source is at about the photosphere (see right panel of Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the toy model, a point-like off-center source at 𝜏𝑠𝑐 = 1 in a spherical envelope is assumed to obtain a first-order estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The relative contribution by the off-center component at day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to the total energy input at the photospheric level is found to be between 5% and 10% (see the right panel of Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Using as reference the axis defined by the center and the location the DDT, a tangential energy source causes a flip in the polarization angle or, in the 𝑄–𝑈 diagram, the polarization axis should rotate by 90◦(Höflich 1995c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, only ∼ 70◦ are observed relative to the layers seen at day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Thus, our toy model predicts a difference between the symmetry axes of the outer structural component and the inner layers which causes a change in PA in the 𝑄–𝑈 diagram of about 20◦ compared to day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This estimate is obtained by the vectorial addition of the polarization contributions by the off-center source and the spherical source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Although a change in position angle from day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 is hard to measure in SN 2019np owing to the very low intrinsic continuum polarization, we suggest that the rotation of the dominant axis fitted to the same optical wavelength range as above (see Figure 7) is compatible with the prediction of an off-center energy source beginning to be exposed to the observer at this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' One of the major effects of the very steep density slope in the outer layers is that even the small ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2% continuum polarization ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days after the SN explosion implies a significant aspherical density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The polarization of SN 2019np at the earliest epoch is higher than that measured in other Type Ia SNe at later phases, which are closer to epoch 2 on day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and epoch 3 on day −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For example, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='07% was observed in SN 2019ein (Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022) on day −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='9, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='12% in SN 2012fr (Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2013) on day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09% continuum polarization of SN 2019np on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days after the explosion) is comparable to 7 As shown in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3, we cannot allow a large-scale density asymmetry in Model 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Estimating the viewing angle 𝜃 from polarization spectra (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As examples, the full polarization spectrum of the off-center DDT Model 25 at day −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 is shown for inclinations 𝜃 of about 0◦ (blue, equator- on), 45◦ (red), and 60◦ (cyan), and, as reference, the intrinsic polarization of SN 2019np at a corresponding resolution (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The estimate of 𝜃 is based on all epochs, and the error in 𝜃 is constrained by the uncertainty of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For ±90o the polarization is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note the sensitivity of the ratio between line and continuum polarization to 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the marginal detection of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='20%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='13% continuum polarization in SN 2018gv on day −13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, ∼ 5 days after the explosion, (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At that moment, the density exponent in SN 2018gv had dropped to ∼ 9–10 (see the left panel of Figure 12 and Figure 21 of Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This leads to a ∼ 10%—35% deviation from spherical symmetry within the outermost ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5–2) ×10−2 MWD for an equator-on configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The cases of SNe 2019np and 2018gv may provide a hint that any asphericity in the outer layers of normal- bright Type Ia SNe becomes apparent in polarization only during the very earliest phases and thereafter quickly almost vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' There- fore, given the high density gradient near the surface layers of the ejecta of Type Ia SNe, a low but nonzero continuum polarization measured in the first few days after the explosion does not necessar- ily imply a low deviation from sphericity in their outermost layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 Polarization Spectra The spectral evolution of SN 2019np is similar to that of other normal-bright Type Ia SNe (Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022), enabling us to compare the polarization spectra of SN 2019np with the models for normal Type Ia SNe discussed by Höflich (1995a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The inclination was ob- tained by comparing the direction-dependent synthetic polarization spectra to those of SN 2019np at all epochs and minimising the 𝜒2 averaged over 100 Å-wide bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To fit the evolution of the polar- ization spectra (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 6 and 9), we find that a viewing angle of 𝜃 = +45◦ ±10◦ is the most plausible approximation of the actual case8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In Figure 16, the off-center delayed-detonation model viewed at this angle is in good overall agreement with the observations of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In Model 25, the polarization across Si ii 𝜆6355 is formed within an extended geometrical structure between 9000 and 27,000 km s−1 which undergoes complete and incomplete oxygen burning in veloc- ity (middle panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 11, where the velocity of the photosphere at the time of the observations is indicated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the outermost region of 8 For example, at maximum light, compared to 𝜃 = +45◦, the polarization in Si ii 𝜆6355 is larger by ∼ +50% at 𝜃 ≈ +35◦, vanishes at +90◦, and becomes small for negative angles depending on the phase, whereas the overall level of 𝑝 peaks at 𝜃 ≈ +10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' partial explosive oxygen and carbon burning, the polarization of Si ii 𝜆6355 is weaker since its abundance diminishes with increasing ve- locity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At early times, this line forms close to the region with 𝜏sc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Because the polarization by electron scattering is mostly formed in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 < 𝜏sc < 1, the polarization across Si ii lines is generally low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The Si polarization increases as the photosphere continuously recedes and, without the structural component, reaches its peak when the photosphere enters the layers with quasi-equilibrium conditions around the Si group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Thus, the polarization in Si ii 𝜆6355 increases with growing distance between the optical depth at a given wave- length and the layer with 𝜏sc ≈ 1, which is always more internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Any aspherical distribution is expected to be most prominent around this phase, when the photosphere passes the inner boundary of the ex- plosive C- and O-burning, and the QSE(Si)/NSE interface becomes exposed (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' After peak luminosity, the polariza- tion of Si ii 𝜆6355 decreases because the quasi-continuum opacities increasingly dominate the electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Apart from Si ii 𝜆6355, the polarization over a quasi-continuum wavelength range also increases in the same region that forms various other spectral features, which are resolved (see Figures 14, 15, and 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This concerns the entire wavelength range occupied by blends of the Fe group, Si ii, S ii, and O i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Depending on time, features in the polarization spectra appear around (for example) 4400, 4800, 5400, 5800, 6800, 7200, 7500, 8300, and 9000 Å (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Over- all, thousands of overlapping lines are involved (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Figures 1 and 2 of Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The variations in this quasi-continuum depend on the velocity gradients, the abundances, and the ionisation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The polarization is very sensitive to this pattern as it influences the thermalisation optical depth by individual components because spectral lines mostly depolarize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In flux spectra, these variations are mostly blurred because photons are absorbed and emitted, but they are visible in the line-formation radii traced by spectropolarimetry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As a result, spectropolarimetry is effectively much more sensitive to spectral lines than flux spectroscopy because its observ- able signatures are much less volatile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Nevertheless, some individ- ual patterns can be identified by comparing the line identifications (Figures 1 to 5 and 15) and, from the models, by variations in the wavelength-dependent radii of line formation as presented in Fig- ure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For instance, the rather persistent feature at 9000 Å can be attributed to a strong Fe ii + Co ii blend which becomes obvious as a change in the thermalisation radius and appears in both observed and synthetic spectra (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the models, a maximum or minimum in polarization is produced if the thermalisation optical depth is above or below 𝜏sc ≈ 1, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Owing to the sensitivity of the polarization, maxima in the observations can be minima in the synthetic spectra for moderately strong blends which appear in the optical depth (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' One example is the Fe/Co blend at ∼ 9000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This feature toggles with time between maxima and minima in both theory and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At most epochs, the changes in model and observations are synchro- nised, except for days −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Another example is the S/Fe blend at ∼ 5800 Å, for which the simulations mostly reproduce the observations but on day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Both features can be identified as an elevation in the radius of photon decoupling as shown by Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For these examples, the time to toggle from high to low polarization can be estimated from the rate with which the photosphere recedes over the abundance gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The gradients typically extend over ∼ 500–1000 km s−1 (Figure 11) and so correspond to a timescale of ∼ 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From spectral analysis, a similar timescale of a few days is well established for changes in ionisation stages (Branch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Mazzali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Höflich 1995c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Lentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Baron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Dessart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This is faster than our observing MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Model 25 at day -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 seen from different directions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6 00 +45° +60° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 0 5000 6000 7000 8000 9000 Rest-frame Wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='Geometry of Type Ia SN 2019np 13 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Flux and polarization spectra observed in SN 2019np compared to the spectral formation radius computed with Model 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Top panels: Scaled flux spectrum (black curve) and degree of polarization (red histogram) observed on days −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 (left), and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Bottom panels: Photon-decoupling radius 𝑅 for 𝜏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 and 1 computed with delayed-detonation Model 25 at similar phases (left panel for day −7 and right panel for day +0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Major spectral lines are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that the polarization is mostly produced by Thomson scattering between 𝜏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Even strong features such as Si ii 𝜆6355, the Ca ii NIR3 complex, and various blended features below 5500 Å are formed in the same region, setting a qualitative limit to the picture of line polarization being produced by chemically selective blocking of an underlying scattering-dominated photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Spectral formation as a function of radius at maximum light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We compare the observation with the spectrum of Model 25 (upper panel) and the radii corresponding to optical depths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 and 1, between which the absorption features and the polarization are formed (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The observed and synthetic spectra generally agree without fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For strong lines like Si ii 𝜆6355, the flux minima and the blue wings typically correspond to an optical depth of 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This is also true for other strong features such as Ca ii NIR3 at ∼ 8200 Å and many of the line blends below 5500 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Strong spectral features form in the same region where the continuum polarization is also produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The high-frequency structure in the radius of formation of Si ii 𝜆6355 is due to Fe ii transitions which do not appear in the flux spectra, but show up in the line profiles as discussed in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note the reduced effect of the quasi-continuum, justifying the wavelength range used for the determination of 𝑝cont in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' frequency of SN 2019np and may reflect an insignificant phase shift in the evolution of the models relative to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This phase shift may also point toward small-scale structures such as Rayleigh- Taylor fingers or Kelvin-Helmholtz instabilities not included in our models, which would reveal themselves in short-term variations in the polarization spectra, but are not resolved in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The numerous wiggles (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 and Figure 16) are not resolved in the current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' They may possibly be understood in the same way as resolved features: namely in terms of atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Many coincide with features produced by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Some of them may indicate genuine small-scale structures, and others may be just noise in the data or ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Their true nature cannot be determined with the current observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This ambiguity points toward a need for high-cadence observations to separate small scale instabilities from imprints governed by atomic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The change of the polarization profiles of (for example) Si ii 𝜆6355 can also be understood within the same framework, leading to a new diagnostic (Figures 14 and 16) of substructures in lines, although the spectral resolution of our polarimetry may not be sufficient to fully reveal the underlying velocity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Overall, the location of the peak (its Doppler shift) agrees between observations and synthetic profiles including the evolution of the line width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This supports the interpretation by large-scale asphericity in the abundance distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From the models, this evolution can be understood even though, at higher granularity, some discrepancies need to be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (a) Binning of the data may introduce artifacts, in particular at very early times when the SNR is low in the current data as on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='59 or on day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, when the polarization peak in Si ii 𝜆6355 occupies just one wavelength bin whereas the associated change in the position angle takes place over three bins (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (b) Some discrepancies between observations and model profiles may also hint at the model 9 On day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, Si ii 𝜆6355 shows multiple components at 25 Å binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 14 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polarization calculated with off-center DDT Model 25 (blue histograms) compared to the intrinsic polarization of SN 2019np (red his- tograms) from days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 to +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The grey dotted curve in each panel shows the observed scaled flux spectrum at the given epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the model calculations, an inclination of ∼ 45◦ was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Fe opacities being too weak between 6000 and 6500 Å, possibly ow- ing to a lack of Rayleigh-Taylor mixing, too low excitation of the atomic levels, or slightly too low a metallicity in the progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As discussed above, even the strong lines are blended with many weak lines, which do not appear in the flux but in the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the simulations, the strength of the polarization depends on the thermalisation depth in the atmosphere and the density profile (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' If at some wavelength the thermalisation depth is close to the Thomson optical depth of 1, the polarization peaks at that wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' It becomes smaller with decreasing thermalisation depth, and reaches the asymptotic value for large depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As a result, the line profile is broader at early times when, owing to steep density profiles combined with decreasing abundances in the region of in- complete oxygen burning, namely around day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, the radii of the photon decoupling regions are similar and, thus, the resulting profiles are broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' With time, the density slope flattens and, to first order, the profile becomes narrower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Note that, in an expanding atmosphere, the absorption is determined by the Sobolev optical depth which is not inherently spherically symmetric in wavelength (see Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By days −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the Si profile is formed in the QSE region and a flat density gradient leads to an increasing blueshift of the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, Si ii 𝜆6355 is blended with Fe ii 𝜆𝜆6293, 6358, 6497 and weaker Fe ii and Fe iii transitions from excited levels, leading to a more complicated profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the models, the iron blends seem to be weaker than in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Because line absorption depolarizes, this can explain the lack of depolarization in the model profile in both the blue and red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the same reason, at days −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the observed profile has a steep decline whereas the synthetic profile shows a long red tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The imprint of Fe ii 𝜆6497 may be seen in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A similar shape of the Si ii 𝜆6355 polarization profile has also been seen in SN 2018gv around peak luminosity (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the models and the observations after peak luminosity of SN 2019np, Si ii 𝜆6355 becomes progressively blended with sev- eral strong Fe ii lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The Si line has vanished by day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 as the photosphere recedes into the NSE region, which displays strong Fe- group elements that form both a quasi-continuum and discrete lines in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The feature at ∼ 6200 Å, which is conventionally attributed to Si ii 𝜆6355, becomes increasingly dominated by Fe ii lines and, as a consequence, the corresponding polarization across this wavelength range also disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Overall, the quasi-continuum on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 is produced by numerous overlapping Fe-group lines from Fe, Co, Ni, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Without fine-tuning of our model, the polarization spectra across several major spectral features can also be reproduced and generally agree with the observed polarization spectra (see Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For the calculation of the continuum polarization, we use the wavelength range 6400–7000 Å applied in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The global asphericity in the electron density distribution on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 is not accounted for in the hydro simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, we imposed an overall ellip- tical distribution with an axis ratio of 2 to obtain the overall level of the polarization over the entire wavelength range observed (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The choice of the axis ratio is motivated by Figure 12 and results quantitatively from the most likely viewing angle, 𝜃 ≈ 45◦ (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3) and the relation 𝑝 ∝ sin2𝜃 (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At all later epochs, the continuum polarization is calculated di- rectly without modifying the hydro model (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The con- tinuum polarization produced by the detailed off-center model (Fig- ure 16) is within the 1𝜎 error range of the observed values (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' On day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, the asphericity in the electron density is caused by the aspherical abundance distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='10 Its value is ill-defined because of steep changes of the synthetic polarization at the edges of the 6400–7000 Å wavelength range (see second panel from top of Figure 16), although we used the same range as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 to minimise the effect of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='16%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04% returned by the models is somewhat larger than the observed level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='099%± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='080%, but well within the error range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The synthetic continuum po- larization on days −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='05%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='075±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='080%), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='110±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='100%), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='22% (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='186±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='104%), respectively, with the observed values given in brackets11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Most of the discrete polarization features are at the level of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the models, they are produced by depolarization or the fre- quency variation in the thermalisation optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Whether they appear as local maxima or minima in the polarization spectrum de- pends on the scattering optical depth of the corresponding region of formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Many of these wiggles in the observed polarization 10 In SNe Ia and unlike SNe II, the resulting asphericity in the electron dis- tribution remains rather small, 5–10%, because the free electrons per nucleon are about equal for Si/S II and Fe/Co II-III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For example, in the hydrogen- rich envelope of SNe IIP, the opacity drops by 4 orders of magnitudes over the recombination front of hydrogen, causing highly aspherical Thomson- scattering dominated photospheres even in case of slightly aspherical 56Ni distributions or rotation (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Leonard & Filippenko 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 11 Note that the variations in the observed continuum polarization are on a 2𝜎 level (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, they also coincide with a change in the dominant axis in the 𝑄–𝑈 diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 7), and 𝜎 includes spectral variations by lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 15 spectra (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3) coincide with features in the synthetic polar- ization spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Discrepancies may be due to small-scale structures that the observations of SN 2019np do not resolve in time and wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This may hint at the possibility of significant detection of numerous weakly polarized lines in future higher SNR observations with FORS2 at ESO’s VLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Since some patterns do not have mutual counterparts, such observations should also aim for higher spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Simulations with matching resolution are feasible with moderate additional effort (see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Although our models reproduce many (even relatively minor) as- pects of the observations, some limitations are also apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For instance, on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the synthetic polarization spectra exhibit fairly similar overall patterns, but the models do not show the large- amplitude fluctuations with wavelength observed in the polarization spectra of the strongest resonance lines, namely in Ca ii NIR3 (see Figure 8 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4) and possibly in Si ii 𝜆6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='12 The involve- ment of Ni, Co, or Fe seems to be ruled out because, in the spectral region strongly affected by iron-group elements (𝜆 ≲ 4700 Å and 𝜆 ≳ 8500 Å), similar patterns do not exist and observations and synthetic spectra agree well for our model with solar abundances in the outer C/O layers and Fe, Co, and Ni mass fractions of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='002, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='00005, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='0001, respectively (Anders & Grevesse 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1, an amount of 𝑀(56Ni) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='03 M⊙ in the outer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 M⊙ (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 in mass fraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' alternatively produced in a sub-𝑀Ch explosion) is needed to explain the early bumps in light curves of SNe 2017cbv and 2018oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The associated spectra are dominated by Fe, Co, and Ni lines (Höflich 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Magee & Maguire 2022), traces of which are likely just barely seen in all spectra of SN 2019np (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Another example for the shortcomings of our current model is the increased polarization in Ca ii NIR3 around and after maximum light, which is not reproduced by our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This resonance line has by far the largest cross-section and is optically thick even in regions with solar abundances, so that very minor inhomogeneities can have a big impact on the polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Extended, inhomogeneous radial components in the Ca distribution may be expected from Rayleigh- Taylor instabilities, interactions with a companion star, and/or sheet- like/caustic structures, which may develop within 5–10 days after the explosion as the result of mixing of radioactive 56Ni and electron- capture elements (Marietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Maeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Additionally, the late rise of the Ca ii NIR3 polarization may also be caused by the alignment of calcium atoms in the presence of a magnetic field as recently suggested by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' If combined with near- and mid-infrared nebular spectra, later-epoch polarimetry of the Ca ii NIR3 feature will allow us to discriminate between various possibilities concerning the nature of the progenitor and the explosion mechanism as discussed by Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2004), Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2015), Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2021), and Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2021), since the spatial distribution of radioactive Co and stable Fe, Ni, and Co can be probed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 SN 2019np in Polarimetric Context with other Type Ia SNe The polarization properties of some Type Ia SNe are remarkably different from those that are typical for normally bright thermonu- clear SNe (Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Patra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022) and SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An 12 On day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5, the feature at ∼ 6000 Å (within the Si ii 𝜆6355 profile, Figure 16), which occupies a single bin with 30 Å binning, breaks up into two components with peaks at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='48% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='39% if 40 Å bins are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' example is SN 2004dt, which exhibited exceptionally high polariza- tion in some spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For instance, the peak polarization across Si ii 𝜆6355 reached ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4% and ∼ 3% after binning to 50 Å and 25 Å, respectively (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Although the continuum polarization was as low as ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2%–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3% around peak brightness (Leonard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006), many features of Si, S, and Mg in the synthetic polarization spectrum13 had their equivalent in the observations (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006b) as commonly found in Type Ia SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These findings might be accounted for by either a violent merger of two C-O WDs (Bulla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2016a) or an off-center delayed detonation model within a continuum of param- eters consisting of especially the position of the delayed-detonation transition, the amount of burning during the deflagration phase, and the viewing angle of the observer (Höflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For normally bright Type Ia SNe, the polarization of Si ii 𝜆6355 five days before 𝐵-band maximum light (𝑝−5 d Si ii ), which is repre- sentative of the maximum value (𝑝max Si ii ), correlates with the light- curve stretch parameter measured as the decline in magnitude within 15 days after 𝐵 maximum (Δm15(𝐵);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Cikota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For SN 2019np, Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) measured Δm15(𝐵) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04 mag, and 𝑝max Si ii amounted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='03% on day +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (with 30 Å binning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, we conclude that SN 2019np is consistent with the 𝑝max Si ii –Δm15(𝐵) relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In a follow-up study, Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2010b) investigated Si ii 𝜆6355 observations of a sample of nine normal Type Ia SNe and found that 𝑝−5 d Si ii is also correlated with the temporal velocity gradient �𝑣Si ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Accordingly, the deceleration of the SN expansion is also correlated with the degree of chemical asphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The interpolated velocity gradient of SN 2019np was 21±5 km s−1 day−1 on day +10 (Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By interpolating the Si ii 𝜆6355 velocity evolution estimated from our VLT observations, we estimated velocity gradients of 53±19 and 22±9 km s−1 day−1 on days +0 and +10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This means that SN 2019np was also consistent with the 𝑝−5 d Si ii –�𝑣Si ii relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Subluminous Type Ia SNe exhibit substantially different polariza- tion properties than discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For instance, a polarization of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7% in the optical continuum but only ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3% across Si ii 𝜆6355 were observed in SNe 1999by (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2001) and 2005ke (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012) at ∼ 0 and −7 days relative to maximum light, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The high degree of continuum polarization can be explained by a global asphericity of as much as 15% (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These two events are outliers from the correlation proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2007) between the Si ii 𝜆6355 polarization five days before 𝐵-band maximum and Δm15, nor do they match the relation between the velocity gradient of Si ii 𝜆6355 and the associated peak polarization (Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2010b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The mismatch may be due to SNe 1999by and 2005ke perhaps being typical representatives of underluminous Type Ia SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Their spectroscopic and polarimetric properties can be un- derstood within the frameworks of delayed detonations originating from a rapidly rotating WD or WD-WD mergers (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 5 CONCLUSIONS At five epochs between days −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 from maximum light, we have obtained high-quality optical VLT spectropolarimetry of the normal Type Ia SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The first epoch of our observation is the earliest such measurement carried out to date for any Type Ia SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The data have been analysed with detailed radiation-hydrodynamic non- LTE simulations in the framework of an off-center delayed detonation 13 The continuum polarization in the off-center DDT is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1%–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 16 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' which produces aspherical distributions in the burning products and, in particular, an aspherical 56Ni core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The observations are also compatible with the presence of a central energy source that deviates from spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The understanding of SN 2019np that we have achieved with our simulations can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (1) A viewing angle of ∼ 45◦ provides the best fit to the amplitude and temporal evolution of the polarization spectra including Si ii 𝜆6355 and the continuum (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As discussed in point (3) below, the continuum polarization at the first epoch on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 requires a separate component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2) The five epochs roughly cover the time interval in which the photosphere receded through the layers of incomplete carbon burn- ing, complete carbon burning, incomplete and complete (QSE14) ex- plosive oxygen burning, and incomplete Si burning at the interface to NSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The cadence of the observations corresponds to a resolution of ∼ 6000 km s−1 in expansion velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Higher-cadence observations than obtained for SN 2019np are required to resolve any structures in the ejecta at smaller scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For instance, considering the stratifica- tion in expansion velocity of the abundance layers and the recession speed of the photosphere, a cadence of ∼ 1 day would be essential to map out the interfaces between different chemical layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (3) The outermost ≲ 4 × 10−3 M⊙ region seen during the first ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days after the SN explosion is consistent with a C/O-rich layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The change of the polarization position angle (Figure 7) and the ro- tation of the dominant axis (Figure 8) from the first to later epochs suggest a different orientation of the outermost layer compared to the inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To account for the outer asymmetry, a separate structure had to be added to the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This renders it relatively unlikely that SN 2019np originated from a sub-𝑀Ch double detona- tion induced by a helium shell at the surface of the WD because the initial shape of the core of the WD can be expected to be symmetric, being governed by gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A deformation of the core may occur in a rapidly, differentially rotating WD with extreme specific angular momentum close to the center (Eriguchi & Mueller 1993), but is not likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (4) Although the continuum polarization on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 was only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='21%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='09%, the hydrodynamic modeling of the polarization with an evolving density profile suggests the presence of a remarkably aspherical outermost layer of the SN, comprising a fraction of ∼ (2–3) ×10−3 of the WD-progenitor mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In conjunction with the inclination of ∼ 45◦, the axis ratio of the electron density distribution predicted by the models amounts to ∼ 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' That is, in the presence of a steep density gradient in the outermost layers, a low but nonzero continuum polarization cannot be taken as an indicator of a high level of sphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (5) The rise of the continuum polarization to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='19%±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='07% about two weeks after peak luminosity is consistent with an aspherical 56Ni distribution in the core as predicted by off-center delayed-detonation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From the models, a change of the polarization position angle by ∼ 20◦ can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The time-evolving distribution of the polarimetric signals in the polar plots (Figure 8) and the rotating dominant axis on the 𝑄–𝑈 plane (Figure 7 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3) are consistent with this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Around maximum light, the off-center contribution causes a continuum polarization of ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (6) Small 𝑝cont seems to be a characteristic of off-center DDT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This justifies the assumption of negligible intrinsic polariza- tion made for the determination of the ISP (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This method differs from those commonly applied to core-collapse SNe (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The difference between the continuum polarization 𝑝cont and 14 QSE: Quasi-Statistical -Equilibrium the average polarization over the entire spectrum with many line- dominated spectral regions remains small, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1% from −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days for our normally bright off-center DDT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The differ- ence is consistent with the observations of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To understand the physical reason for being small, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Despite much stronger line polarization, a similar small value of 𝑝cont and a small difference in the continuum is found for the Type Ia SN 2004dt when analysed within the framework of off-center DDT models (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (7) The polarization observed across Si ii 𝜆6355 on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 is higher than in our model (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The dominant axes fitted to this line and the optical continuum are both not well defined at this phase (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' From our full-star models, it is hard to deduce whether the asphericity of these two components in the outermost layers has a common origin — for instance, owing to interaction with a low-mass accretion disc, which is a free parameter and remains unconstrained in the current modeling process of the full star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the toy model for the continuum polarization (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2) demands different symmetry axes for the density and the abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (8) High asphericity is found to be confined to the very outer layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' It emerges from the fitting when introduced as an additional free parameter not included in the hydrodynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Possible physical causes may include a short-lived interaction between the SN ejecta with a low-mass accretion disc (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007) or a companion star (Marietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2000), or surface burning as found in sub-𝑀Ch explosions (Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012) but see item (3), or the imprint of the burning of H/He-rich material originating from the surface (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Interaction with a donor star seems less likely because it would affect not only the outermost layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As shown in Figures 1 to 3 and discussed in Section 4, the sub-𝑀Ch model is in tension with the observation of the Ca ii NIR3 feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A similar early- time polarization has been observed in the normal-bright SN 2018gv (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, in the underluminous SN 2005ke, a significant polarization was observed in the outer ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 M⊙, hinting toward rapid rotation or a dynamical merger (Patat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (9) The continuum polarization of SN 2019np vanished by day −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 and remained consistent with zero within one 𝜎 until the SN reached its peak luminosity, indicating a high degree of spherical symmetry between the outermost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='03 MWD and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 MWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the case of a highly aspherical configuration extending into deeper layers, the continuum polarization should increase with time since the density distribution flattens significantly and the scattering opti- cal depth decreases from the optically thick regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, such an increase in continuum polarization was not observed in SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This partly invalidates the conventional assumption that a low con- tinuum polarization provides evidence of low asphericity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the presence of a steep density gradient as in the outermost layers of SN 2019np, major deviations from spherical symmetry are well pos- sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' High-cadence observations are needed to distinguish between these alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (10) The increased continuum polarization on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 can be explained by abundance asphericities in an off-center delayed det- onation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The direction of the dominant axis of SN 2019np has also changed between days +0 and +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 (Figure 7), which may indicate a small, off-center distribution of the central energy source, 5%–10% of the total amount of 56Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The model also predicts a change of the polarization position angle of the quasi-continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Although quali- tatively indicated by the observations, the change in the polarization position angle of the continuum is hard to quantify from the available observations because the intrinsic continuum polarization is very low (Section 3), and the numerous small wiggles at this low level may also cause a problem (Figures 14–16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 17 (11) None of the possible mechanisms causing the rise of the Ca ii NIR3 polarization on day +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 has been included in our sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Its origin remains uncertain, as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (12) In the optical domain, spectral line formation and polar- ization by electron scattering take place in the same region of the expanding atmosphere (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 and Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By contrast, the canonical polarization-by-obscuration picture (Wang & Wheeler 2008) requires that spectral lines are formed mainly above the last continuum-scattering surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To first order, this approximation can be used to place an upper limit on the peak polarization from large- scale asymmetries, which are produced by the large Sobolev optical depth over the entire photosphere, if the formation of the quasi- continuum is dominated by Thomson scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In SNe Ia, the Si ii 𝜆𝜆6348, 6373 doublet provides an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' It originates from a low-excitation state, and Si accounts for ∼60% of the total mass fraction corresponding to up to ∼ 2, 500 times the solar value found in the H-rich envelopes of CC-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The expansion velocities of Si- rich layers range from ∼ 9000 to more than 22,000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the presence of large-scale asphericity in Si, this line will be significantly polarized unless the region of asymmetric density or Si abundance is hidden behind an extended photospheric region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For small-scale or multiple structures, the resulting polarization depends sensitively on 𝜏sc at their location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (13) Overall, the polarization spectra and their temporal evolution can be understood as a variable thermalisation optical depth and partial blocking of the photosphere at a given geometric depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (14) There are strong high-amplitude fluctuations in the polariza- tion spectra on day −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' They can be attributed to the strongest S/Si and the Ca ii lines, in particular Ca ii NIR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The lack of similar patterns and the agreement between the observations and synthetic spectra in the spectral region dominated by iron-group elements, namely the 𝑈 and 𝐵 bands and longward of 8500 Å, seem to rule out Ni, Co, or Fe as being affected by the same mechanism(s) responsi- ble for the fluctuations just mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='. For details and their possible origin see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (15) At the same time, there are also flocculent structures in the polar diagram for Ca ii NIR3 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4, Figure 16) with a broader distribution in position angle and a lower polarization degree (see Figure 8), suggesting a more complex structure of the outermost layers than adopted in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (16) Many relatively weak polarization features within the wave- length range ∼ 4500–6000 Å and beyond 6800 Å can be interpreted as signatures of spectral line blends and unresolved wiggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The general agreement between data and model suggests that many of these weak polarization features are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Some discrepancies do not necessarily invalidate this conclusion, but likely point toward small- scale structures (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the noise in the data for SN 2019np sets a limit to probing those scales in this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (17) Polarization is able to pick up spectral signatures not visible in the flux spectra because lines depolarize but both absorb and emit photons (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Spectropolarimetry of sufficient spectral resolution can reveal spectral lines that are undetectable in flux spec- tra, and, at proper cadence, their depth of formation can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This added diagnostic power is independent of any asphericity and can be important to discriminate explosion scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (18) The asphericity in the 56Ni distribution is significant (Fig- ure 11) although the continuum polarization is relatively low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2, a low thermalisation depth results in van- ishingly low continuum polarization, except for photons that graze the photosphere at low optical Thomson-scattering depth (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As a corollary, high polarization can be expected if the off-center component dominates, but polarization may fail to detect even significant asphericity in the density produced by a Fe/Co/Ni core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (19) The polarization profiles of strong isolated features provide new diagnostics to probe for mixing on small and medium scales, and to explore the chemical stratification (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 6 FUTURE OPPORTUNITIES 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 The Diagnostic Power of Spectropolarimetry of Type Ia SNe Asphericity holds a key to the understanding of the nature of ther- monuclear SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' It involves three main components: (1) the exploding WD, (2) all other matter bound in the progenitor system, which may include a companion star and any bound CSM such as a common en- velope or an accretion disc, and the inner parts of several winds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', from the WD, a companion star or its Roche lobe, and an accretion disc), and (3) the unbound CSM consisting of the outer faster and less dense parts of the winds and ultimately the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A detailed discussion of the effects of these three constituents on the geometry is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A broad intro- duction to the geometrical signatures in polarization and nebular spectra expected from all commonly considered explosion models and progenitor channels was recently given by Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Reviews on various topics can be found among the articles collected in Alsabti & Murdin (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' a) To properly plan observing sequences, timescales are critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For all explosion paths, the initial phase of the explosion takes a few seconds to a minute, and the main spatial dimension is given by the exploding WD or the two merging WDs and ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5×108 to 109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The hydrodynamical interaction of the explosively expand- ing envelope with a companion star, any accretion disc, and the inner parts of the wind(s) takes place within ∼ 1010–1013 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Considering the velocities and masses in the outer layers of the explosion, this corresponds to timescales of minutes to about an hour for interaction with the bound matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Interaction with the wind(s) and the ISM can extend over days to many years, and is only limited by the transition to the supernova-remnant phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' b) The impact of the various components on the geometrical struc- ture and the associated polarization depends on the mass of the com- panion and any bound CSM relative to that of the ejecta which differs strongly between the explosion processes (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='6–2 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' c) The mass-loss rate from the system may range from ∼ 10−6 to 10−4 M⊙ yr−1 and can be due to any model-dependent combina- tion of the wind from the WD, the wind from a companion, super- Eddington accretion onto the WD (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Nomoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1976), the Roche lobe in a single-degenerate system, or the high-velocity wind from accretion discs in cataclysmic variables, or it may be produced during the final phase of dynamical mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Upper limits to the total mass content of these winds integrated over the time considered for early polarimetry are ∼ 10−4–10−7 M⊙ (Dragulin & Hoeflich 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Chevalier & Fransson 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Even at the earliest times when sig- nificant polarization can still be expected, the dynamical effects of the impact of the ejecta on the unbound CSM are likely to be small (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Nevertheless, in early-time photometry, some small ad- ditional blue flux may appear owing to energy released during the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For very high wind densities, the hard radiation at the shock discontinuity and the reverse shock may lead to enhanced ioni- sation in the photosphere of the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In SN 2019np, we see no obvious evidence of such effects (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) 18 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' d) The CSM bound in the system may include matter in a Roche lobe and/or an accretion disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Observational evidence has been re- ported by (for example) Aldering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2006), and, from high- velocity Ca ii absorptions in early-phase spectra, its mass has been estimated to be of order 10−2–10−3 M⊙ (Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This mass is comparable to that in the layers probed by our early polarime- try of SN 2019np, and it is compatible with the large asymmetries proposed (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The small-scale structures depend on the scale height of the material (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', the Roche lobe or the disc) and the sound-crossing time during the hydrodynamical phase of the inter- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, since the mass of the bound CSM is much larger than that contained in the wind, the structure produced by the inter- action of the ejecta with this matter can be expected to be conserved in the subsequent possible interaction with the outer winds and the ISM, although the temperatures and sound speed are likely to be high (Margutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, polarimetry will provide a unique way to explore the bound CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As discussed above (Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 & 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3), some of the wiggles in polarization spectra and flocculent structures in polar plots seen in the spectropolarimetry of SN 2019np may have their origin in Rayleigh-Taylor or Kelvin- Helmholtz instabilities and crossing shock waves produced during the injection with an orientation imprinted by the bound CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Po- larization measurements with a latency between hours and one day are needed (Figure 17) to learn whether the polarization position angle persists, which would suggest a large-scale structure, domi- nated by instabilities imprinting their characteristic size as wiggles and large-amplitude fluctuations in the polarization spectra ans well as flocculent structures in the polar diagrams, or a combination of large and small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These structures become most prominent in Ca ii NIR3, which is an excellent tracer of structure owing to its large atomic cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' e) Alternatively, the early wiggles and the flocculent structures in SN 2019np may be related to the explosion mechanism, in which case they would have an origin internal to the SN proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Poten- tial sources are explosive surface He-burning in sub-𝑀Ch explosions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Shen & Moore 2014) or in NSE-rich material rising from the central region in gravitationally confined thermonuclear explosions (Kasen & Plewa 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In both cases, some 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 M⊙ of 56Ni may reach near-surface regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This is well within the mass range that can be probed by polarization as in SN 2019np (Figure 11) be- cause the characteristics and especially the structures associated with these processes are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A central distinguishing criterion is the presence of products from low-density burning in sub-𝑀Ch explo- sions and of NSE-dominated material from high-density burning and a mixture of NSE and QSE in gravitationally confined detonations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Prompted by detections of early excess luminosity in optical light curves, Piro & Kollmeier (2018) suggested 56Ni as a possible energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At a first glance, early photometry of iPTF16abc (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2018) and SN 2019np (Sai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022) makes it plausible that both light curves can be well explained by 56Ni mixed into the outer layers of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A more detailed study has been presented by Magee & Maguire (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The authors proposed that the bump in the light curve can be understood by a Ni shell of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='02–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='03 M⊙ in the outer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 M⊙ but also noted that the colours are too blue, and the spectra would be dominated by Fe/Co/Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Since both of these side effects are not supported by the observations, the authors suggested as a remedy that the inferred 10% of 56Ni is concentrated in a small clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the observed flux and polarization spectra safely rule out 56Ni-powered early-time light curves (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Moreover, the proposed small clump is unlikely to be the correct explanation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the luminosity would be dominated by a plume resulting in a flip in polarization position angle (Figure 12) because even as little as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='02 M⊙ of 56Ni would dominate the energy input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In both the sub- 𝑀Ch and the gravitationally confined detonation, the hypothetical 56Ni would be in the surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' It would contribute 50% of the total heating and be equivalent to the energy output from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1 M⊙ of 56Co, and may not solve the blue-colour problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The latter may be an opacity effect in a rapidly expanding atmosphere instead of a heat indicator (Ashall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The presence of burned material from a He-triggered sub-𝑀Ch explosion can be clearly demonstrable with spectropolarimetry because it can detect iron-group elements down to solar metallicity (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3), which is not possible from total-flux spectra alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Furthermore, polarimetry is the ideal tool to distinguish explosions without surface burning and He-triggers in sub-𝑀Ch WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' f) Interaction of the SN ejecta with a companion star is a conse- quence of most explosion models except for the violent, dynamical, and secular mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The early bump identified in the photometry of SN 2017cbv appears blue and has been modeled by interactions with a subgiant star at a distance of 56 solar radii from the exploding WD (Hosseinzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the model also predicts a stronger ultraviolet flux than was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' All SN mechanisms with an internally triggered explosion seem to have a problem explaining the early blue excess flux and for the explanation to resort to inter- action with some CSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' An interaction with a relatively high-mass object can be expected for many progenitor systems, and it would leave its imprint not only in the surface layer but all the way down to the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' So large a structure is likely to become visible in polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' But it may also produce small-scale structures in the abundances and, depending on the donor star, the density (Marietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Interaction with a companion causes a tight connection between outer and inner regions of the expanding envelope, including a common and persistent symmetry axis and, initially, a cone with an opening angle of ∼ 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Although our modeling of SN 2019np did not include a companion, the observed change in polarization angle (Figures 1–4) probably disfavours a dominant effect of an ejecta/companion interaction on the polarization (Figure 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Any such effect depends on the size of, and distance to, the companion star, and the large-scale asymmetry imposed by a small companion can be expected to be largest in deep layers (Marietta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Furthermore, in contrast to an off-center DDT, a small-scale structure produced by Rayleigh-Taylor instabilities will occupy a cone instead of a spherical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Unfortunately, in our study of SN 2019np, we lack the cadence to resolve small scales and their distribution from the outer to the inner regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' g) For deflagration fronts, we must expect small-scale Rayleigh- Taylor instabilities and the imprint of the thermonuclear runaway, the caustic distribution of the burning processes, and magnetic fields as discussed throughout this paper (especially Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3 and 5) and by Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Polarimetry of Ca ii NIR3 is a highly sensitive tool to probe for the associated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The polarization of this triplet in the central region is strong evidence for mixing from the outside because burning to hot NSE destroys Ca (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, the huge atomic cross section of Ca ii NIR3 desensitises it to abundance effects (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' When observations of this feature and the continuum polarization with similar characteristics as ours of SN2019 np but with higher cadence are combined with high-resolution nebular spectra in the near- and mid-infrared at later epochs, a fairly complete three-dimensional model of the structure of thermonuclear SNe can be assembled (Kotak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Telesco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' h) Off-center DDT explosions of 𝑀Ch WDs can generally be thor- oughly investigated by spectropolarimetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For sub-𝑀Ch explosions, the location of the secondary ignition of the C/O core can only be ex- MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 19 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Rate of recession of the photosphere in the spherical high- resolution Model 25 as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The recession rate 𝑑𝑣phot/𝑑𝑡 for Model 25 (see Section 4) was calculated using the Rosseland mean opacity in the optical and a range of ±2 days to determine 𝑣phot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The exact values depend on the explosion model used (see Figure 9 of Quimby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' pected to show an imprint on the polarization spectrum if it happens in the regime of distributed burning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' i) For SN 2019np, we found evidence of an off-center component in the 56Ni distribution, which disfavours sub-𝑀Ch explosions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' How- ever, we also showed that large-scale asymmetries in the extended central Fe/Co/Ni region cannot be excluded (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Contrary to normal-bright SNe and SN 2019np in particular, polarimetry of underluminous Type Ia SNe provided strong evidence for signifi- cant asphericity even in deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This may favour fast rotating WDs or dynamical mergers (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A possible discriminator may be the presence of Rayleigh-Taylor instabilities in models with deflagration burning, and their absence in pure detonation models like dynamical mergers (García-Berro & Lorén-Aguilar 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As explained above, time-resolved polarimetry can establish the actual facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' j) The Introduction has discussed polarization of Type Ia SNe as a diagnostic tool, also in connection with minority events like vio- lent mergers (Pakmor 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Kushnir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Such events should exhibit prominent polarization signatures because the off-center com- ponent would dominate (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, the amplitude in the continuum polarization with time would be larger by a factor of 5–6, and large polarization can also be expected in spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 Implications for the Design of Spectropolarimetric Observing Sequences and Simulations Spectropolarimetry with adequate cadence from the earliest possible moment and (to potentially expose the EC layers) up to ∼ 3 weeks after maximum brightness is necessary to get a comprehensive pic- ture of normal Type Ia SNe on all scales and to discriminate between competing interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The SNR must be high, and the spectral resolution should be better than that of our SN 2019np data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In our observations of SN 2019np, the combination of a spectral resolving power of 𝑅 ≈ 440 and a ∼ 4–6 day cadence only achieved a reso- lution of ∼ 6000 km s−1 in comoving-frame velocity between mass elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This limited our analysis of any smaller-scale structure in the SN ejecta and the surface layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Observations with low cadence disqualify for several quantitative comparisons with models, most importantly the identification of individual clumps (predicted by Rayleigh-Taylor instabilities), layers of specific chemical elements, the interface between the Ca-rich and the inner NSE-region (Fig- ure 11), and the origin of the asphericity in the outermost layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To devise a more optimised observing strategy, the evolution of the recession velocity of the photosphere needs to be considered, which Figure 17 illustrates for Model 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Starting ∼ 1 week after explo- sion, during the rising and photospheric phases of Type Ia SNe, this velocity declines from ∼ 1000 to ∼ 200 km s−1 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Accordingly, a 2–4 day observing cadence suffices to locate the interfaces between various chemical constituents relevant for the internal structure of the explosion (items (e)–(j) above and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' With one-day cadence and a matching spectral resolving power of 1000 (corresponding to a transversal resolution of 300 km s−1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the morphology and origin of the individual structures discussed in items (e)–(j) can also be explored,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' namely the properties of the burning front including sec- ondary detonations or a DDT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' shear instabilities due to interactions with a companion star,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the imprint of the thermonuclear runaway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' the consequence of mixing of radioactive and EC elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' and possibly magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' To also determine the shape of the outermost layers of the ex- ploding WD calls for rapid-response and high-cadence observations because the radial density index governing this shape changes early and rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Some 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 days after the explosion, a one-day cadence corresponds to a radial resolution of ∼ 3500 km s−1 and 10−4 M⊙ in mass (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Therefore, such observations resolve both the bound CSM (item (d) above) and the He layer and 56Ni mass (item (e)) to be expected for sub-𝑀Ch or confined detonation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At an expansion velocity of 24,000 km s−1 (a typical value of high-velocity components in Ca ii NIR3 and the cutoff velocity of the blue wings in Si ii 𝜆6355;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Quimby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Gerardy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2007), the transver- sal resolution (which is set by the spectral resolution of 300 km s−1) exceeds the radial resolution by a factor ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' These resolutions de- scribe cones with approximate opening angles of 20◦ and 2◦ for the radial and transversal directions (respectively), and are respectively governed by the cadence and the spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='15 The apices of the cones are at the center of the WD and the numerical grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The opening angles are comparable to the angle suspended by (for example) the region affected by an interaction with a companion star, or about 20% and 2% of the photospheric radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' With such data, the interaction of the ejecta with an accretion disc, a Roche lobe (item (d) above) and a progenitor star (item (f)) can be significantly distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' They may also be sufficient to characterise the structures and underlying physical mechanism(s) which produce the large-amplitude fluctuations in the polarization profile and flocculent features in polar diagrams of strong spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The numerical models applied in this study have an effective spa- tial resolution of ∼ 600 km s−1 for clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='16 This is just sufficient to resolve the largest Rayleigh-Taylor instabilities, for example, but already appears to be higher than all SN spectropolarimetry obtained 15 The opening angles are given by the arcsin of the structure size divided by the photospheric radius in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 16 The effective resolution is estimated from the grid resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', the domain size (2 × 25, 000 km s−1) divided by the number of grid points (330) in a differential scheme (contributing a factor of ∼ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' For spherical clumps as the simplest structures, it is multiplied by another factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This results in 75 km s−1 × 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Model 25 4 3 2 d 0 0 10 20 30 Days Since Explosion [days]20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Higher-resolution models to resolve small-scale structures can be achieved with sub-star instead of full-star simulations or, bet- ter, a space-domain implementation17 for our Variable Eddington Tensor solver in HYDRA as discussed by Hristov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2021, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Investigation of the effects caused by details of the thermonuclear runaway in a 𝑀Ch explosion or any secondary C/O ignition would require spectropolarimetry at late times in com- bination with late-time near- and mid-infrared flux spectra (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Spectropolarimetry will provide a tomographic sam- pling of the geometric properties of the layer near the Si/Fe interface in the inner regions of the WD remnant, while nebular flux profiles at sufficient spectral resolution will probe the physical conditions and kinematics of this region through unblended line profiles, which are also sensitive to the aspect angle of the observer (Hoeflich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The combination of these two datasets will subject models to critical consistency checks because they investigate the same re- gion from different perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Analogous simulations should be employed to test other models and make use of hydro-dynamical simulations to investigate points (e)–(j) in the list above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A detailed discussion of many other scenarios is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Obviously, similar simulations can be applied to a wide variety of explosion scenarios and other transients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Leonard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Maund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Dessart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Buckley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2021), and those will be performed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' By using details previously not considered in combination with extensive modelling, we obtained new insights into the formation of 𝑝 and obtained many results for SN 2019np in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' However, we pushed the analysis to the limit of current data and modelling efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This study should be regarded as a pathfinder for a new approach to the analysis of SNe Ia data, to evaluate the limits and to identify the potential and shortcomings of current observations and theoretical models, to evaluate methods to correct for the ISM, and to develop future polarization programs (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Acknowledgements: We are grateful to the European Organisation for Astronomical Research in the Southern Hemisphere (ESO) for the generous allocation of observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' We especially thank the staff at Paranal for their proficient and highly motivated support of this project in service mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' acknowledges support by the National Science Foundation (NSF) through grant AST-1715133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='F.’s supernova group at U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Berkeley is grateful for financial assistance from the Christopher R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Redlich Fund and many individual donors, including Gary and Cynthia Bengier, Clark and Sharon Winslow, Sanford Robertson, Sunil Nagaraj, Landon Noll, and Sandy Ottelini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The research of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' has also been supported through a Benoziyo Prize Postdoctoral Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Facilities: The observations were obtained with FORS2 and the Very Large Telescope at the European Southern Observatory’s La Silla Paranal Observatory in Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The simulations have been performed on the computer cluster of the astro-group at Florida State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Software: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', under cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' HYDRA and various modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' OpenDx, an open-source graphics package by IBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 17 As part of an ongoing PhD project, the domain deposition is currently being implemented using “pancake slicing” and cycling between the three Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Data Avaibility Statement: All data are available on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' REFERENCES Aldering G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 2006, ApJ, 650, 510 Alsabti A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Murdin P.' metadata={'source': 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+page_content=', Mar- tel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', eds, American Institute of Physics Conference Series Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 586, 20th Texas Symposium on relativistic astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' pp 459–471, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='1419593 MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 21 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Log of spectropolarimetry of SN 2019np .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Epoch Object MJD Date Phase𝑎 Exposure Grism / Airmass (UT) (day) (s) Resol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Power Range 1𝑏 SN 2019np 58495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='270 2019-01-12 06:30 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 4×550 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='84–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='73 58495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='298 2019-01-12 07:10 4×550 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='73–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70 2 SN 2019np 58498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='302 2019-01-15 07:14 −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 4×600 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='71–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70 3 SN 2019np 58503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='305 2019-01-20 07:18 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='4 4×210 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='71 HD 93621𝑐 58503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='322 2019-01-20 07:43 – 1×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='51 300V/440 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='13 4 SN 2019np 58510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='272 2019-01-27 06:31 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 4×360 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='71–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70 5 SN 2019np 58524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='188 2019-02-10 04:31 +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='5 4×360 300V/440 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='71–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='70 𝑎 Relative to the estimated peak on MJD 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7 / UT 2019-01-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MJD and Date are given as the start time of the CCD exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 𝑏Epoch 1 observation consists of two sets of exposures at four half-wave plate angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 𝑐Flux standard, observed at a half-wave plate angle of 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Values of least-squares fitting parameters on the 𝑄–𝑈 plane for SN 2019np #Epoch 𝑄cont 𝑝cont 𝛼 / 𝛼∗ 𝜃𝑑 / 𝜃∗ 𝑑 𝛼Si II𝜆6355 𝜃Si II𝜆6355 𝑑 𝛼Ca II NIR3 𝜃Ca II NIR3 𝑑 Phase𝑎 𝑈cont [%] 𝛽 / 𝛽∗ (deg) 𝛽Si II𝜆6355 (deg) 𝛽Ca II NIR3 (deg) # 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='194±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='209±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='087 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='061±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='142 / +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='478±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='131 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='8+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='9 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2 / −33.' metadata={'source': 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B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Makhviladze G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', Sivashinskil G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 1970, Journal of Applied Mechanics and Technical Physics, 11, 264 APPENDIX A: EXPLOSION TIME OF SN 2019np In order to estimate the phases of the FORS2 observations relative to the explosion time of SN 2019np, we modeled the early flux of SN 2019np as a function of time by fitting its 𝑟-band flux with a power law, 𝑓 (𝑡) ∝ (𝑡 − 𝑡0)𝑚 , (A1) where 𝑡0 denotes the time of the first light from the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' This formalism assumes that the WD exploded as an expanding fireball with constant temperature and velocity (Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The ear- liest photometric dataset of SN 2019np consists of the SDSS 𝑔- and 𝑟-band light curves generated by the Zwicky Transient Facility (ZTF) alert packets (Patterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Owing to the lack of 𝑔-band pho- tometry at the earliest phases of SN 2019np, we fitted the power law to the 𝑟-band flux, for which we considered two subsets, namely ob- servations before days ∼ −14 and ∼ −12, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The fitting is also constrained by the last nondetection on day −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' As shown by the filled black and open orange circles in the upper-left panel of Figure A1, after arbitrarily scaling the 𝑔 and 𝑟 light curves of SN 2019np, the evolution of the flux in both bandpasses is consistent between days −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7 and −12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7, when photometry is available for both filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The best fit to the observations before day −14 gives 𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04 and a rise time 𝑡0 = −17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The sta- tistical error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='06 day in 𝑡0 is much smaller than the systematic uncertainty of the time of the 𝐵-band light-curve maximum, which amounts to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='51 day in our analysis and has to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Figure A1 also compares the flux evolution of SN 2019np to its fit with Equation A1 and includes data for selected other SNe with well-sampled photometry at similarly early phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The 𝑔-band light curve of the normal Type Ia SN 2011fe is well approximated by an expanding fireball model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=', 𝑚 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Nugent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Two cases with an early flux excess, namely SN 2017cbv (Hosseinzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2017) and iPTF16abc (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' 2018), exhibit a fast rise within the first ∼ 5 days after the explosion and favour a power-law index around unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' At 𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='04, SN 2019np is similar to both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022) Geometry of Type Ia SN 2019np 23 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' Best-fit 𝑓 ∝ (𝑡 − 𝑡0)𝑚 model to describe the early flux evolution of SN 2019np compared to that of selected other SNe with well-sampled early photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' All flux distributions are normalised to the peak magnitude measured for each SN in the given bandpasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the upper-left panel, the black and the brown solid lines fit the 𝑟-band flux (filled-black circles) of SN 2019np before −14 and −12 days relative to the 𝐵-band maximum on MJD 58509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' In the inset, the inner to outer contours represent the 1𝜎, 2𝜎, and 3𝜎 confidence levels of the power-law parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The open orange circles mark the 𝑔-band photometry of SN 2019np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The residuals of the fits to the 𝑟-band light curves before −14 and −12 days are shown by the black and brown dots, respectively, in the bottom-left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The upper-right panel compares the fit of SN 2019np to the 𝑔 light curves of SNe 2017cbv, 2011fe, and iPTF16abc within the first four days after explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' SN 2019np exhibits a similar power-law index as SN 2017cbv and iPTF16abc, for which a blue excess has been identified within the first ∼ 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' The residuals are shown in the bottom-right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} +page_content=' (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf'} diff --git a/RtE0T4oBgHgl3EQf1wIT/content/2301.02702v1.pdf b/RtE0T4oBgHgl3EQf1wIT/content/2301.02702v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ee46e329b473b495f1f1029cdb6a17bfeb126e7a --- /dev/null +++ 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sha256:620b8ddece9a157a46faf6d99ca7ef6275dfde4f4bdf83ac72ebf9f7944dea4d +size 150774 diff --git a/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/2301.05168v1.pdf.txt b/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/2301.05168v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d6e2929fcd39d2c336785558ab0a5f62a88a4ab0 --- /dev/null +++ b/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/2301.05168v1.pdf.txt @@ -0,0 +1,2260 @@ +1 +A Novel Modular, Reconfigurable Battery Energy +Storage System: Design, Control, and +Experimentation +Amir Farakhor, Student Member, IEEE, Di Wu, Senior Member, IEEE, Yebin Wang, Senior Member, IEEE, and +Huazhen Fang, Member, IEEE +Abstract—This paper presents a novel modular, reconfigurable +battery energy storage system. The proposed design is charac- +terized by a tight integration of reconfigurable power switches +and DC/DC converters. This characteristic enables isolation of +faulty cells from the system and allows fine power control for +individual cells toward optimal system-level performance. An +optimal power management approach is developed to extensively +exploit the merits of the proposed design. Based on receding- +horizon convex optimization, this approach aims to minimize +the total power losses in charging/discharging while allocating +the power in line with each cell’s condition to achieve state-of- +charge (SoC) and temperature balancing. By appropriate design, +the approach manages to regulate the power of a cell across its +full SoC range and guarantees the feasibility of the optimization +problem. We perform extensive simulations and further develop +a lab-scale prototype to validate the proposed system design and +power management approach. +Index Terms—Battery management systems, cell balancing, +convex optimization, reconfigurable battery energy storage sys- +tems (RBESSs). +NOMENCLATURE +Variables +S +Reconfiguration switches–binary (1/0) variable +V ∗ +t +RBESS reference output voltage +V max +C +Maximum output voltage of DC/DC converters +Imax +C +Maximum output current of DC/DC converters +v +Cell voltage +u +Cell open-circuit voltage +iL +Cell current +Pb +Cell internal power +P +Cell output power +E +Cell energy +Pl +Module power losses +J +Total power losses +Pout +RBESS output power +Iout +RBESS output charging/discharging current +˙Qcnd +Conductive heat transfer rate +This work was supported in part by the U.S. National Science Foundation +under Awards CMMI-1763093 and CMMI-1847651, and in part by the U.S. +Department of Energy, Office of Electricity through the Energy Storage +Program. +A. Farakhor and H. Fang (corresponding author) are with the Department of +Mechanical Engineering, University of Kansas, Lawrence, KS, USA (Email: +fang@ku.edu, a.farakhor@ku.edu). +D. Wu is with the Pacific Northwest National Laboratory, Richland, WA, +USA (Email: di.wu@pnnl.gov). +Y. Wang is with the Mitsubishi Electric Research Laboratories, Cambridge, +MA, USA (Email: yebinwang@merl.com). +˙Qconv +Convective heat transfer rate +ξ +Slack variable +z +Optimization variables +Parameters +n +Number of battery cells +ns +Number of cells in series +np +Number of cells in parallel +J +Set of in-service cells +L +Inductor for DC/DC converters +C +Output capacitor for DC/DC converters +¯Q +Cell capacity +q, qmin, qmax SoC of a cell, and its lower and upper limits +qavg +Average SoC +∆q +SoC imbalance tolerance +∆E +Energy imbalance tolerance +α +Intercept coefficient of the SoC/OCV line seg- +ment +β +Slope coefficient of the SoC/OCV line segment +R +Cell internal resistance +RC +Resistance to capture the power losses of +DC/DC converters +Rcnd +Conductive thermal resistance +Rconv +Convective thermal resistance +Cth +Thermal capacitance +T, T max +Cell temperature and its upper limit +Tavg +Average temperature +Tenv +Environmental temperature +∆T +Temperature imbalance tolerance +λ +Penalty weight for the multi-objective opti- +mization +∆t +Sampling time +H +Optimization horizon +I. INTRODUCTION +L +ITHIUM-ION battery energy storage systems (BESSs) +have proven themselves as an enabling technology for +various applications, including electric cars, electric aircraft, +smart grid, and space systems[1–4]. Despite their high energy +density and long cycle life, lithium-ion batteries suffer from +safety risks, which trace to the high reactivity of lithium +and flammability of the commonly used electrolyte solutions +and are exacerbated by side reactions, aging, and degradation +[5]. Hence, it is imperative to ensure their safe and reliable +arXiv:2301.05168v1 [eess.SY] 12 Jan 2023 + +2 +operation, particularly in safety-critical applications [6]. Re- +configurable BESS (RBESS) have attracted much attention as +a promising means to achieve this end. An RBESS characteris- +tically uses power electronics switches to make the connection +among the constituent cells reconfigurable, providing the capa- +bility to bypass faulty cells without interrupting the operation +of the system [7]. This feature overcomes the vulnerability of +conventional hardwired BESS to single-cell failures due to the +fixed configuration [8]. This paper proposes a novel RBESS +design that integrates reconfigurable switches with DC/DC +converters. The new design uses the switches to reconfigure the +connectivity of the cells for the sake of safety and meanwhile, +leverages the converters to achieve robust power management +and supply from cell to system level. Further, we present an +optimal power management algorithm and develop a lab-scale +prototype to validate the proposed RBESS design and control. +A. Literature Review +This paper centers around the RBESS circuit design and +power management. Therefore, we survey the literature on the +two dimensions one by one. The review will also encompass +some recent studies about hardwired BESS due to the rele- +vance. +1) Review of RBESS circuit design: The literature has pre- +sented two main ways to design RBESS circuit architectures. +The first one builds and integrates a circuit of controllable +power electronics switches with the cells. By controlling the +switches, a cell can be put into or cut off from the connection +with other cells when a fault occurs. The circuit topology +plays a key role in the level of reconfigurability, functional +flexibility, and circuit complexity. The study in [9] shows a +switching circuit for a series-connected battery pack, which +uses only two switches for every cell. More sophisticated +topologies can provide more versatile reconfiguration, though +at the expense of using larger numbers of switches. The work +in [10] considers a string of modules of parallel-connected +cells in series and allows switch-based bypass of any cell +or module. The circuit topologies proposed in [11, 12] place +five and six switches, respectively, around each cell to realize +arbitrary series and parallel connection among the cells, and +the one shown in [13] uses only four to achieve the same +end. It is interesting to point out that reconfigurable switching +circuits can enable more functions than just isolating faulty +cells. A battery system designed in [14] dynamically connects +some battery modules in series or parallel to produce multi- +level and even AC voltage output. In [15], an inverter based on +switching circuits combines batteries and supercapacitors for +hybrid energy storage. While easily reconfigurable, switching +circuits are unable to do cell-level charging/discharging power +regulation. This limitation will lead to unbalanced use of the +cells following the reconfiguration-based bypass of a cell. +Another important way for RBESS design uses converters. +A converter can not only connect a cell into the circuit, but also +charge or discharge it at a controlled current, voltage, or power. +This capability offers a higher degree of freedom for cell-level +control to enhance the balanced use of the cells. In [16], a +centralized converter interfaces with a few cells, and it contains +a selector to select and put the cells into operation. The work in +[17] pairs converters, which by design include power switches +to offer a bypass mode, with individual cells one by one to +form a reconfigurable pack. However, converter-based RBESS +architectures are unable to offer flexible topology changes, +compared to the switching circuits. As such, only series- +connected RBESS is considered in [16, 17]. Converters have +also found their way into hardwired BESS. The studies in [18– +20] integrate DC/DC converters with the cells to enable cell +balancing and power loss minimization. However, the fixed +hardwired connections among the cells make them susceptible +to single-cell faults. +2) Review of RBESS control and power management: For +a given switching-circuit-based RBESS, an essential question +is how to control its topology. A main approach lies in +finding out a connection topology to meet load requirements. +For example, a heuristic method in [10] groups battery cells +into modules and chooses a minimum set of modules on a +load request to perform charging/discharging. The method +of dynamic programming finds use in identifying an opti- +mal configuration topology in [11], and rule-based bypassing +mechanisms are proposed in [12] to control the switches for +stable or responsive voltage supply. The study in [13] exploits +the switching circuit to achieve cell equalization through a +hierarchical strategy that combines intra-module equalization +and system-level reconfiguration. Considering the switching +circuits in [9, 10], the study in [21] proposes to sort cells +according to their capacity and then reconfigure them into +serial strings to restrain the cell imbalance. +On a related note, the literature has included a few stud- +ies about power management for hardwired BESS based on +converters. For converter-based RBESS, the main subject of +investigation is developing control approaches to manage the +cell-to-system-level operation. Optimal control has shown as +a useful solution in this regard. However, due to a lithium- +ion battery’s nonlinear characteristics, optimization problems +posed for the power management are often non-convex, and +thus defy the computation of global optima. To mitigate the +issue, the studies in [18, 19] choose to leverage a battery model +convexification approach proposed in [22] to formulate convex +optimal power control problems. The convexification therein +involves a linear approximation of the relationship between +the state-of-charge (SoC) and the open-circuit-voltage (OCV), +thus restricting the proposed methods to only limited operating +ranges. It is also possible to use a linear battery model to +achieve optimal control that even admits closed-form solutions +[23], but the method is also just applicable to operating ranges +in which the linear model is accurate. +As another valuable approach, distributed power manage- +ment treats the cells as independent agents and makes them +perform individual control toward a global goal. One can +decompose a global optimization task and distribute it among +individual cells for charging or load sharing, as shown in +[20]. As an alternative way, distributed consensus control is +applied in [24] to achieve SoC balancing among the cells. +Not requiring numerical optimization, this method offers fast +computation. But the lack of optimality makes it in need of +more time to converge. + +3 +TABLE I +CHARACTERISTICS OF DIFFERENT BESS STRUCTURES. +Switch-based RBESS +Converter- +based +RBESS +Converter-based +hardwired BESS +Proposed design +[9] +[10] +[11] +[12] +[13] +[14] +[15] +[16] +[17] +[18] +[19] +[20] +Safety (Bypass of faulty cells) +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +× +× +× +✓ +Output voltage regulation +× +× +× +× +× +× +× +✓ +✓ +× +× +× +✓ +SoC balancing +✓ +✓ +✓ +✓ +✓ +× +✓ +✓ +✓ +✓ +✓ +✓ +✓ +Temperature balancing +× +× +× +× +× +× +× +× +× +✓ +✓ +✓ +✓ +State-of-health balancing +× +× +× +× +× +× +× +× +× +× +× +× +✓ +Optimal cell-level power control +× +× +× +× +× +× +× +× +× +✓ +✓ +✓ +✓ +Compatibility with cells of different types +× +× +× +× +× +× +✓ +× +✓ +× +✓ +✓ +✓ +Flexibility in series/parallel connection +L +M +H +H +H +M +L +L +L +× +× +× +H +L: Low, M: Medium, H: High +Table I summarizes the main BESS architectures in the +literature for comparison. In this context, we propose the +presented work to improve the state of the art. +B. Statement of Contributions +Despite recent advances, the RBESS technology remains far +from reaching a level of maturity in both design and control. +The existing RBESS architectures use either only switches +or only converters to enable good reconfigurability or good +power regulation capability, but not both. There is also a lack +of power control approaches sophisticated enough to maximize +the operating performance of RBESS. To overcome these +substantial shortcomings, our study presents the following +specific contributions. +1) We propose a new modular, reconfigurable power electron- +ics architecture for RBESS. Differing from the existing +ones, the proposed architecture, for the first time, integrates +power switches with DC/DC converters for a combination +of their respective merits, while using the fewest number of +switches for each cell to our best knowledge. Among the +various benefits that the proposed design brings are high +reconfigurability—arbitrary bypass and parallel or series +connection among the adjacent cells—and high regulata- +bility in power supply from cell to system level to satisfy +exogenous power demands, even under the occurrence of +cell failures, and ensure equal use of the cells simultane- +ously. +2) We develop a power management approach based on +optimal control for the proposed RBESS to minimize the +system-wide energy loss while supplying demanded power +and equalizing the cells in SoC and temperature. Compared +to the prior methods, our approach is different in two +aspects. First, we adopt piecewise linear SoC/OCV ap- +proximation in the battery model convexification to enable +control across low to high SoC regions. Second, previous +methods may be subject to infeasibility when the cells’ +actual conditions make the pre-set operating constraints +unsatisfiable. Here, we introduce slack variables to relax +the optimization problem to ensure the feasibility and +practical applicability of our approach. Finally, we present +a reconfiguration method to modify the switching circuit +topology after a faulty cell is isolated. +3) We develop an experimental prototype and conduct a series +of experiments to validate and assess the performance +of the proposed RBESS. The experiments involve var- +ious scenarios under non-uniform cell conditions, fault +occurrence, and reconfiguration. The results demonstrate +the effectiveness of the proposed architecture and power +management approach. +Based on the above contributions, our RBESS design +presents significant advantages to distinguish itself from the +literature, as shown in Table I. A preliminary conference +version of this work appeared in [25] to report the RBESS +architecture design. Here, we introduce substantial expansions +in optimal power management and experimental validation. +C. Organization +The remainder of the paper is organized as follows. Section +II presents the power electronics architecture of the proposed +RBESS and discusses its unique features. Section III develops +the optimal power management approach, which covers the +modeling, optimization problem formulation, and convexifica- +tion. This section also presents a switching circuit reconfigura- +tion method. Section IV provides extensive simulation results, +and Section V proceeds to develop an experimental prototype +to validate the proposed RBESS design and control approach. +Finally, Section VI concludes the study. +II. ARCHITECTURE OF THE PROPOSED RBESS +This section elaborates the proposed RBESS. Fig. 1 shows +the power electronics architecture. As illustrated, every cell is +connected with a DC/DC converter to make up a module. Here, +we employ synchronous DC/DC converters, even though other +types of converter topologies are also allowed. The converter, +which comprises two power switches, an inductor, and an +output capacitor, provides bi-directional power processing to +control the charging and discharging of the cell. As such, +the module has the capability of cell-level power control. +The modules are connected via a switching circuit, which +places three power switches between every two adjacent +modules. With an appropriate reconfiguration of the switches, +the switching circuit can bypass a module subject to faults and +achieve arbitrary series or parallel connection among adjacent + +4 +S12 +C1 +S11 +S13 +Q11 +Q12 +L1 +S22 +C2 +S21 +S23 +S(n-1)2 +Cn-1 +S(n-1)1 +S(n-1)3 +Q21 +Q22 +L2 +Q(n-1)1 +Q(n-1)2 +Ln-1 +Cn +Qn1 +Qn2 +Ln +Terminal (Load/Charger) +Cell 1 +Cell 2 +Cell n-1 +Cell n +Module 1 +A +Current +Sensor +SoC Estimation +Measurement Bus (Cell voltages, currents, and temperatures) +Fault Detection +Optimal Power Management +Fault status +PWM control signals +to DC/DC converters +Local Controller #1 ++ +- +PI +Local Controller #2 +Local Controller #n +Control signals to +reconfiguration switches +Optimal cell currents +Output power +SoC of cells +BMS +Fig. 1. The proposed modular DC/DC converter-integrated RBESS. +modules. We label the three switches connecting modules i +and i + 1 as Sij for j = 1, 2, 3. For switch Sij, Sij = 1 when +it is on, and Sij = 0 when it is off. To bypass and isolate cell +i for 1 ≤ i ≤ n − 1 from the battery pack, we let Si1 = 1 +and Si2 = Si3 = 0. To bypass cell n, we let S(n−1)2 = 1 and +S(n−1)3 = S(n−1)1 = 0. Modules i and i + 1 are configured +in series when Si1 = Si2 = 0 and Si3 = 1, and in parallel +when Si1 = Si2 = 1 and Si3 = 0. +The proposed design uses only 3(n − 1) power switches +for n cells. To our knowledge, this is more economical than +any other RBESS design in the literature to provide the same +level of reconfigurability. The circuit simplicity further results +in convenient operation and reconfiguration. Specifically, a cell +requires only one switch to be on for the series and bypass +configurations, and a parallel configuration needs only two +switches to be on, as seen from above. +Fig. 1 illustrates how the proposed design dovetails with the +battery management system (BMS). The BMS adopts a two- +layer control strategy. At the higher level, the optimal power +management generates optimal charging/discharging currents +for the cells; at the lower level, PI-based local current mode +controllers perform reference tracking for the cell currents, as +is common in control of DC/DC converters [26]. +Our proposed RBESS can provide high reconfigurability +and control flexibility, which lead to distinct benefits for +practical applications. A summary of the major ones is as +follows. +1) The proposed design allows to bypass and isolate any faulty +module. Hence, the battery pack can continue to operate +rather than shut down as a whole, despite safety threats and +anomalies. Further, following the bypass of a module, the +switching circuit can reconfigure to redirect the power flow +and share the load equally among the remaining in-service +cells to promote balanced use of them. +2) In the proposed design, the embedded DC/DC converters +take on responsibility for the external power electronic +devices in the conventional designs to control the charg- +ing/discharging of the cells. The DC/DC converters would +yield useful functions with their capability of power con- +version and control. First, they can regulate their output +voltage so that the RBESS can supply desired or refer- +ence voltage. The voltage supply can remain consistent +before and after a fault-induced reconfiguration. Second, +the converters apply individual current or voltage control +to the cells, thus making it possible to customize and +optimize the charging/discharging for each cell based on its +present condition. One can translate this strength into cell +balancing, e.g., by charging (resp., discharging) the cells +with high SoC less (resp., more) relative to the cells with +low SoC. It is also viable to balance the cells in temperature +and state-of-health. +3) Even though beyond the scope of this paper, the proposed +design can accommodate the heterogeneity of the cells. For +example, one can leverage it to integrate heterogeneous +cells from different manufacturers or even based on dif- +ferent electrochemistries to form a workable system. In +a similar vein, the design can potentially enable hybrid +energy storage consisting of battery cells, supercapacitors, +and even solar cells. +Note that the use of the embedded DC/DC converters can +mitigate the use of exogenous power electronics devices, and +that one can connect a string or pack of cells, rather than a +single cell, to each DC/DC converter in practical adoptions. +These factors would help make the overall implementation cost +of the proposed RBESS manageable. Finally, the diverse new +functions and safety improvements brought by the design are +valuable especially for high-stakes, safety-critical applications +including electric vehicles and aircraft. +III. MODELING AND OPTIMAL CONTROL OF THE +PROPOSED RBESS +This section investigates optimal power management for +the proposed RBESS. We will begin with the electrical and +thermal modeling for the modules. We then proceed to present +a model-based optimal control problem for the RBESS and + +5 +convexify it for computational tractability. Here, the model- +ing and optimal control formulation are a refinement of the +methodology in [18, 19], and improvements are introduced to +expand the operating SoC range of the RBESS and ensure the +feasibility of the optimization problem. Finally, we present +a switching circuit reconfiguration mechanism that dovetails +with the power management method. +A. Electrical and Thermal Modeling +Consider the proposed RBESS consisting of n modules. +Each module includes a cell, a DC/DC converter, and three +power switches connected in cascade, as shown in Fig. 2 (a). +We denote the set of all the in-service modules as J . The +electrical model of module j for j ∈ J is schematically shown +in Fig. 2 (b). It includes two parts. The first part is the Rint +model to describe the cell’s electrical dynamics [27], which +comprises an OCV source uj in series with an internal resistor +Rj. The model’s governing equations are: +˙qj(t) = − 1 +¯Qj +iLj(t), +(1a) +vj(t) = uj(qj(t)) − RjiLj(t), +(1b) +where vj, iLj, uj, ¯Qj, and qj are the terminal voltage of +the cell, applied current in Ampere, OCV, capacity, and SoC, +respectively. The internal power of the battery cell is given by +Pbj = uj(qj(t))iLj(t). +(2) +The DC/DC converter is modeled as an ideal DC/DC +transformer along with a series resistor RC to capture power +losses. The reconfiguration switches are also modeled by the +ideal switches with series resistors RSji. For module j, we +assume that we can collect the power losses on RSji for +i = 1, 2, 3 in a single resistor RSj. Thus, the module’s output +power Pj can be calculated as +Pj = uj(qj(t))iLj(t) − (Rj + RC + RSj)i2 +Lj(t), +(3) +where Rji2 +Lj(t), RCi2 +Lj(t), and RSji2 +Lj(t) represent the inter- +nal power losses of the cell, the converter, and the reconfigu- +ration switches, respectively. +We adopt a lumped thermal model in [18] to describe the +thermal dynamics of module j. The thermal model is depicted +in Fig. 2 (c). The model captures the heat transfer due to the +convection between module j and the environment, ˙Qconv,j, +and the conduction between module j and its adjacent cells, +˙Qcnd,j. Meanwhile, the power loss caused by the internal +resistor, Rji2 +Lj, translates into heat generation, which becomes +the main heating source. Combining all, the thermal model is +governed by +Cth,j ˙Tj(t) = Rji2 +Lj(t) − ˙Qcnd,j − ˙Qconv,j, +(4a) +˙Qconv,j(t) = (Tj(t) − Tenv)/Rconv, +(4b) +˙Qcnd,j(t) = (2Tj(t) − Tj+1(t) − Tj−1(t))/Rcnd, +(4c) +where Tj and Tenv are the cell’s and environmental temper- +atures, respectively. In addition, the term Cth,j represents the +thermal capacitance of the cell; Rcnd and Rconv are the thermal +resistances between neighboring cells and between cell j and +Qj1 +Qj2 +Cj +Lj +Sj1 +Sj2 +Sj3 +(a) +Sj1 +Sj2 +Sj3 +Battery Cell +DC/DC converter +(b) ++ +- ++ +- ++ +- ++ +- +Conduction +Conduction +Convection +(c) +Fig. 2. The proposed DC/DC converter-integrated cell module. (a) The circuit +structure of the proposed module. (b) The electrical model of the proposed +module. (c) The thermal model of the cell j. +the environment, respectively. Here, Rconv depends inversely +on the external surface area of the cell Aj and the convective +heat transfer coefficient between the cell’s surface and the +environment h. For instance, one can consider a parallel forced +air cooling approach for the proposed design [28], which +allows every cell to experience the same amount of cooling +air. +The above electro-thermal model is concise but expressive +and computationally efficient. Putting them together for all +the modules, one can obtain a complete description of the +dynamics of the RBESS, which allows us to perform optimal +power management design subsequently. +B. Problem Formulation +The aim of our RBESS power management is to distribute +the power load among the cells so that the power losses +can be minimized under some key safety, balancing and +power demand satisfaction constraints. To begin with, we will +formulate the optimization problem. +The total power losses of the RBESS can be expressed by +J(t) = +� +j∈J +(Rj + RC + RSj)i2 +Lj(t). +(5) +We use the following objective function to encompass the total +power losses over a horizon: +� H +0 +J(t)dt, +(6) + +6 +where H is the planning horizon length. For the sake of safety, +we require each cell to operate within some favorable current, +SoC, and temperature ranges: +imin +Lj ≤ iLj ≤ imax +Lj , +(7a) +qmin +j +≤ qj ≤ qmax +j +, +(7b) +Tj ≤ T max +j +, +(7c) +where imin/max +Lj +, qmin/max +j +, and T max +j +are the lower/upper safety +bounds for the current, SoC, and temperature of cell j, respec- +tively. It is important to note that imin +Lj can be set to be zero +as the zero current means the bypass of the module. Further, +we impose the following SoC and temperature balancing +constraints to equalize the cells and make an even usage of +them: +|qj(t) − qavg(t)| ≤ ∆q, +(8) +|Tj(t) − Tavg(t)| ≤ ∆T, +(9) +Here, qavg(t) and Tavg(t) represent the average SoC and tem- +perature of all the cells that belong to J . They are calculated +as +Xavg(t) = +1 +card(J ) +� +j∈J +Xj(t), +where X = q and T, and card(J ) is the cardinality of J . The +SoC and temperature thresholds ∆q and ∆T determine the +tolerated deviation of each cell’s SoC and temperature from +the average. Here, ∆q and ∆T are tunable parameters, and +one can tune them to meet the SoC and temperature balancing +requirements for a specifically considered application. While +lower ∆q and ∆T values force a more balanced SoC and +temperature distribution among the cells, higher values allow +more deviation for the cells’ SoC and temperature from the +average. To make the RBESS meet the power demands, we +present the following output power satisfaction constraint: +� +j∈J +Pbj − (Rj + RC + RSj)i2 +Lj = Pout, +(10) +where Pout is the total power demanded of the RBESS. +Summing up the above, our power management approach +is based on addressing the constrained nonlinear optimization +problem as follows: +min +iLj ,j∈J +� H +0 +J(t)dt, +s.t. +(1b), (4a), (7) − (10). +(11) +This optimization problem pursues predictive minimization of +the power losses while complying with the constraints that +promote safety, SoC and temperature balancing, and power +supply-demand match. Note that the optimization problem (11) +is non-convex due to the nonlinearity of the equality constraint +(10). Thus, the solution to this problem is neither trivial nor +computationally cheap. To overcome the issue, we relax the +problem slightly to formulate a convex optimization problem, +as suggested in [22]. The convexification is described in detail +as follows. +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +SoC +3.2 +3.3 +3.4 +3.5 +3.6 +3.7 +3.8 +3.9 +4 +4.1 +4.2 +4.3 +OCV (V) +Line 1 +Line 2 +Line 3 +Fig. 3. +The SoC/OCV curve of the simulated cells and the multi-segment +linearization. +C. Convex Problem Formulation +For the sake of convexification, we begin with linearizing +the SoC/OCV curve. The existing studies, e.g., [18, 19] per- +form the linearization for only the medium SoC range, where +the OCV is closely linear with SoC for lithium-ion batteries. +However, this treatment excludes the use of the low and high +SoC ranges. To address the issue, we introduce multi-segment +linearization based on different SoC ranges to approximate the +complete SoC/OCV curve: +uj(qj(t)) = αi +j(qj(t)) + βi +j(qj(t))qj(t), +(12) +where αi +j and βi +j are the intercept and slope coefficients of the +i-th line segment for cell j. Fig. 3 illustrates an example of +the linearization, where the SoC/OCV curve taken from a real +cell is approximated by three line segments. Differing from +the literature, the αi +j and βi +j values are SoC-dependent, and +the multi-segment linear approximation spans the SoC/OCV +curve from 0 to 100% SoC. To ease the notation, we will drop +the superscript i from αi +j and βi +j in the sequel without causing +confusion. Next, we present a convex model by introducing +the notion of accumulated energy Ej to take the place of SoC. +The accumulated energy of a battery cell can be expressed as +Ej(t) = 1 +2Cju2 +j(qj(t)) − E0 +j , +(13) +where Cj = ¯Qj/βj and E0 +j = +1 +2Cju2 +j(qj(0)) is the initial +energy. Inserting (12) to (13) and using (1b), the dynamic +equation of the cell’s accumulated energy can be derived as +˙Ej(t) = −Pbj. +(14) +In the above, we extract a desirably linear dynamic model +to represent the evolution of Ej(t) driven by Pbj. Based on +(14), we will reformulate the optimization problem to be one +with respect to Pbj, as will be seen later. Proceeding forward, +we consider module j’s power loss, Plj(t), which can be +expressed in terms of Pbj as +Plj(t) = +(Rj + RC + RSj)CjP 2 +bj(t) +2(Ej(t) + E0 +j ) +. +(15) +As our optimization goal is to minimize the total power loss, +(15) serves as an equality constraint. Since Plj(t) is not a +linear function of Pbj, the resulting optimization problem + +7 +would be non-convex due to the nonlinearity of (15). Thanks +to the fact that the objective function minimizes the total loss +of the battery pack, we can relax (15) to comply with the +convexity requirement +Plj(t) ≥ +(Rj + RC + RSj)CjP 2 +bj(t) +2(Ej(t) + E0 +j ) +, +(16) +by which the optimization problem will practically reduce +Plj(t) to its lower bound. The safety constraints (7a)-(7b) can +also be reformulated in terms of the Pbj and Ej as follows: +� +2 +Cj +(Ej + E0 +j )imin +Lj ≤ Pbj ≤ +� +2 +Cj +(Ej + E0 +j )imax +Lj , +(17a) +1 +2Cju2 +j(qmin +j +(t)) ≤ Ej + E0 +j ≤ 1 +2Cju2 +j(qmax +j +(t)). +(17b) +Similarly, the SoC balancing constraint (8) translates into the +following one: +����� +2 +Cj +Ej(t) − +1 +card(J ) +� +i∈J +2 +Ci +Ei(t) +����� ≤ ∆Ej, +(18) +where ∆Ej = (αj + βj∆q)2 − α2 +j. It is worth noting that +the SoC balancing constraint, either (8) or (18), may result in +infeasibility for the optimization problem, when ∆q or ∆E +fails to bound the cells’ initial difference in SoC. The same +issue applies to the temperature balancing constraint (9). Once +happening, the infeasibility will cause the power optimization +procedure to abort. While it is possible to make ∆E and +∆T large enough to forestall the issue, this will sacrifice the +achievable performance in both power loss minimization and +cell balancing. To guarantee the feasibility, we introduce slack +variables to modify the constraints in (18) and (9) as follows: +����� +2 +Cj +Ej(t) − +1 +card(J ) +� +l∈J +2 +Cl +El(t) +����� ≤ ∆Ej + ξ(E) +j +, +(19) +|Tj(t) − Tavg(t)| ≤ ∆T + ξ(T ) +j +, +(20) +where ξ(E) +j +, ξ(T ) +j +≥ 0 denote the SoC and temperature slack +variables, respectively. The slack variables will be included +into the objective function to penalize potential constraint +violations. As such, if a cell’s SoC or temperature is beyond +the constraints, it will be driven close to the constraints by +heavily penalizing the corresponding slack variables, without +compromising the feasibility. Besides, the use of the slack +variables will improve the power control flexibility. This will +be discussed in the simulation study in Section IV. +Based on the above, we are now ready to make a convex +relaxation of the problem in (11). Here, we also turn our +focus to discrete-time optimization for the sake of compu- +tation, through applying the forward Euler method to (4) +and (14) with the sampling time of ∆t. We use the vector +zj += [Pbj, Plj, Ej, Tj, ξ(E) +j +, ξ(T ) +j +]⊤, j ∈ J to collect all +the optimization variables and propose the following convex +optimization problem for the RBESS power management: +min +zj,j∈J +H +� +k=0 +� +j∈J +Plj[k] + λ(E)ξ(E) +j +[k] + λ(T )ξ(T ) +j +[k], +Safety constraints: +� +2 +Cj +(Ej[k] + E0 +j )imin +Lj ≤ Pbj[k] ≤ +� +2 +Cj +(Ej[k] + E0 +j )imax +Lj , +Tj ≤ T max +j +1 +2Cju2 +j(qmin +j +[k]) ≤ Ej[k] + E0 +j ≤ 1 +2Cju2 +j(qmax +j +[k]), +Balancing constraints: +����� +2 +Cj +Ej[k] − +1 +card(J ) +� +l∈J +2 +Cl +El[k] +����� ≤ ∆Ej + ξ(E) +j +[k], +|Tj[k] − Tavg[k]| ≤ ∆T + ξ(T ) +j +[k], +Power loss constraint: +Plj[k] ≥ +(Rj + RC + RSj)CjP 2 +bj[k] +2(Ej[k] + E0 +j ) +, +Energy dynamics: +Ej[k + 1] − Ej[k] = −Pbj[k]∆t, +Thermal dynamics: +Tj[k + 1] = Tj[k] + ∆t +Cth,j +� +Plj[k] − (Tj[k] − Tenv)/Rconv +− (2Tj[k] − Tj+1[k] − Tj−1[k])/Rcnd +� +, +Power supply-demand balance: +� +j∈J +Pbj[k] − Plj[k] = Pout[k], +(21) +where λ(E) and λ(T ) are the respective penalty weights for +ξ(E) and ξ(T ). The above problem is verifiably convex as a +result of the convex cost function and constraints. The convex- +ity makes it advantageous in practice as robust algorithms are +available to find out its global optimum with efficient com- +putation. The problem setup is similar to [18, 19]. However, +we crucially introduce the slack variables here to make the +problem always feasible. This improvement eliminates the risk +of no solution to satisfy the hard constraints, further enhancing +the practical aspects of power management. +The problem is designed to be implemented in a receding- +horizon manner. This will bring three benefits. First, predictive +optimization over a limited time horizon rather than the whole +mission duration will make the computation more manageable. +Second, the receding-horizon power control can better respond +to changes that occur to the RBESS in operation, e.g., fault- +triggered cell bypass and switching circuit reconfiguration. +Finally, the SoC change in each receding horizon is slight, +so the optimization only needs to consider a single SoC/OCV +linear segment and hence runs more efficiently. +D. Reconfiguration +The proposed RBESS allows dynamic switching of the +power switches to bypass faulty cells, ensuring continuous + +8 +Algorithm 1 Power management of the proposed RBESS +1: for Run-time do +2: +if a fault occurs to cell i then +3: +Bypass cell i +4: +Determine the set of the in-service cells J +5: +Calculate ns and np from (22) +6: +Reconfigure the switch circuit Si1:3, i = 1, 2, ...n +7: +end if +8: +Run the optimal power management strategy (21) +9: +return Pbj +10: +if Pbj == 0 for any j then +11: +Bypass cell j +12: +end if +13: end for +system operation. Following the bypass, an important question +is how to reconfigure the connection topology among the +cells. However, it is not easy to identify a complete answer, +as the large discrete reconfiguration decision space due to +the use of switches would defy an exhaustive search for an +optimal topology. In addition, inappropriate reconfiguration +may produce poor topologies to cause short circuits or other +issues. Note that the power management approach in (21) +determines the optimal charging/discharging power of the cells +individually and is not affected by any arbitrary series or +parallel connection among them. This makes its run and the +reconfiguration procedure separable but contiguous. +Here, we leverage an efficient heuristic to address the +question and outline it as below. Suppose that all the remaining +cells are approximately uniform in SoC and temperature at +the time of the reconfiguration, since the power management +based on (21) has driven cell balance. The reconfiguration +then should yield a topology that facilitates a balanced use +of the cells and makes every cell take an even power load. +A straightforward topology design to fulfill this need is one +based on ns serially connected modules with each module +consisting of np cells in parallel connection. We denote this +topology as npPnsS. We can determine ns and np by +ns = V ∗ +t +V max +C +, +np = Iout +imax +C +, +(22) +where V ∗ +t +is the desired terminal voltage; V max +C +and imax +C +are the maximum output voltage and current stresses of +the DC/DC converters, and Iout = Pout/V ∗ +t +is the output +charging/discharging current of the battery pack. Subsequently, +the RBESS can follow the series/parallel switching analysis in +Section II to reconfigure the switch circuit. +This heuristic-based reconfiguration mechanism is compu- +tationally fast, fail-safe, and easy to implement. Further, it +promotes system-wide cell balance and fits together with the +power management in (21). This leads us to the overall RBESS +management approach as shown in Algorithm 1. +IV. SIMULATION RESULTS +This section presents simulation results to evaluate the +proposed RBESS design and power management approach. +Table II summarizes the specifications of the RBESS under +TABLE II +SPECIFICATIONS OF THE PROPOSED RBESS +Symbol +Parameter +Value [Unit] +n +Number of battery cells +15 +v +Cell nominal voltage +3.6 [V] +¯Q +Cell nominal capacity +2.5 [A.h] +R +Cell internal resistance +31.3 [mΩ] +[qmin, qmax] +Cell SoC limits +[0.05,0.95] +[imin, imax] +Cell current limits +[-10,10] [A] +vcut-off +Cell cut-off voltage +3.3 [V] +Cth +Thermal capacitance +40.23 [J/K] +Rconv +Convection thermal resistance +41.05 [K/W] +Rcnd +Conductance thermal resistance +26.6 [K/W] +Tenv +Environment temperature +298 [K] +∆q +SoC balancing threshold +1% +∆T +Temperature balancing threshold +0.5 [K] +∆t +Sampling time +1 [s] +simulation. The battery cells are assumed to be Samsung +INR18650-25R, and we have identified their parameters (see +Table II) and SoC/OCV relationship (see Fig. 3) from ex- +periments using the approach in [29]. We approximate the +SoC/OCV curve using a piecewise linear function with three +segments that together span from zero to 100% SoC. The +power load profile for Pout is obtained by repeating the +scaled Urban Dynamometer Driving Schedule (UDDS). We +use the CVX package [30] to configure and solve the convex +optimization problem in (21) to compute Pb. The optimization +runs over a receding horizon of 20 seconds, i.e., H = 20. +The initial SoC of the cells is drawn from a normal +distribution with mean of 90% and variance of 3%. Similarly, +the initial temperature of the cells follows a normal distribution +with mean of 308 K and variance of 3 K. In order to +investigate whether the power management can handle the +cells’ heterogeneity, a white Gaussian noise with variance of +4 mΩ is added to the internal resistance value of each cell. +Furthermore, it is assumed that cells 4, 8, and 14 are bypassed +and isolated from the battery pack at the 2,000th, 4,000th, and +6,000th seconds, respectively. +Fig. 4 depicts the SoC and temperature balancing perfor- +mance of the proposed power management approach. The +tolerated SoC and temperature deviation bounds, ∆q and +∆T, are 1% and 0.5 K, respectively. According to Fig. 4 +(a), the cells are different in their initial SoC. Among them, +cell 2 has the lowest initial SoC of 86.48%, and cell 15 +has the highest SoC of 92.61%. The difference is beyond +the desired error bounds. However, the power management +approach successfully drives the SoC of the cells to reach +within the bounds after 200 seconds and continues to regulate +the charging/discharging power of the cells to ensure SoC +balance in the battery pack. Both cells 2 and 15 end up with +the same SoC of 8.2% when the simulation is finished. It is +important to note the key of incorporating the slack variables +in guaranteeing the feasibility of the power optimization. Fig. 4 + +9 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Time (s) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +SoC +Bounds +Cell 1 +Cell 2 +Cell 3 +Cell 4 +Cell 5 +Cell 6 +Cell 7 +Cell 8 +Cell 9 +Cell 10 +Cell 11 +Cell 12 +Cell 13 +Cell 14 +Cell 15 +50 +100 +150 +200 +0.86 +0.88 +0.9 +0.92 +Cell 8 is bypassed +Cell 14 is bypassed +Cell 5 is bypassed +(a) +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Time (s) +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.1 +SoC diff. from average +Threshold +Cell 1 +Cell 2 +Cell 3 +Cell 4 +Cell 5 +Cell 6 +Cell 7 +Cell 8 +Cell 9 +Cell 10 +Cell 11 +Cell 12 +Cell 13 +Cell 14 +Cell 15 +7000 +7100 +7200 +7300 +7400 +7500 +6 +8 +10 +12 +10-3 +Cell 5 is bypassed +Cell 8 is bypassed +Cell 14 is bypassed +(b) +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Time (s) +20 +22 +24 +26 +28 +30 +32 +34 +36 +38 +40 +Temperature (C) +Bounds +Cell 1 +Cell 2 +Cell 3 +Cell 4 +Cell 5 +Cell 6 +Cell 7 +Cell 8 +Cell 9 +Cell 10 +Cell 11 +Cell 12 +Cell 13 +Cell 14 +Cell 15 +100 +200 +300 +400 +500 +30 +32 +34 +36 +38 +7000 +7050 +7100 +28 +30 +32 +Cell 5 is bypassed +Cell 8 is bypassed +Cell 14 is bypassed +(c) +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Time (s) +0 +1 +2 +3 +4 +5 +6 +Temp. diff. from average (C) +Threshold +Cell 1 +Cell 2 +Cell 3 +Cell 4 +Cell 5 +Cell 6 +Cell 7 +Cell 8 +Cell 9 +Cell 10 +Cell 11 +Cell 12 +Cell 13 +Cell 14 +Cell 15 +7000 +7100 +7200 +7300 +7400 +7500 +0.4 +0.5 +0.6 +0.7 +Cell 5 is bypassed +Cell 8 is bypassed +Cell 14 is bypassed +(d) +Fig. 4. Simulation results of the SoC and temperature balancing. (a) The SoC of the cells. (b) The SoC difference of the cells from the average. (c) The +temperature of the cells. (d) The temperature difference of the cells from the average. +(b) illustrates the deviation of the cells’ SoC from the average. +The tolerance bound is set to be 1%. Some of the cells +initially are beyond this bound—for example, cell 15 deviates +from the average SoC by 3.5%. In this case, the optimization +problem would have been infeasible, but this issue is avoided +as the slack variable ξ(E) permits slight violation of the SoC +balancing constraints with a negligible compromise to physical +safety of the cells. Meanwhile, the penalization of ξ(E) as in +(21) in the cost function forces the cells to remain within the +tolerated error bound once after they enter the bound, keeping +the SoC balanced. The SoC of the bypassed cells remains +unchanged after isolation as the cells are no longer used. +Fig. 4 (c) shows the evolution of the cells’ temperature. +Similar to the SoC initialization, the initial temperatures of +the cells stretch beyond the desired bounds where cells 1 +and 2 have the highest and lowest temperature of 37.92℃ +and 29.73℃, respectively. The power management approach +effectively controls the cell temperatures to reach a balanced +temperature after 500 seconds. Note that a cell’s temperature +is still affected by the temperature of its adjacent cells and +the environment after it is bypassed. As shown in Fig. 4 (c), +when the average temperature of the battery pack increases, +the temperature of the bypassed cells also rises due to the con- +ductive heat transfer among adjacent cells. Here, even though +the cells’ initial temperature difference exceeds the bound +of 0.5℃ (see Fig. 4 (d)), the optimization for temperature +balancing maintains feasibility as a result of introducing the +slack variable ξ(T ). +To further investigate the role of the slack variables in the +50 +100 +150 +200 +250 +Time (s) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +5 +10 +15 +20 +25 +Sum of the SoC balancing slack variables +Sum of the temperature balancing slack variables +Fig. 5. The change of the slack variables. +formulated optimization problem, Fig. 5 depicts their evolution +through time. When the cells’ SoC or temperature lies outside +the balancing constraints, the slack variables will take nonzero +values to relax the balancing constraints gently, thus turning +the nominally infeasible optimization problem into a feasible +one. The slack variables will decrease and approach zero as the +cells are increasingly balanced in SoC and temperature. When +they are zero, the SoC and temperature balancing constraints +are fully satisfied. Penalizing the slack variables restricts over- +relaxation of the constraints and tightens the bounds as the +SoC and temperature get closer to or into the constraints. The +penalization weights associated with the slack variables are +subject to tuning so as to achieve the performance desired +by a user. In general, heavier penalization will lead to less +constraint relaxation and more time to achieve balancing. + +10 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +8000 +Time (s) +-40 +-30 +-20 +-10 +0 +10 +20 +30 +40 +Cell Power (W) +Cell 1 +Cell 2 +Cell 3 +Cell 4 +Cell 5 +Cell 6 +Cell 7 +Cell 8 +Cell 9 +Cell 10 +Cell 11 +Cell 12 +Cell 13 +Cell 14 +Cell 15 +1565 +1567 +25 +26 +27 +28 +5757 +5760 +24 +28 +32 +Fig. 6. The output power profiles of the cells. +The output power profiles of the cells are shown in Fig. 6. +We can see that the power of the individual cells is regulated +to vary from one to another. This is because the cells have dif- +ferent conditions in SoC, temperature and internal resistances +and must collectively minimize the overall power losses while +complying with safety and balancing constraints. The cells +can also adjust their own output on the bypass of a faulty +cell. The peak power of battery cells are around 28 W for +1, 560 < t < 1, 570 s before any cells are bypassed from the +pack. However, when three cells are bypassed from the pack, +the peak power of the remaining cells is increased to around +33 W for 5, 750 < t < 5, 760 s to compensate for the bypassed +cells and to ensure a continuous power supply to the load. +It is of our interest to investigate whether the proposed +RBESS is more capable of reducing the total power losses +than conventional hardwired battery systems. Fig. 7 shows a +comparison of the resultant power losses, which focuses on +1, 000 < t < 2, 000 s for the purpose of visual illustration. +The hardwired pack is found to constantly suffer more power +dissipation, because the it neglects that the cells have different +internal resistances (associated with the higher power losses, +the pack also faces higher operating temperatures as well +as significant SoC and temperature imbalance). By contrast, +the proposed RBESS is able to optimally allocate the charg- +ing/discharging power among the cells to gain more power +efficiency. +We further assess whether the proposed power management +approach can distribute power among the cells relative to their +state-of-health (SoH), which is important to reduce the cell +aging and degradation. To this end, we consider the root-mean- +square (RMS) of the output power of the cells, and use the +internal resistance as the SoH indicator—overall, the higher +the internal resistance, the more degraded the cell is. Fig. 8 +illustrates the normalized RMS of the output power of the +battery cells in comparison to their internal resistance values. +We observe that the cells with lower resistances are allocated +more power load overall, see the groups of cells 1-4, cells +6-7, and cells 9-10. While the pattern is obvious, the power +distribution also depends on each cell’s SoC and temperature +and thus shows certain perturbations. We can argue that the +power management approach contributes to a balanced use of +the battery cells in terms of SoH. +1000 +1100 +1200 +1300 +1400 +1500 +1600 +1700 +1800 +1900 +2000 +Time (s) +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +Power loss (W) +Conventional hardwired design +Proposed design +1400 +1420 +1440 +1460 +1480 +1500 +0 +1 +2 +3 +4 +5 +6 +Fig. 7. The power loss comparison. +1 +2 +3 +4 +6 +7 +9 +10 +11 +12 +13 +15 +Cells +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Normalized cell internal resistances +Normalized RMS of cell powers +Fig. 8. The normalized internal resistance and RMS of the output power of +the cells. +V. EXPERIMENTAL RESULTS +We develop a lab-scale prototype of the proposed RBESS +for experimental validation. Fig. 9 (a) shows the experimental +setup, and Figs. 9 (b)-(c) illustrate the circuit boards of the +RBESS prototype based on the design in Fig. 1. The RBESS is +a pack of five cells integrated with five converters (Fig. 9 (b)) +and 12 relay switches for reconfigurable connection (Fig. 9 +(c)). Table III lays out the specifications of the key components +of the prototype. Type K thermocouples are attached on the +surface of each cell to measure their temperature. A National +Instruments PCIe-6321 DAQ board with LabVIEW is used +to collect the cells’ voltages, temperatures, and output power +data. Using the CVX package, we then solve the optimal +power management problem using MATLAB every minute +(i.e., ∆t = 60 s). The optimal power values of the cells are +then fed to local controllers using DSP TMS320F28335. The +local controllers based on STM8S003F3P6 microcontrollers, +generate 250 kHz PWM signals to DC/DC converters. The +prototype is connected to a 20 Ω resistance load with a total +output discharge power of 50 W. +The cells, labeled from 1 to 5 in order, have an initial SoC of +87%, 89%, 82%, 91%, and 93%, respectively. The experiment +lasts for 30 minutes with a sampling time of ∆t = 60 s. +Each cell’s output current is limited to 5 A. To investigate the +effect of fault occurrence, a fault is assumed for cell 3 after 15 +minutes of discharging in the experiment. The obtained results +are shown in Figs. 10-11. + +11 + + + +LabVIEW Software for +Data Acquisition +Data Acquisition +Proposed RBESS +Optimization +(a) + +DC/DC Converters +Thermocouple +Local Controller +(b) + +Reconfiguration Switches +Local Controller +(c) +Fig. 9. +Lab-scale prototype of the proposed RBESS. (a) The experimental +setup. (b) Top circuit view. (c) Bottom circuit view +Fig. 10 (a) shows the SoC of the battery cells. The initial +SoC values of the cells are not within the desired tolerance +bound. However, the optimal power management approach +successfully distributes the discharging power among the cells +such that the cells reach the SoC balancing bounds after about +four minutes. The corresponding output power profiles of the +cells are shown in Fig. 11. It can be seen that cell 3, with +the lowest initial SoC, is assigned zero power load (and thus +bypassed by reconfiguration) in the first two minutes, while +cell 5, with the highest SoC, delivers the maximum allowed +power. Not only does the proper distribution of the output +power lead to cell balancing, but the reconfiguration capability +of the proposed design also helps cell balancing. Fig. 10 (b) +also shows the SoC deviation of the cells from their average +value. We point out that, without the inclusion of the slack +variables, the optimization would have been infeasible at the +very initial moment when the SoC deviation goes beyond the +SoC balancing constraint. Fig. 10 (c) depicts the temperature +TABLE III +LIST OF KEY COMPONENTS +Device +Model (Value) +MOSFET +CSD86356Q5D +Relay switch +TE OJT-SS-105HM +Gate driver +TPS28225 +Inductor +SER2915H-333KL (33 µH) +Capacitor +(10 µF) +Local controller +STM8S003F3P6 +Main controller +TMS320f28335 +Battery cell +Samsung INR18650-25R +of the cells. The initial temperature of all the cells is 20.7℃. +Due to the uneven power distribution among the cells for SoC +balancing, the cells will see their temperature rise and slightly +drift away from each other. However, the deviation remains +within the desired bound without violating the temperature +balancing constraint. Fig. 10 (d) also shows the temperature +deviation of the cells from the average value. The difference +increases from zero to the pre-specified bound of 0.5℃ in the +beginning. But afterwards, it shows a declining trend and is +well bounded. +When a fault occurs to cell 3 after 15 minutes of discharg- +ing, the cell is bypassed and isolated, as indicated in Fig. 11. +Right after this happens, the other four cells that remain in +service increase their discharging power accordingly, contin- +uing to supply a total output power of 50 W as demanded. +This highlights the benefit of the proposed RBESS in ensuring +robust and consistent operation despite cell faults. +VI. CONCLUSION AND FUTURE WORK +The RBESS technology offers an important way to enhance +the safe use of lithium-ion batteries. In this paper, we pro- +posed a novel modular RBESS design, which distinguishes +itself by the integration of reconfigurable power switches and +DC/DC converters. The design harnesses the switching circuit +reconfiguration to bypass any defective cells, and exploits +the DC/DC converters to facilitate optimal power distribution +at the cell level and ensure consistent power storage/supply +at the system level. Based on the design, we developed a +power management approach to achieve power-loss-minimized +operation of the RBESS along with SoC and temperature +balancing among the cells. Compared to existing methods, +this approach allows wide-SoC-range operation of the cells +by multi-segment SoC/OCV approximation and guarantees the +feasibility of the optimization problem via mild relaxation. We +conducted extensive simulations and then developed a lab- +scale prototype of the RBESS design to perform validation +experiments. The results substantiate the effectiveness of the +proposed design and the power management approach. 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Fan, “Evaluation of lithium-ion battery equivalent circuit +models for state of charge estimation by an experimental approach,” Energies, +vol. 4, no. 4, pp. 582–598, 2011. +[28] A. A. Pesaran, “Battery thermal models for hybrid vehicle simulations,” Journal +of Power Sources, vol. 110, no. 2, pp. 377–382, 2002. +[29] N. Tian, Y. Wang, J. Chen, and H. Fang, “One-shot parameter identification of +the Thevenin’s model for batteries: Methods and validation,” Journal of Energy +Storage, vol. 29, p. 101282, 2020. +[30] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex program- +ming, version 2.1,” http://cvxr.com/cvx, Mar. 2014. +Amir Farakhor (Student Member, IEEE) received +the B.Sc. and M.Sc. degrees in electrical engineer- +ing from the Azarbaijan Shahid Madani Univer- +sity, Tabriz, Iran in 2012 and 2014, respectively, +and Ph.D. degree in Power Electronics from the +University of Tabriz in Feb 2019. He is currently +a Ph.D. student in mechanical engineering at the +University of Kansas, Lawrence, KS, USA. His +research interests include power electronics, battery +management systems, renewable energies, and dis- +tributed generation. +Di Wu (Senior Member, IEEE) is a Chief Research +Engineer and a Team Leader within the Optimization +and Control Group at the Pacific Northwest National +Laboratory (PNNL). He received the B.S. and M.S. +degrees in electrical engineering from Shanghai Jiao +Tong University, China, in 2003 and 2006, respec- +tively, and the Ph.D. in electrical and computer +engineering from Iowa State University, Ames, in +2012. At PNNL, Dr. Wu leads research work in +the areas of energy storage analytics, building-to- +grid integration, microgrid design, and hybrid energy +systems. Dr. Wu is a Senior Member of IEEE and a member of the IEEE +Power and Energy Society and the Control System Society. He serves as an +Editor for the IEEE Open Access Journal of Power and Energy and IEEE +Transactions on Energy Markets, Policy and Regulation. +Yebin Wang (Senior Member, IEEE) received the +B.Eng. degree in mechatronics engineering from +Zhejiang University, Hangzhou, China, in 1997, the +M.Eng. degree in control theory and control engi- +neering from Tsinghua University, Beijing, China, in +2001, and the Ph.D. degree in electrical engineering +from the University of Alberta, Edmonton, AB, +Canada, in 2008. He has been with Mitsubishi Elec- +tric Research Laboratories, Cambridge, MA, USA, +since 2009, where he is currently a Senior Principal +Research Scientist. From 2001 to 2003, he was a +Software Engineer, the Project Manager, and the Manager of the Research and +Development Department in Industries, Beijing, China. His current research +interests include nonlinear control and estimation, optimal control, adaptive +systems, and their applications, including mechatronic systems. +Huazhen Fang (Member, IEEE) received the B.Eng. +degree in computer science and technology from +Northwestern Polytechnic University, Xi’an, China, +in 2006, the M.Sc. degree in mechanical engineering +from the University of Saskatchewan, Saskatoon, +Canada, in 2009, and the Ph.D. degree in mechanical +engineering from the Department of Mechanical and +Aerospace Engineering, University of California, +San Diego, La Jolla, CA, USA, in 2014. He is an +Associate Professor of mechanical engineering with +the University of Kansas, Lawrence, KS, USA. His +research interests include control and estimation theory with application to +energy management and robotics. Dr. Fang received the National Science +Foundation CAREER Award in 2019. He currently serves as an Associate +Editor for IEEE Transactions on Industrial Electronics, IEEE Open Journal of +the Industrial Electronics Society, IEEE Control Systems Letters, IEEE Open +Journal of Control Systems, and Information Sciences. + diff --git a/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/load_file.txt b/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4dde1939f4af3dd3766edea67fe2c7f77fb288c3 --- /dev/null +++ b/T9E4T4oBgHgl3EQfmg0h/content/tmp_files/load_file.txt @@ -0,0 +1,1158 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf,len=1157 +page_content='1 A Novel Modular, Reconfigurable Battery Energy Storage System: Design, Control, and Experimentation Amir Farakhor, Student Member, IEEE, Di Wu, Senior Member, IEEE, Yebin Wang, Senior Member, IEEE, and Huazhen Fang, Member, IEEE Abstract—This paper presents a novel modular, reconfigurable battery energy storage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The proposed design is charac- terized by a tight integration of reconfigurable power switches and DC/DC converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This characteristic enables isolation of faulty cells from the system and allows fine power control for individual cells toward optimal system-level performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' An optimal power management approach is developed to extensively exploit the merits of the proposed design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on receding- horizon convex optimization, this approach aims to minimize the total power losses in charging/discharging while allocating the power in line with each cell’s condition to achieve state-of- charge (SoC) and temperature balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' By appropriate design, the approach manages to regulate the power of a cell across its full SoC range and guarantees the feasibility of the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We perform extensive simulations and further develop a lab-scale prototype to validate the proposed system design and power management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Index Terms—Battery management systems, cell balancing, convex optimization, reconfigurable battery energy storage sys- tems (RBESSs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='NOMENCLATURE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Reconfiguration switches–binary (1/0) variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='V ∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='RBESS reference output voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='V max ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Maximum output voltage of DC/DC converters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Imax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Maximum output current of DC/DC converters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell open-circuit voltage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='iL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell current ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Pb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell internal power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell output power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell energy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Pl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Module power losses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Total power losses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Pout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='RBESS output power ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Iout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='RBESS output charging/discharging current ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='˙Qcnd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Conductive heat transfer rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' National Science Foundation under Awards CMMI-1763093 and CMMI-1847651, and in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Department of Energy, Office of Electricity through the Energy Storage Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Farakhor and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fang (corresponding author) are with the Department of Mechanical Engineering, University of Kansas, Lawrence, KS, USA (Email: fang@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='edu, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='farakhor@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Wu is with the Pacific Northwest National Laboratory, Richland, WA, USA (Email: di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='wu@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='gov).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Wang is with the Mitsubishi Electric Research Laboratories, Cambridge, MA, USA (Email: yebinwang@merl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ˙Qconv Convective heat transfer rate ξ Slack variable z Optimization variables Parameters n Number of battery cells ns Number of cells in series np Number of cells in parallel J Set of in-service cells L Inductor for DC/DC converters C Output capacitor for DC/DC converters ¯Q Cell capacity q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' qmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' qmax SoC of a cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' and its lower and upper limits qavg Average SoC ∆q SoC imbalance tolerance ∆E Energy imbalance tolerance α Intercept coefficient of the SoC/OCV line seg- ment β Slope coefficient of the SoC/OCV line segment R Cell internal resistance RC Resistance to capture the power losses of DC/DC converters Rcnd Conductive thermal resistance Rconv Convective thermal resistance Cth Thermal capacitance T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' T max Cell temperature and its upper limit Tavg Average temperature Tenv Environmental temperature ∆T Temperature imbalance tolerance λ Penalty weight for the multi-objective opti- mization ∆t Sampling time H Optimization horizon I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' INTRODUCTION L ITHIUM-ION battery energy storage systems (BESSs) have proven themselves as an enabling technology for various applications, including electric cars, electric aircraft, smart grid, and space systems[1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Despite their high energy density and long cycle life, lithium-ion batteries suffer from safety risks, which trace to the high reactivity of lithium and flammability of the commonly used electrolyte solutions and are exacerbated by side reactions, aging, and degradation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Hence, it is imperative to ensure their safe and reliable arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='05168v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='SY] 12 Jan 2023 2 operation, particularly in safety-critical applications [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Re- configurable BESS (RBESS) have attracted much attention as a promising means to achieve this end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' An RBESS characteris- tically uses power electronics switches to make the connection among the constituent cells reconfigurable, providing the capa- bility to bypass faulty cells without interrupting the operation of the system [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This feature overcomes the vulnerability of conventional hardwired BESS to single-cell failures due to the fixed configuration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This paper proposes a novel RBESS design that integrates reconfigurable switches with DC/DC converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The new design uses the switches to reconfigure the connectivity of the cells for the sake of safety and meanwhile, leverages the converters to achieve robust power management and supply from cell to system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Further, we present an optimal power management algorithm and develop a lab-scale prototype to validate the proposed RBESS design and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Literature Review This paper centers around the RBESS circuit design and power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Therefore, we survey the literature on the two dimensions one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The review will also encompass some recent studies about hardwired BESS due to the rele- vance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1) Review of RBESS circuit design: The literature has pre- sented two main ways to design RBESS circuit architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The first one builds and integrates a circuit of controllable power electronics switches with the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' By controlling the switches, a cell can be put into or cut off from the connection with other cells when a fault occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The circuit topology plays a key role in the level of reconfigurability, functional flexibility, and circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The study in [9] shows a switching circuit for a series-connected battery pack, which uses only two switches for every cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' More sophisticated topologies can provide more versatile reconfiguration, though at the expense of using larger numbers of switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The work in [10] considers a string of modules of parallel-connected cells in series and allows switch-based bypass of any cell or module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The circuit topologies proposed in [11, 12] place five and six switches, respectively, around each cell to realize arbitrary series and parallel connection among the cells, and the one shown in [13] uses only four to achieve the same end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is interesting to point out that reconfigurable switching circuits can enable more functions than just isolating faulty cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A battery system designed in [14] dynamically connects some battery modules in series or parallel to produce multi- level and even AC voltage output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In [15], an inverter based on switching circuits combines batteries and supercapacitors for hybrid energy storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' While easily reconfigurable, switching circuits are unable to do cell-level charging/discharging power regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This limitation will lead to unbalanced use of the cells following the reconfiguration-based bypass of a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Another important way for RBESS design uses converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A converter can not only connect a cell into the circuit, but also charge or discharge it at a controlled current, voltage, or power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This capability offers a higher degree of freedom for cell-level control to enhance the balanced use of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In [16], a centralized converter interfaces with a few cells, and it contains a selector to select and put the cells into operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The work in [17] pairs converters, which by design include power switches to offer a bypass mode, with individual cells one by one to form a reconfigurable pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, converter-based RBESS architectures are unable to offer flexible topology changes, compared to the switching circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As such, only series- connected RBESS is considered in [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Converters have also found their way into hardwired BESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The studies in [18– 20] integrate DC/DC converters with the cells to enable cell balancing and power loss minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, the fixed hardwired connections among the cells make them susceptible to single-cell faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2) Review of RBESS control and power management: For a given switching-circuit-based RBESS, an essential question is how to control its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A main approach lies in finding out a connection topology to meet load requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For example, a heuristic method in [10] groups battery cells into modules and chooses a minimum set of modules on a load request to perform charging/discharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The method of dynamic programming finds use in identifying an opti- mal configuration topology in [11], and rule-based bypassing mechanisms are proposed in [12] to control the switches for stable or responsive voltage supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The study in [13] exploits the switching circuit to achieve cell equalization through a hierarchical strategy that combines intra-module equalization and system-level reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Considering the switching circuits in [9, 10], the study in [21] proposes to sort cells according to their capacity and then reconfigure them into serial strings to restrain the cell imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' On a related note, the literature has included a few stud- ies about power management for hardwired BESS based on converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For converter-based RBESS, the main subject of investigation is developing control approaches to manage the cell-to-system-level operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Optimal control has shown as a useful solution in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, due to a lithium- ion battery’s nonlinear characteristics, optimization problems posed for the power management are often non-convex, and thus defy the computation of global optima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To mitigate the issue, the studies in [18, 19] choose to leverage a battery model convexification approach proposed in [22] to formulate convex optimal power control problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The convexification therein involves a linear approximation of the relationship between the state-of-charge (SoC) and the open-circuit-voltage (OCV), thus restricting the proposed methods to only limited operating ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is also possible to use a linear battery model to achieve optimal control that even admits closed-form solutions [23], but the method is also just applicable to operating ranges in which the linear model is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As another valuable approach, distributed power manage- ment treats the cells as independent agents and makes them perform individual control toward a global goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' One can decompose a global optimization task and distribute it among individual cells for charging or load sharing, as shown in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As an alternative way, distributed consensus control is applied in [24] to achieve SoC balancing among the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Not requiring numerical optimization, this method offers fast computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' But the lack of optimality makes it in need of more time to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3 TABLE I CHARACTERISTICS OF DIFFERENT BESS STRUCTURES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Switch-based RBESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Converter- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='RBESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Converter-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='hardwired BESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Proposed design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[10] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[11] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[13] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[14] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[15] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[16] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[18] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[19] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='[20] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Safety (Bypass of faulty cells) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Output voltage regulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='SoC balancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Temperature balancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='State-of-health balancing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Optimal cell-level power control ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Compatibility with cells of different types ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Flexibility in series/parallel connection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='L: Low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' M: Medium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' H: High Table I summarizes the main BESS architectures in the literature for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In this context, we propose the presented work to improve the state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Statement of Contributions Despite recent advances, the RBESS technology remains far from reaching a level of maturity in both design and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The existing RBESS architectures use either only switches or only converters to enable good reconfigurability or good power regulation capability, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' There is also a lack of power control approaches sophisticated enough to maximize the operating performance of RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To overcome these substantial shortcomings, our study presents the following specific contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1) We propose a new modular, reconfigurable power electron- ics architecture for RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Differing from the existing ones, the proposed architecture, for the first time, integrates power switches with DC/DC converters for a combination of their respective merits, while using the fewest number of switches for each cell to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Among the various benefits that the proposed design brings are high reconfigurability—arbitrary bypass and parallel or series connection among the adjacent cells—and high regulata- bility in power supply from cell to system level to satisfy exogenous power demands, even under the occurrence of cell failures, and ensure equal use of the cells simultane- ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2) We develop a power management approach based on optimal control for the proposed RBESS to minimize the system-wide energy loss while supplying demanded power and equalizing the cells in SoC and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Compared to the prior methods, our approach is different in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' First, we adopt piecewise linear SoC/OCV ap- proximation in the battery model convexification to enable control across low to high SoC regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Second, previous methods may be subject to infeasibility when the cells’ actual conditions make the pre-set operating constraints unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, we introduce slack variables to relax the optimization problem to ensure the feasibility and practical applicability of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, we present a reconfiguration method to modify the switching circuit topology after a faulty cell is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3) We develop an experimental prototype and conduct a series of experiments to validate and assess the performance of the proposed RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The experiments involve var- ious scenarios under non-uniform cell conditions, fault occurrence, and reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The results demonstrate the effectiveness of the proposed architecture and power management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on the above contributions, our RBESS design presents significant advantages to distinguish itself from the literature, as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A preliminary conference version of this work appeared in [25] to report the RBESS architecture design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, we introduce substantial expansions in optimal power management and experimental validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Organization The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Section II presents the power electronics architecture of the proposed RBESS and discusses its unique features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Section III develops the optimal power management approach, which covers the modeling, optimization problem formulation, and convexifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This section also presents a switching circuit reconfigura- tion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Section IV provides extensive simulation results, and Section V proceeds to develop an experimental prototype to validate the proposed RBESS design and control approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, Section VI concludes the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ARCHITECTURE OF THE PROPOSED RBESS This section elaborates the proposed RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1 shows the power electronics architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As illustrated, every cell is connected with a DC/DC converter to make up a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, we employ synchronous DC/DC converters, even though other types of converter topologies are also allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The converter, which comprises two power switches, an inductor, and an output capacitor, provides bi-directional power processing to control the charging and discharging of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As such, the module has the capability of cell-level power control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The modules are connected via a switching circuit, which places three power switches between every two adjacent modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' With an appropriate reconfiguration of the switches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' the switching circuit can bypass a module subject to faults and achieve arbitrary series or parallel connection among adjacent 4 S12 C1 S11 S13 Q11 Q12 L1 S22 C2 S21 S23 S(n-1)2 Cn-1 S(n-1)1 S(n-1)3 Q21 Q22 L2 Q(n-1)1 Q(n-1)2 Ln-1 Cn Qn1 Qn2 Ln Terminal (Load/Charger) Cell 1 Cell 2 Cell n-1 Cell n Module 1 A Current Sensor SoC Estimation Measurement Bus (Cell voltages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' currents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' and temperatures) Fault Detection Optimal Power Management Fault status PWM control signals to DC/DC converters Local Controller #1 + PI Local Controller #2 Local Controller #n Control signals to reconfiguration switches Optimal cell currents Output power SoC of cells BMS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The proposed modular DC/DC converter-integrated RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We label the three switches connecting modules i and i + 1 as Sij for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For switch Sij, Sij = 1 when it is on, and Sij = 0 when it is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To bypass and isolate cell i for 1 ≤ i ≤ n − 1 from the battery pack, we let Si1 = 1 and Si2 = Si3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To bypass cell n, we let S(n−1)2 = 1 and S(n−1)3 = S(n−1)1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Modules i and i + 1 are configured in series when Si1 = Si2 = 0 and Si3 = 1, and in parallel when Si1 = Si2 = 1 and Si3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The proposed design uses only 3(n − 1) power switches for n cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To our knowledge, this is more economical than any other RBESS design in the literature to provide the same level of reconfigurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The circuit simplicity further results in convenient operation and reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Specifically, a cell requires only one switch to be on for the series and bypass configurations, and a parallel configuration needs only two switches to be on, as seen from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1 illustrates how the proposed design dovetails with the battery management system (BMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The BMS adopts a two- layer control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' At the higher level, the optimal power management generates optimal charging/discharging currents for the cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' at the lower level, PI-based local current mode controllers perform reference tracking for the cell currents, as is common in control of DC/DC converters [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Our proposed RBESS can provide high reconfigurability and control flexibility, which lead to distinct benefits for practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A summary of the major ones is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1) The proposed design allows to bypass and isolate any faulty module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Hence, the battery pack can continue to operate rather than shut down as a whole, despite safety threats and anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Further, following the bypass of a module, the switching circuit can reconfigure to redirect the power flow and share the load equally among the remaining in-service cells to promote balanced use of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2) In the proposed design, the embedded DC/DC converters take on responsibility for the external power electronic devices in the conventional designs to control the charg- ing/discharging of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The DC/DC converters would yield useful functions with their capability of power con- version and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' First, they can regulate their output voltage so that the RBESS can supply desired or refer- ence voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The voltage supply can remain consistent before and after a fault-induced reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Second, the converters apply individual current or voltage control to the cells, thus making it possible to customize and optimize the charging/discharging for each cell based on its present condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' One can translate this strength into cell balancing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', by charging (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', discharging) the cells with high SoC less (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', more) relative to the cells with low SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is also viable to balance the cells in temperature and state-of-health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3) Even though beyond the scope of this paper, the proposed design can accommodate the heterogeneity of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For example, one can leverage it to integrate heterogeneous cells from different manufacturers or even based on dif- ferent electrochemistries to form a workable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In a similar vein, the design can potentially enable hybrid energy storage consisting of battery cells, supercapacitors, and even solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Note that the use of the embedded DC/DC converters can mitigate the use of exogenous power electronics devices, and that one can connect a string or pack of cells, rather than a single cell, to each DC/DC converter in practical adoptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' These factors would help make the overall implementation cost of the proposed RBESS manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, the diverse new functions and safety improvements brought by the design are valuable especially for high-stakes, safety-critical applications including electric vehicles and aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' MODELING AND OPTIMAL CONTROL OF THE PROPOSED RBESS This section investigates optimal power management for the proposed RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We will begin with the electrical and thermal modeling for the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We then proceed to present a model-based optimal control problem for the RBESS and 5 convexify it for computational tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, the model- ing and optimal control formulation are a refinement of the methodology in [18, 19], and improvements are introduced to expand the operating SoC range of the RBESS and ensure the feasibility of the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, we present a switching circuit reconfiguration mechanism that dovetails with the power management method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Electrical and Thermal Modeling Consider the proposed RBESS consisting of n modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Each module includes a cell, a DC/DC converter, and three power switches connected in cascade, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We denote the set of all the in-service modules as J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The electrical model of module j for j ∈ J is schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It includes two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The first part is the Rint model to describe the cell’s electrical dynamics [27], which comprises an OCV source uj in series with an internal resistor Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The model’s governing equations are: ˙qj(t) = − 1 ¯Qj iLj(t), (1a) vj(t) = uj(qj(t)) − RjiLj(t), (1b) where vj, iLj, uj, ¯Qj, and qj are the terminal voltage of the cell, applied current in Ampere, OCV, capacity, and SoC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The internal power of the battery cell is given by Pbj = uj(qj(t))iLj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (2) The DC/DC converter is modeled as an ideal DC/DC transformer along with a series resistor RC to capture power losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The reconfiguration switches are also modeled by the ideal switches with series resistors RSji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For module j, we assume that we can collect the power losses on RSji for i = 1, 2, 3 in a single resistor RSj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Thus, the module’s output power Pj can be calculated as Pj = uj(qj(t))iLj(t) − (Rj + RC + RSj)i2 Lj(t), (3) where Rji2 Lj(t), RCi2 Lj(t), and RSji2 Lj(t) represent the inter- nal power losses of the cell, the converter, and the reconfigu- ration switches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We adopt a lumped thermal model in [18] to describe the thermal dynamics of module j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The thermal model is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The model captures the heat transfer due to the convection between module j and the environment, ˙Qconv,j, and the conduction between module j and its adjacent cells, ˙Qcnd,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Meanwhile, the power loss caused by the internal resistor, Rji2 Lj, translates into heat generation, which becomes the main heating source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Combining all, the thermal model is governed by Cth,j ˙Tj(t) = Rji2 Lj(t) − ˙Qcnd,j − ˙Qconv,j, (4a) ˙Qconv,j(t) = (Tj(t) − Tenv)/Rconv, (4b) ˙Qcnd,j(t) = (2Tj(t) − Tj+1(t) − Tj−1(t))/Rcnd, (4c) where Tj and Tenv are the cell’s and environmental temper- atures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In addition, the term Cth,j represents the thermal capacitance of the cell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Rcnd and Rconv are the thermal resistances between neighboring cells and between cell j and Qj1 Qj2 Cj Lj Sj1 Sj2 Sj3 (a) Sj1 Sj2 Sj3 Battery Cell DC/DC converter (b) + + + + Conduction Conduction Convection (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The proposed DC/DC converter-integrated cell module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (a) The circuit structure of the proposed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (b) The electrical model of the proposed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (c) The thermal model of the cell j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' the environment, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, Rconv depends inversely on the external surface area of the cell Aj and the convective heat transfer coefficient between the cell’s surface and the environment h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For instance, one can consider a parallel forced air cooling approach for the proposed design [28], which allows every cell to experience the same amount of cooling air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The above electro-thermal model is concise but expressive and computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Putting them together for all the modules, one can obtain a complete description of the dynamics of the RBESS, which allows us to perform optimal power management design subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Problem Formulation The aim of our RBESS power management is to distribute the power load among the cells so that the power losses can be minimized under some key safety, balancing and power demand satisfaction constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To begin with, we will formulate the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The total power losses of the RBESS can be expressed by J(t) = � j∈J (Rj + RC + RSj)i2 Lj(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (5) We use the following objective function to encompass the total power losses over a horizon: � H 0 J(t)dt, (6) 6 where H is the planning horizon length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' For the sake of safety, we require each cell to operate within some favorable current, SoC, and temperature ranges: imin Lj ≤ iLj ≤ imax Lj , (7a) qmin j ≤ qj ≤ qmax j , (7b) Tj ≤ T max j , (7c) where imin/max Lj , qmin/max j , and T max j are the lower/upper safety bounds for the current, SoC, and temperature of cell j, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is important to note that imin Lj can be set to be zero as the zero current means the bypass of the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Further, we impose the following SoC and temperature balancing constraints to equalize the cells and make an even usage of them: |qj(t) − qavg(t)| ≤ ∆q, (8) |Tj(t) − Tavg(t)| ≤ ∆T, (9) Here, qavg(t) and Tavg(t) represent the average SoC and tem- perature of all the cells that belong to J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' They are calculated as Xavg(t) = 1 card(J ) � j∈J Xj(t), where X = q and T, and card(J ) is the cardinality of J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The SoC and temperature thresholds ∆q and ∆T determine the tolerated deviation of each cell’s SoC and temperature from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, ∆q and ∆T are tunable parameters, and one can tune them to meet the SoC and temperature balancing requirements for a specifically considered application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' While lower ∆q and ∆T values force a more balanced SoC and temperature distribution among the cells, higher values allow more deviation for the cells’ SoC and temperature from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To make the RBESS meet the power demands, we present the following output power satisfaction constraint: � j∈J Pbj − (Rj + RC + RSj)i2 Lj = Pout, (10) where Pout is the total power demanded of the RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Summing up the above, our power management approach is based on addressing the constrained nonlinear optimization problem as follows: min iLj ,j∈J � H 0 J(t)dt, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (1b), (4a), (7) − (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (11) This optimization problem pursues predictive minimization of the power losses while complying with the constraints that promote safety, SoC and temperature balancing, and power supply-demand match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Note that the optimization problem (11) is non-convex due to the nonlinearity of the equality constraint (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Thus, the solution to this problem is neither trivial nor computationally cheap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To overcome the issue, we relax the problem slightly to formulate a convex optimization problem, as suggested in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The convexification is described in detail as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='95 1 SoC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 OCV (V) Line 1 Line 2 Line 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The SoC/OCV curve of the simulated cells and the multi-segment linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Convex Problem Formulation For the sake of convexification, we begin with linearizing the SoC/OCV curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The existing studies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', [18, 19] per- form the linearization for only the medium SoC range, where the OCV is closely linear with SoC for lithium-ion batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, this treatment excludes the use of the low and high SoC ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To address the issue, we introduce multi-segment linearization based on different SoC ranges to approximate the complete SoC/OCV curve: uj(qj(t)) = αi j(qj(t)) + βi j(qj(t))qj(t), (12) where αi j and βi j are the intercept and slope coefficients of the i-th line segment for cell j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3 illustrates an example of the linearization, where the SoC/OCV curve taken from a real cell is approximated by three line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Differing from the literature, the αi j and βi j values are SoC-dependent, and the multi-segment linear approximation spans the SoC/OCV curve from 0 to 100% SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To ease the notation, we will drop the superscript i from αi j and βi j in the sequel without causing confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Next, we present a convex model by introducing the notion of accumulated energy Ej to take the place of SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The accumulated energy of a battery cell can be expressed as Ej(t) = 1 2Cju2 j(qj(t)) − E0 j , (13) where Cj = ¯Qj/βj and E0 j = 1 2Cju2 j(qj(0)) is the initial energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Inserting (12) to (13) and using (1b), the dynamic equation of the cell’s accumulated energy can be derived as ˙Ej(t) = −Pbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (14) In the above, we extract a desirably linear dynamic model to represent the evolution of Ej(t) driven by Pbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on (14), we will reformulate the optimization problem to be one with respect to Pbj, as will be seen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Proceeding forward, we consider module j’s power loss, Plj(t), which can be expressed in terms of Pbj as Plj(t) = (Rj + RC + RSj)CjP 2 bj(t) 2(Ej(t) + E0 j ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (15) As our optimization goal is to minimize the total power loss, (15) serves as an equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Since Plj(t) is not a linear function of Pbj, the resulting optimization problem 7 would be non-convex due to the nonlinearity of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Thanks to the fact that the objective function minimizes the total loss of the battery pack, we can relax (15) to comply with the convexity requirement Plj(t) ≥ (Rj + RC + RSj)CjP 2 bj(t) 2(Ej(t) + E0 j ) , (16) by which the optimization problem will practically reduce Plj(t) to its lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The safety constraints (7a)-(7b) can also be reformulated in terms of the Pbj and Ej as follows: � 2 Cj (Ej + E0 j )imin Lj ≤ Pbj ≤ � 2 Cj (Ej + E0 j )imax Lj , (17a) 1 2Cju2 j(qmin j (t)) ≤ Ej + E0 j ≤ 1 2Cju2 j(qmax j (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (17b) Similarly, the SoC balancing constraint (8) translates into the following one: ����� 2 Cj Ej(t) − 1 card(J ) � i∈J 2 Ci Ei(t) ����� ≤ ∆Ej, (18) where ∆Ej = (αj + βj∆q)2 − α2 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is worth noting that the SoC balancing constraint, either (8) or (18), may result in infeasibility for the optimization problem, when ∆q or ∆E fails to bound the cells’ initial difference in SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The same issue applies to the temperature balancing constraint (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Once happening, the infeasibility will cause the power optimization procedure to abort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' While it is possible to make ∆E and ∆T large enough to forestall the issue, this will sacrifice the achievable performance in both power loss minimization and cell balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To guarantee the feasibility, we introduce slack variables to modify the constraints in (18) and (9) as follows: ����� 2 Cj Ej(t) − 1 card(J ) � l∈J 2 Cl El(t) ����� ≤ ∆Ej + ξ(E) j , (19) |Tj(t) − Tavg(t)| ≤ ∆T + ξ(T ) j , (20) where ξ(E) j , ξ(T ) j ≥ 0 denote the SoC and temperature slack variables, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The slack variables will be included into the objective function to penalize potential constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As such, if a cell’s SoC or temperature is beyond the constraints, it will be driven close to the constraints by heavily penalizing the corresponding slack variables, without compromising the feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Besides, the use of the slack variables will improve the power control flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This will be discussed in the simulation study in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on the above, we are now ready to make a convex relaxation of the problem in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, we also turn our focus to discrete-time optimization for the sake of compu- tation, through applying the forward Euler method to (4) and (14) with the sampling time of ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We use the vector zj = [Pbj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Plj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ξ(E) j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' ξ(T ) j ]⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' j ∈ J to collect all the optimization variables and propose the following convex optimization problem for the RBESS power management: min zj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='j∈J H � k=0 � j∈J Plj[k] + λ(E)ξ(E) j [k] + λ(T )ξ(T ) j [k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Safety constraints: � 2 Cj (Ej[k] + E0 j )imin Lj ≤ Pbj[k] ≤ � 2 Cj (Ej[k] + E0 j )imax Lj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Tj ≤ T max j 1 2Cju2 j(qmin j [k]) ≤ Ej[k] + E0 j ≤ 1 2Cju2 j(qmax j [k]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Balancing constraints: ����� 2 Cj Ej[k] − 1 card(J ) � l∈J 2 Cl El[k] ����� ≤ ∆Ej + ξ(E) j [k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' |Tj[k] − Tavg[k]| ≤ ∆T + ξ(T ) j [k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Power loss constraint: Plj[k] ≥ (Rj + RC + RSj)CjP 2 bj[k] 2(Ej[k] + E0 j ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Energy dynamics: Ej[k + 1] − Ej[k] = −Pbj[k]∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Thermal dynamics: Tj[k + 1] = Tj[k] + ∆t Cth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='j � Plj[k] − (Tj[k] − Tenv)/Rconv − (2Tj[k] − Tj+1[k] − Tj−1[k])/Rcnd � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Power supply-demand balance: � j∈J Pbj[k] − Plj[k] = Pout[k],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (21) where λ(E) and λ(T ) are the respective penalty weights for ξ(E) and ξ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The above problem is verifiably convex as a result of the convex cost function and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The convex- ity makes it advantageous in practice as robust algorithms are available to find out its global optimum with efficient com- putation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The problem setup is similar to [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, we crucially introduce the slack variables here to make the problem always feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This improvement eliminates the risk of no solution to satisfy the hard constraints, further enhancing the practical aspects of power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The problem is designed to be implemented in a receding- horizon manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This will bring three benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' First, predictive optimization over a limited time horizon rather than the whole mission duration will make the computation more manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Second, the receding-horizon power control can better respond to changes that occur to the RBESS in operation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', fault- triggered cell bypass and switching circuit reconfiguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, the SoC change in each receding horizon is slight, so the optimization only needs to consider a single SoC/OCV linear segment and hence runs more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Reconfiguration The proposed RBESS allows dynamic switching of the power switches to bypass faulty cells, ensuring continuous 8 Algorithm 1 Power management of the proposed RBESS 1: for Run-time do 2: if a fault occurs to cell i then 3: Bypass cell i 4: Determine the set of the in-service cells J 5: Calculate ns and np from (22) 6: Reconfigure the switch circuit Si1:3, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='n 7: end if 8: Run the optimal power management strategy (21) 9: return Pbj 10: if Pbj == 0 for any j then 11: Bypass cell j 12: end if 13: end for system operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Following the bypass, an important question is how to reconfigure the connection topology among the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, it is not easy to identify a complete answer, as the large discrete reconfiguration decision space due to the use of switches would defy an exhaustive search for an optimal topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In addition, inappropriate reconfiguration may produce poor topologies to cause short circuits or other issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Note that the power management approach in (21) determines the optimal charging/discharging power of the cells individually and is not affected by any arbitrary series or parallel connection among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This makes its run and the reconfiguration procedure separable but contiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, we leverage an efficient heuristic to address the question and outline it as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Suppose that all the remaining cells are approximately uniform in SoC and temperature at the time of the reconfiguration, since the power management based on (21) has driven cell balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The reconfiguration then should yield a topology that facilitates a balanced use of the cells and makes every cell take an even power load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A straightforward topology design to fulfill this need is one based on ns serially connected modules with each module consisting of np cells in parallel connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We denote this topology as npPnsS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We can determine ns and np by ns = V ∗ t V max C , np = Iout imax C , (22) where V ∗ t is the desired terminal voltage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' V max C and imax C are the maximum output voltage and current stresses of the DC/DC converters, and Iout = Pout/V ∗ t is the output charging/discharging current of the battery pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Subsequently, the RBESS can follow the series/parallel switching analysis in Section II to reconfigure the switch circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This heuristic-based reconfiguration mechanism is compu- tationally fast, fail-safe, and easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Further, it promotes system-wide cell balance and fits together with the power management in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This leads us to the overall RBESS management approach as shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' SIMULATION RESULTS This section presents simulation results to evaluate the proposed RBESS design and power management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Table II summarizes the specifications of the RBESS under TABLE II SPECIFICATIONS OF THE PROPOSED RBESS Symbol Parameter Value [Unit] n Number of battery cells 15 v Cell nominal voltage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 [V] ¯Q Cell nominal capacity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='h] R Cell internal resistance 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 [mΩ] [qmin, qmax] Cell SoC limits [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='95] [imin, imax] Cell current limits [-10,10] [A] vcut-off Cell cut-off voltage 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 [V] Cth Thermal capacitance 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='23 [J/K] Rconv Convection thermal resistance 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='05 [K/W] Rcnd Conductance thermal resistance 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 [K/W] Tenv Environment temperature 298 [K] ∆q SoC balancing threshold 1% ∆T Temperature balancing threshold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 [K] ∆t Sampling time 1 [s] simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The battery cells are assumed to be Samsung INR18650-25R, and we have identified their parameters (see Table II) and SoC/OCV relationship (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 3) from ex- periments using the approach in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We approximate the SoC/OCV curve using a piecewise linear function with three segments that together span from zero to 100% SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The power load profile for Pout is obtained by repeating the scaled Urban Dynamometer Driving Schedule (UDDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We use the CVX package [30] to configure and solve the convex optimization problem in (21) to compute Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The optimization runs over a receding horizon of 20 seconds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', H = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The initial SoC of the cells is drawn from a normal distribution with mean of 90% and variance of 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Similarly, the initial temperature of the cells follows a normal distribution with mean of 308 K and variance of 3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In order to investigate whether the power management can handle the cells’ heterogeneity, a white Gaussian noise with variance of 4 mΩ is added to the internal resistance value of each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Furthermore, it is assumed that cells 4, 8, and 14 are bypassed and isolated from the battery pack at the 2,000th, 4,000th, and 6,000th seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 depicts the SoC and temperature balancing perfor- mance of the proposed power management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The tolerated SoC and temperature deviation bounds, ∆q and ∆T, are 1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 (a), the cells are different in their initial SoC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Among them, cell 2 has the lowest initial SoC of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='48%, and cell 15 has the highest SoC of 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='61%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The difference is beyond the desired error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, the power management approach successfully drives the SoC of the cells to reach within the bounds after 200 seconds and continues to regulate the charging/discharging power of the cells to ensure SoC balance in the battery pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Both cells 2 and 15 end up with the same SoC of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2% when the simulation is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is important to note the key of incorporating the slack variables in guaranteeing the feasibility of the power optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 9 1000 2000 3000 4000 5000 6000 7000 8000 Time (s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 1 SoC Bounds Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9 Cell 10 Cell 11 Cell 12 Cell 13 Cell 14 Cell 15 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='92 Cell 8 is bypassed Cell 14 is bypassed Cell 5 is bypassed (a) 1000 2000 3000 4000 5000 6000 7000 8000 Time (s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 SoC diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' from average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='10-3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 5 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 8 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 14 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Temperature (C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Bounds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7050 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 5 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 8 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Cell 14 is bypassed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Time (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' from average (C) Threshold Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9 Cell 10 Cell 11 Cell 12 Cell 13 Cell 14 Cell 15 7000 7100 7200 7300 7400 7500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 Cell 5 is bypassed Cell 8 is bypassed Cell 14 is bypassed (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Simulation results of the SoC and temperature balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (a) The SoC of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (b) The SoC difference of the cells from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (c) The temperature of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (d) The temperature difference of the cells from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (b) illustrates the deviation of the cells’ SoC from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The tolerance bound is set to be 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Some of the cells initially are beyond this bound—for example, cell 15 deviates from the average SoC by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In this case, the optimization problem would have been infeasible, but this issue is avoided as the slack variable ξ(E) permits slight violation of the SoC balancing constraints with a negligible compromise to physical safety of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Meanwhile, the penalization of ξ(E) as in (21) in the cost function forces the cells to remain within the tolerated error bound once after they enter the bound, keeping the SoC balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The SoC of the bypassed cells remains unchanged after isolation as the cells are no longer used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 (c) shows the evolution of the cells’ temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Similar to the SoC initialization, the initial temperatures of the cells stretch beyond the desired bounds where cells 1 and 2 have the highest and lowest temperature of 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='92℃ and 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='73℃, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The power management approach effectively controls the cell temperatures to reach a balanced temperature after 500 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Note that a cell’s temperature is still affected by the temperature of its adjacent cells and the environment after it is bypassed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 (c), when the average temperature of the battery pack increases, the temperature of the bypassed cells also rises due to the con- ductive heat transfer among adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Here, even though the cells’ initial temperature difference exceeds the bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5℃ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 4 (d)), the optimization for temperature balancing maintains feasibility as a result of introducing the slack variable ξ(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To further investigate the role of the slack variables in the 50 100 150 200 250 Time (s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 1 0 5 10 15 20 25 Sum of the SoC balancing slack variables Sum of the temperature balancing slack variables Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The change of the slack variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' formulated optimization problem, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 5 depicts their evolution through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' When the cells’ SoC or temperature lies outside the balancing constraints, the slack variables will take nonzero values to relax the balancing constraints gently, thus turning the nominally infeasible optimization problem into a feasible one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The slack variables will decrease and approach zero as the cells are increasingly balanced in SoC and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' When they are zero, the SoC and temperature balancing constraints are fully satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Penalizing the slack variables restricts over- relaxation of the constraints and tightens the bounds as the SoC and temperature get closer to or into the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The penalization weights associated with the slack variables are subject to tuning so as to achieve the performance desired by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In general, heavier penalization will lead to less constraint relaxation and more time to achieve balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10 1000 2000 3000 4000 5000 6000 7000 8000 Time (s) 40 30 20 10 0 10 20 30 40 Cell Power (W) Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 6 Cell 7 Cell 8 Cell 9 Cell 10 Cell 11 Cell 12 Cell 13 Cell 14 Cell 15 1565 1567 25 26 27 28 5757 5760 24 28 32 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The output power profiles of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The output power profiles of the cells are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We can see that the power of the individual cells is regulated to vary from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This is because the cells have dif- ferent conditions in SoC, temperature and internal resistances and must collectively minimize the overall power losses while complying with safety and balancing constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The cells can also adjust their own output on the bypass of a faulty cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The peak power of battery cells are around 28 W for 1, 560 < t < 1, 570 s before any cells are bypassed from the pack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, when three cells are bypassed from the pack, the peak power of the remaining cells is increased to around 33 W for 5, 750 < t < 5, 760 s to compensate for the bypassed cells and to ensure a continuous power supply to the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It is of our interest to investigate whether the proposed RBESS is more capable of reducing the total power losses than conventional hardwired battery systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 7 shows a comparison of the resultant power losses, which focuses on 1, 000 < t < 2, 000 s for the purpose of visual illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The hardwired pack is found to constantly suffer more power dissipation, because the it neglects that the cells have different internal resistances (associated with the higher power losses, the pack also faces higher operating temperatures as well as significant SoC and temperature imbalance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' By contrast, the proposed RBESS is able to optimally allocate the charg- ing/discharging power among the cells to gain more power efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We further assess whether the proposed power management approach can distribute power among the cells relative to their state-of-health (SoH), which is important to reduce the cell aging and degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To this end, we consider the root-mean- square (RMS) of the output power of the cells, and use the internal resistance as the SoH indicator—overall, the higher the internal resistance, the more degraded the cell is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 8 illustrates the normalized RMS of the output power of the battery cells in comparison to their internal resistance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We observe that the cells with lower resistances are allocated more power load overall, see the groups of cells 1-4, cells 6-7, and cells 9-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' While the pattern is obvious, the power distribution also depends on each cell’s SoC and temperature and thus shows certain perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We can argue that the power management approach contributes to a balanced use of the battery cells in terms of SoH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 Time (s) 0 2 4 6 8 10 12 14 16 18 Power loss (W) Conventional hardwired design Proposed design 1400 1420 1440 1460 1480 1500 0 1 2 3 4 5 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The power loss comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1 2 3 4 6 7 9 10 11 12 13 15 Cells 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 Normalized cell internal resistances Normalized RMS of cell powers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The normalized internal resistance and RMS of the output power of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' EXPERIMENTAL RESULTS We develop a lab-scale prototype of the proposed RBESS for experimental validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 9 (a) shows the experimental setup, and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 9 (b)-(c) illustrate the circuit boards of the RBESS prototype based on the design in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The RBESS is a pack of five cells integrated with five converters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 9 (b)) and 12 relay switches for reconfigurable connection (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 9 (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Table III lays out the specifications of the key components of the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Type K thermocouples are attached on the surface of each cell to measure their temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A National Instruments PCIe-6321 DAQ board with LabVIEW is used to collect the cells’ voltages, temperatures, and output power data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Using the CVX package, we then solve the optimal power management problem using MATLAB every minute (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=', ∆t = 60 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The optimal power values of the cells are then fed to local controllers using DSP TMS320F28335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The local controllers based on STM8S003F3P6 microcontrollers, generate 250 kHz PWM signals to DC/DC converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The prototype is connected to a 20 Ω resistance load with a total output discharge power of 50 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The cells, labeled from 1 to 5 in order, have an initial SoC of 87%, 89%, 82%, 91%, and 93%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The experiment lasts for 30 minutes with a sampling time of ∆t = 60 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Each cell’s output current is limited to 5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' To investigate the effect of fault occurrence, a fault is assumed for cell 3 after 15 minutes of discharging in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The obtained results are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 11 LabVIEW Software for Data Acquisition Data Acquisition Proposed RBESS Optimization (a) DC/DC Converters Thermocouple Local Controller (b) Reconfiguration Switches Local Controller (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Lab-scale prototype of the proposed RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (a) The experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (b) Top circuit view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (c) Bottom circuit view Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10 (a) shows the SoC of the battery cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The initial SoC values of the cells are not within the desired tolerance bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, the optimal power management approach successfully distributes the discharging power among the cells such that the cells reach the SoC balancing bounds after about four minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The corresponding output power profiles of the cells are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' It can be seen that cell 3, with the lowest initial SoC, is assigned zero power load (and thus bypassed by reconfiguration) in the first two minutes, while cell 5, with the highest SoC, delivers the maximum allowed power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Not only does the proper distribution of the output power lead to cell balancing, but the reconfiguration capability of the proposed design also helps cell balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10 (b) also shows the SoC deviation of the cells from their average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We point out that, without the inclusion of the slack variables, the optimization would have been infeasible at the very initial moment when the SoC deviation goes beyond the SoC balancing constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10 (c) depicts the temperature TABLE III LIST OF KEY COMPONENTS Device Model (Value) MOSFET CSD86356Q5D Relay switch TE OJT-SS-105HM Gate driver TPS28225 Inductor SER2915H-333KL (33 µH) Capacitor (10 µF) Local controller STM8S003F3P6 Main controller TMS320f28335 Battery cell Samsung INR18650-25R of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The initial temperature of all the cells is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Due to the uneven power distribution among the cells for SoC balancing, the cells will see their temperature rise and slightly drift away from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' However, the deviation remains within the desired bound without violating the temperature balancing constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10 (d) also shows the temperature deviation of the cells from the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The difference increases from zero to the pre-specified bound of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5℃ in the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' But afterwards, it shows a declining trend and is well bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' When a fault occurs to cell 3 after 15 minutes of discharg- ing, the cell is bypassed and isolated, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Right after this happens, the other four cells that remain in service increase their discharging power accordingly, contin- uing to supply a total output power of 50 W as demanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' This highlights the benefit of the proposed RBESS in ensuring robust and consistent operation despite cell faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK The RBESS technology offers an important way to enhance the safe use of lithium-ion batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' In this paper, we pro- posed a novel modular RBESS design, which distinguishes itself by the integration of reconfigurable power switches and DC/DC converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The design harnesses the switching circuit reconfiguration to bypass any defective cells, and exploits the DC/DC converters to facilitate optimal power distribution at the cell level and ensure consistent power storage/supply at the system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on the design, we developed a power management approach to achieve power-loss-minimized operation of the RBESS along with SoC and temperature balancing among the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Compared to existing methods, this approach allows wide-SoC-range operation of the cells by multi-segment SoC/OCV approximation and guarantees the feasibility of the optimization problem via mild relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' We conducted extensive simulations and then developed a lab- scale prototype of the RBESS design to perform validation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The results substantiate the effectiveness of the proposed design and the power management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' The study can benefit and potentially drive the use of lithium-ion batteries for safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Based on the study, several interesting research questions are worth pursing further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' One is how to quantify and assess the reliability of the proposed RBESS given the failure rates of the batteries, switches, and converters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' A subsequent question is how to optimize the RBESS architecture and size under specified reliability metrics and power requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Finally, it is important to enable fast computation in large-size RBESS power management, and how to scale up the optimization approach in this paper is open to exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Affanni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Bellini, G.' 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186502500mAh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7V 186502500mAh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7V 186502500mAh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7V 186502500mAh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7V Rd11 Rd21 首Rd31 Rd41 612 O O STM1 LED_ST_1 M2 LEO_STM_2 NRST STM3 LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='STM3 LED_STM4 NRST 口 4R 口 国 伽 LED STM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 UCA 3 LED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='STM3 UCAP UCAP LEDSTM4· U12 J34 Program_2 UCAP_2C_STM_2 NRST_2 R_SW1 IMS3 R_SW2 C_SW2 R_SW3 EMSTO RSW4 CSW4 E012 200 400 600 800 1000 1200 1400 1600 1800 Time (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 1 SoC bounds Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 0 100 200 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='95 Cell 3 is bypassed due to a fault (a) 200 400 600 800 1000 1200 1400 1600 1800 Time (s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='1 SoC diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' from average Threshold Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 3 is bypassed due to a fault (b) 200 400 600 800 1000 1200 1400 1600 1800 Time (s) 15 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 20 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 25 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='5 30 Temperature (C) bounds Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 3 is bypassed due to a fault (c) 200 400 600 800 1000 1200 1400 1600 1800 Time (s) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='8 1 Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' from average (C) Threshold Cell 1 Cell 2 Cell 3 Cell 4 Cell 5 Cell 3 is bypassed due to a fault (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Experimental results of the proposed RBESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (a) The SoC of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (b) The temperature of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (c) The SoC difference of the cells from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' (d) The temperature difference of the cells from the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='com/cvx, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Amir Farakhor (Student Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degrees in electrical engineer- ing from the Azarbaijan Shahid Madani Univer- sity, Tabriz, Iran in 2012 and 2014, respectively, and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in Power Electronics from the University of Tabriz in Feb 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He is currently a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' student in mechanical engineering at the University of Kansas, Lawrence, KS, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' His research interests include power electronics, battery management systems, renewable energies, and dis- tributed generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Di Wu (Senior Member, IEEE) is a Chief Research Engineer and a Team Leader within the Optimization and Control Group at the Pacific Northwest National Laboratory (PNNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degrees in electrical engineering from Shanghai Jiao Tong University, China, in 2003 and 2006, respec- tively, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' in electrical and computer engineering from Iowa State University, Ames, in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' At PNNL, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Wu leads research work in the areas of energy storage analytics, building-to- grid integration, microgrid design, and hybrid energy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Wu is a Senior Member of IEEE and a member of the IEEE Power and Energy Society and the Control System Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He serves as an Editor for the IEEE Open Access Journal of Power and Energy and IEEE Transactions on Energy Markets, Policy and Regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Yebin Wang (Senior Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in mechatronics engineering from Zhejiang University, Hangzhou, China, in 1997, the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in control theory and control engi- neering from Tsinghua University, Beijing, China, in 2001, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in electrical engineering from the University of Alberta, Edmonton, AB, Canada, in 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He has been with Mitsubishi Elec- tric Research Laboratories, Cambridge, MA, USA, since 2009, where he is currently a Senior Principal Research Scientist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' From 2001 to 2003, he was a Software Engineer, the Project Manager, and the Manager of the Research and Development Department in Industries, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' His current research interests include nonlinear control and estimation, optimal control, adaptive systems, and their applications, including mechatronic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Huazhen Fang (Member, IEEE) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in computer science and technology from Northwestern Polytechnic University, Xi’an, China, in 2006, the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in mechanical engineering from the University of Saskatchewan, Saskatoon, Canada, in 2009, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' degree in mechanical engineering from the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, USA, in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He is an Associate Professor of mechanical engineering with the University of Kansas, Lawrence, KS, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' His research interests include control and estimation theory with application to energy management and robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' Fang received the National Science Foundation CAREER Award in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} +page_content=' He currently serves as an Associate Editor for IEEE Transactions on Industrial Electronics, IEEE Open Journal of the Industrial Electronics Society, IEEE Control Systems Letters, IEEE Open Journal of Control Systems, and Information Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf'} diff --git a/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/2301.01327v1.pdf.txt b/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/2301.01327v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..75d3582b49c0ca82b2eff894d414437403c73c53 --- /dev/null +++ b/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/2301.01327v1.pdf.txt @@ -0,0 +1,1741 @@ +OPERATOR THEORY, KERNELS, AND FEEDFORWARD +NEURAL NETWORKS +PALLE E.T. JORGENSEN, MYUNG-SIN SONG, AND JAMES TIAN +Dedicated to the memory of Professor Ka-Sing Lau. +Abstract. In this paper we show how specific families of positive definite +kernels serve as powerful tools in analyses of iteration algorithms for multiple +layer feedforward Neural Network models. Our focus is on particular kernels +that adapt well to learning algorithms for data-sets/features which display +intrinsic self-similarities at feedforward iterations of scaling. +Contents +1. +Introduction +1 +2. +Neural networks (NN), and reproducing kernel Hilbert spaces (RKHS) +2 +2.1. +Basis approach +9 +2.2. +Dual pairs of operators +10 +2.3. +Functions and Operators +12 +2.4. +The case when K = L +13 +3. +Neural Network-activation functions from p.d. kernels +15 +4. +Applications to fractal images +18 +References +21 +1. Introduction +Recently many authors have offered diverse approaches to feedforward Neural +Network (NN) algorithms [ZC23, AK23, HL23, DWZ+23], as well as optimization +terms based on kernels. Here we establish some new results in operator theory, +and we bring them to bear on the problem. The list of applications of feedforward +NN models includes a variety of machine learning settings, and deep NN based on +kernels [GK23, MK23, BSW23, SHO22, GKNV22, GPR+21, Kut20, GKP20]. +A common theme in feedforward NN models is specific prescribed iterations +which entail (i) ReLu functions [JR23, OSZ22, CKM22, CCK22, JR22, CL22], (ii) +substitution from prescribes systems of affine mappings. Moreover, (iii) each step +is then linked to the next with a choice of an activation function. In this paper +we show that there are natural positive definite kernels associated with the three +steps going into feedforward NN constructions, as well as to their iteration. We +2000 Mathematics Subject Classification. 41A30, 46E22, 47B32, 68T07, 92B20. +Key words and phrases. algorithms, multipliers, spectral resolutions, normal operators, it- +erated function systems, fractal measures, feedforward neural network, explicit kernels, ReLU, +reproducing kernel Hilbert spaces, positive definite kernels, composition operators. +1 +arXiv:2301.01327v1 [cs.LG] 3 Jan 2023 + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +2 +believe that this then yields a more direct tool for kernel-based feedforward NN +models. This advantage of our approach is based on two facts. First, we identify +a direct notion of kernel iteration which accounts for traditional function theoretic +feedforward NN steps. Secondly, our approach offers a more direct and natural +choices of kernels which govern approximations involved in deep NN models, for +example graph NN constructions. +While positive definite kernels and their associated reproducing kernel Hilbert +spaces have found diverse applications in pure and applied mathematics, we shall +focus here on a new role of kernels in feedforward network models. In more detail, +the main purpose of our paper is a presentation of choices of particular families of +positive definite kernels which serve as powerful tools in analyses of multiple layer +feedforward Neural Networks. +In general, reproducing kernel constructions, and the corresponding RKHSs are +powerful tools in diverse applications. In the present framework of kernel neural +networks (KNNs) , their role may be summarized as follows: Starting with the +problem at hand, when we build our RKHS(µ) via IFS iterations (e.g., via Cantor- +like fractal limits), then the Cantor-like µ activation functions arise as relative +reproducing dipole functions for RKHS(µ) as in Figure 3.1 below. +2. Neural networks (NN), and reproducing kernel Hilbert spaces +(RKHS) +A main theme in our paper is a development of new tools for design of feedforward +Neural Network constructs. For this purpose we point out the use of positive definite +kernels, and associated generating function for the NN algorithms. These kernel +based functions include the more familiar ReLu functions, see Theorems 3.3 and 3.4 +below. We stress that the particular RKHS constructs will be relative in the sense +of Theorems 3.3 and 3.4, i.e., the inner product reproduces differences of function +values. +Our approach to the use of kernels and functions for feedforward Neural Network +(NN) algorithms, is based on a systematic study of two classes of operators. They +act as follows: (i) between prescribed kernel Hilbert spaces, and (ii) other operators +acting at indexed levels in the network, i.e., operators acting at fixed levels, so within +choices of kernel Hilbert spaces. Case (i) includes a systematic study of composition +operators (see Corollaries 2.7 and 2.9) in the context of kernel Hilbert spaces; and +case (ii), the study of multiplier operators and their adjoints, see e.g., Theorem 2.15. +We emphasize that the two classes of operations discussed below depend on choices +of kernels at each level in particular NN-network models. Together these families of +operators allow for realizations of black box filter-entries in associated generalized +multi-resolutions systems, including operators which consist of composition followed +by multiplication. Specific 3D applications are presented in the subsequent sections, +secs 3 and 4. +Conventions. Inside the paper we shall work with Hilbert spaces of functions, +e.g., reproducing kernel Hilbert spaces (RKHSs), L2 spaces, and Sobolev spaces. It +will be assumed that these are Hilbert spaces of real valued functions. Inner prod- +ucts will be written ⟨·, ·⟩, and we shall use subscripts on ⟨·, ·⟩ to indicate the Hilbert +space under consideration. Moreover, in our use of differentiation, or differential +operators, we shall mean weak derivatives, i.e., differentiation in the sense of dis- +tributions, or making use of the natural duality for the spaces under consideration. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +3 +Our restriction here to the real valued case is dictated by our present applications +to feedforward Neural Networks. However, many of our general results in Section +2 below extend to complex RKHS theory. The latter in turn are important in the +study of geometry and potential theory of complex domains, see e.g., [Eng96]. +The power of kernel machines derives in part from the following facts. First, +kernel machines serve to map points in a low-dimensional data sets (typically non- +linear) into higher dimensions. The dimensionality of this linear “hyperspace” may +be infinite but is designed for optimization and efficient encoding of features. Hence +the kernel method allows one to find coefficients of separating hyperplanes for the +problem at hand via RKHS-inner products, one selected for each pair of high- +dimensional features. While kernel machines of various types have been used for +decades, it was with the invention of support vector machines (SVMs) that kernels +have now taken center stage (see e.g., [CST01, PORSTS21, HSTHD11, HST10, +RSSST06, STWCK05]). By now, SVMs are used in diverse applications, including +in bioinformatics (for finding similarities between different protein sequences), ma- +chine vision, and handwriting recognition. Deep neural networks (to be discussed +in Sections 3 and 4 below) are made of layers of artificial neurons: input layer, an +output layer, and multiple hidden layers in-between them. Deeper the networks +have more hidden layers. The parameters of the network represent the strengths of +the connections between layers. Training a network yield determination of values of +parameters. Once trained, the ANN represents a model for turning an input (say, +an image) into an output (a label or category). +The variety of uses of forward Neural Network algorithms, the recent literature is +substantial and diverse, especially with regards to applications. See e.g., [ASA+23, +MM23, KG22, CC21, MCA20, Han16]. +The following lemma is a basic result in the theory of RKHSs. For details, see +e.g., [JT22] and the papers cited therein. +Lemma 2.1. Fix a p.d. kernel X × X +K +−−→ R (or C), let HK denote the corre- +sponding RKHS. Then a function F on X is in HK if and only if there exists a +constant CF < ∞, such that the following estimate holds for all n ∈ N, all (ξi)n +i=1, +ξi ∈ R (or C), and all (xi)n +i=1, xi ∈ X: +����� +n +� +i=1 +ξiF (xi) +����� +2 +≤ CF +n +� +i=1 +n +� +j=1 +ξiξjK (xi, xj) . +(2.1) +Remark 2.2. With the construction K �→ HK (referring to a RKHS of a fixed p.d. +kernel K), we arrive at the following two conclusions: +(1) For all x ∈ X, the function Kx := K (·, x) is in HK; and +(2) For all F ∈ HK, and x ∈ X, we have +F (x) = ⟨F, K (·, x)⟩HK , +(2.2) +i.e., the values of functions F in HK are reproduced via the inner product +⟨·, ·⟩HK, and the kernel functions. +In addition to (2.2), we shall also consider relative reproducing kernels, and relative +RKHSs. As noted in [AJV14], the relative reproducing property takes the following +form +F (y) − F (x) = ⟨F, vx,y (·)⟩Hrel , +(2.3) + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +4 +now valid for all pairs of points x, y ∈ X. So this entails double-indexed kernel +functions vx,y ∈ Hrel. +A particular class of Hrel spaces are considered in Theorem 3.4 below. There the +setting is X = R, and the relative kernel functions va,b take the form of activation +functions for classes of feedforward-NN-algorithms, see e.g., Figure 3.1. +A systematic study of (2.3) is undertaken in [AJV14] where it is shown that the +setting of relative reproducing is characterized by conditionally negative definite +functions. +We now recall the RKHS for the standard 1-dimensional Brownian motion. (See +e.g., [JT22, AJ21, JT21, JT20].) +Lemma 2.3. When K is the Brownian motion kernel on R≥ × R≥, i.e., +K (x, y) = x ∧ y = |x| + |y| − |x − y| +2 +, +x, y ∈ R≥, +(2.4) +the corresponding RKHS HK is the Hilbert space of absolutely continuous functions +f on R such that the derivative f ′ = df/dx is in L2 (R), and f (0) = 0. Moreover, +∥f∥2 +HK = +� +R≥ +|f ′ (x)|2 dx, +for all f ∈ HK. +(2.5) +Proof. The key observation is that, if x > 0, the function +R≥ ∋ y �−→ Fx (y) := K (y, x) = +� +y +if y ≤ x +x +if y > x +(2.6) +has weak derivative. Indeed, we have +dFx +dy = χ[0,x], +(2.7) +i.e., the indicator function of the interval [0, x]. +Hence if f is a function with +f ′ ∈ L2 (R) and f (0) = 0, then +f (x) = f (x) − f (0) = +� x +0 +f ′ (y) dy = +� +R +F ′ +x (y) f ′ (y) dy, +(2.8) +and the RHS in (2.8) is the inner product from the Hilbert space defined by the +RHS in (2.5). +The corresponding implication follows from the general theory of RKHS. Recall +that the RKHS of a kernel is a Hilbert space completion of the functions +y �−→ K (x, y) +(2.9) +as x varies over R. Moreover, for K (x, y) = x ∧ y, +⟨K (·, x1) , K (·, x2)⟩HK = K (x1, x2) = x1 ∧ x2, +and we can compute as follows: +� +R +� d +dy K (·, x1) +� � d +dy K (·, x2) +� +dy = +� +R +χ[0,x1] (y) χ[0,x2] (y) dy += λ ([0, x1] ∩ [0, x2]) += x1 ∧ x2 = K (x1, x2) +where λ = dy denotes the Lebesgue measure. +□ + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +5 +Remark 2.4. Note that the functions va,b (called dipoles) in HK which satisfy +f (b) − f (a) = ⟨f, va,b⟩ , +for all f ∈ HK +(see (2.5) and (2.8)) are as follows: +va,b (x) = +� +� +� +� +� +0 +if x < a +x − a +if a ≤ x < b +b − a +if x > b, +as illustrated in Figure 2.1. Also compare with Theorem 3.4 and Figure 3.1, and +the iterations in Section 4. +𝑥 − 𝑎 +𝑏 − 𝑎 +𝑎 +𝑣!,#(⋅) +𝑏 +Figure 2.1. The generating dipole function {va,b} indexed by +pairs a, b such that a < b. Compare with Figure 3.1 below. +Induced metrics +For a general p.d. kernel K on X × X, there is an induced metric on X, +dK : X × X → R+ +defined as (see e.g., [AJ21]) +dK (x, y) = ∥K (·, x) − K (·, y)∥2 +HK . +(2.10) +In particular, +dK (x, y) = K (x, x) + K (y, y) − 2ℜ {K (x, y)} . +Note that d1/2 +K +is also a metric on X × X. +Example 2.5. For K (x, y) = x ∧ y on R × R as in (2.4), +∥K (·, s) − K (·, t)∥2 +HK = +��(· ∧ s)′ − (· ∧ t)′��2 +L2 += +��χ[0,s] − χ[0,t] +��2 +L2 += |s − t| . +The results below deal with a general framework of pairs of sets X and Y , each +equipped with a positive definite kernel, K resp., L, K on X, and L on Y . With view +to realization of feedforward Neural Network-functions, we will present an explicit +framework (see (2.11) and (2.20)) which allows us to pass from (nonlinear) functions +f : X → Y to linear operators Tf acting between the respective RKHSs HK +and HL. This will be a representation in the sense that composition of functions +will map into products of the corresponding linear operators. Some care must be +exercised as the linear operators Tf will in general be unbounded. Nonetheless, +we shall show that the operators still fall in a class where spectral resolutions are +available, see Theorem 2.11 and Corollary 2.12. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +6 +Theorem 2.6. Consider p.d. +kernels X × X +K +−−→ R and Y × Y +L +−−→ R. +Let +f : X → Y be Lipschitz continuous with respect to the induced metrics dK, dL, i.e., +dL (f (x) , f (y)) ≤ cfdK (x, y) , +x, y ∈ X, +for some constant cf. Define the operator Tf : HK → HL by +Tf (Kx) (y) = L (f (x) , y) +and extend it by linearity and density. +Then, for any fixed c ∈ Y , the function +Ff : X → R, +Ff (x) := L (f (x) , c) +(2.11) +is in the RKHS HK if, and only if +Lc ∈ dom(T ∗ +f ), +(2.12) +the domain of the adjoint operator. +Moreover, +Ff,y (x) := L (f (x) , y) ∈ HK, ∀y ∈ Y +⇕ +Tf is closable. +(See also Theorem 2.8.) +Proof. Let the setting be as in the statement of the theorem, i.e., X +f +−−→ Y assumed +continuous with respect to the two metrics, dK on X and dL on Y ; so in particular, +for pairs of points x1, x2 ∈ X, we have +dK (x1, x2) = ∥K (·, x1) − K (·, x2)∥2 +HK +(2.13) += K (x1, x1) + K (x2, x2) − 2K (x1, x2) . +(2.14) +We further fix a point c ∈ Y , and set F = Ff,c, specified as follows: +F (x) = L (f (x) , c) , for all x ∈ X, +so F : X → R+ ∪ {0}. +Now, for every N, and every subset SN = (x1, x2, . . . , xN) ⊂ X, consider the +following matrix operations (in N dimensions): +F +�� +N := +� +�� +F (x1) +... +F (xN) +� +�� +� +�� +� +column vectors +, and +QN := +��F +�� +N ⟩⟨ F +�� +N +�� +� +�� +� +matrix of a rank-1 operator +, +(2.15) +i.e., the rank-1 operator on RN written in Dirac’s notation, defined as +QN (ξ) = ⟨FN, ξ⟩ FN +(2.16) +for all ξ ∈ RN. Set +KN := (K (xi, xj))N +i,j=1 = +� +�� +K (x1, x1) +· · · +K (x1, xN) +... +... +... +K (xN, x1) +· · · +K (xN, xN) +� +�� , +(2.17) +a sample matrix. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +7 +For the convex cone of all positive definite N × N matrices, we introduce the +following familiar ordering, K ≪C K′ iff (Def.) ∃C < ∞ such that +ξT KNξ ≤ CξT K′ +Nξ for all ξ ∈ RN. +(2.18) +Now an application of Lemma 2.1 above shows that the assertion in the theorem +is equivalent to the existence of a finite constant C (independent of SN = (xi)N +i=1) +satisfying QN ≪C KN, i.e., the estimate +����� +� +i +ξL (f (xi) , c) +����� +2 +≤ C +� +i +� +j +ξiK (xi, xj) ξj +(2.19) +holds for all N, all SN = {xi}N +i=1, and all ξ ∈ RN. We get this from the assumption +on f in the theorem. See details below. +□ +Summary of Theorem 2.6: Start with K p.d. on X × X, L p.d. on Y × Y , +and f : X → Y . We introduce the metrics dK on X, and dL on Y , and we consider +f continuous, or Lipchitz. To get the desired conclusion +(x �−→ L (f (x) , y)) ∈ HK, +we must introduce an operator Tf : HK → HL. The right choice is +Ly ∈ dom(T ∗ +f ). +See details below: +Fixing two kernels K and L, assumed p.d. on X × X, and on Y × Y . Pass to +the corresponding RKHS HK and HL. +Problem. Find conditions on functions X +f +−−→ Y with the property that, for +∀y ∈ Y , then the induced function +(X ∋ x �−→ L (f (x) , y)) +� +�� +� +Ff,y(·) as a function on X +∈ HK. +(2.20) +The argument stressed below is via dual operators (bounded) +HK +Tf +� +HL +T ∗ +f +� +; +but the unbounded case is also interesting. +Some remarks on the definition of the operator Tf : HK → HL in the case when +no additional assumptions are placed on X +f +−−→ Y . +We define +Tf (Kx) (y) = L (f (x) , y) ; +and so we extend Tf to linear combinations: +DK := +� � +i +ciKxi +� +�� +� +� +function on X +Tf +−−−−−→ +� +i +ciL (f (xi) , ·) +� +�� +� +function on Y +. +(2.21) +But to make sense of (2.21) so it is well defined, we must be careful with equivalence +classes. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +8 +If f : X → Y is a general function, the (generalized) operator (2.21) may be +non-closable. However, we can still define the adjoint T ∗ +f , but its domain might be +“small”. +Set +DK := span {Kx}x∈X , +(2.22) +then (Definition) a vector ψ ∈ HL is in dom(T ∗ +f ) iff ∃Cψ < ∞ s.t. +���⟨Tfϕ, ψ⟩HL +��� ≤ Cψ ∥ϕ∥HK , +∀ϕ ∈ DK. +(2.23) +We then set T ∗ +f ψ = the solution to +⟨Tfϕ, ψ⟩HL = +� +ϕ, T ∗ +f ψ +� +HK , +∀ϕ ∈ DK. +(2.24) +Let K, L, f be as specified, and assume for some y ∈ Y , that we have Ly ∈ +dom(T ∗ +f ), then (2.24) for Ly ∈ HL +T ∗ +f +−−→ HK, ϕ = Kx, ψ = Ly yields +x �−→ T ∗ +f (Ly) (x) = L (f (x) , y) ∈ HK. +So the conclusion in Theorem 2.6 that +(x �−→ L (f (x) , y)) ∈ HK +(2.25) +holds iff Ly ∈ dom(T ∗ +f ). +In this case there are no difficulties with (2.21) and we get a dual pair Tf and +T ∗ +f , +⟨Tfϕ, ψ⟩HL = +� +ϕ, T ∗ +f ψ +� +HK +(2.26) +for ∀ϕ ∈ DK, and ψ ∈ DL, HK +Tf +� +HL +T ∗ +f +� +. +Setting ϕ = Kx, and ψ = Ly, (2.26) implies +T ∗ +f (Ly) (x) = L (f (x) , y) = (Tf (Kx)) (y) . +(2.27) +But the previous condition Ly ∈ dom(T ∗ +f ) (compare (2.26)) amounts to the asser- +tion that T ∗ +f (Ly) ∈ HK, and by (2.27), this is then the conclusion for Theorem +2.6. +Corollary 2.7 (composition operators). Let X, Y , K and L be as specified above; +in particular, K is a fixed p.d. kernel on X × X, and the RKHS HK is a Hilbert +space of scalar valued functions on X. Similarly, HL is a Hilbert space of scalar +valued functions on Y . Both HK and HL satisfy the defining axioms for RKHSs; +see Lemma 2.1 above. +As noted, every function f, X +f +−−→ Y , induces a linear +operator +HK +Tf +−−→ HL, +(2.28) +with dense domain DK; see the statement of Theorem 2.6. For the adjoint operator +T ∗ +f , +HL +T ∗ +f +−−→ HK, +(2.29) +we have the following: For a function ψ in HL, the two characterizations (2.30) +and (2.31) hold: +ψ ∈ dom(T ∗ +f ) +(2.30) + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +9 +⇕ +ψ ◦ f ∈ HK. +(2.31) +In the affirmative, +T ∗ +f (ψ) = ψ ◦ f : X → R, +(2.32) +i.e., T ∗ +f is the composition operator. +Proof. (2.30)⇒(2.31). Assume (2.30), we then apply (2.23), and get Cψ < ∞ with: +����� +� +i +ci +� +T ∗ +f (ψ) +� +(xi) +����� +2 +≤ Cψ +� +i +� +j +cicjK (xi, xj) . +(2.33) +But +T ∗ +f (ψ) (xi) = +� +Kxi, T ∗ +f (ψ) +� +HK +(2.34) += ⟨Tf (Kxi) , ψ⟩HL += ⟨L (f (xi) , ·) , ψ⟩HL += ψ (f (xi)) , +where we used the RKHS property for HL in the last step. Substitution into (2.33) +yields +����� +� +i +ciψ (f (xi)) +����� +2 +≤ Cψ +� +i +� +j +cicjK (xi, xj) , +(2.35) +and, so by Lemma 2.1 applied to F = ψ ◦ f, conclusion in (2.31) follows. +(2.31)⇒(2.30). Assume (2.31), we then reverse the above reasoning to get +��� +� +Tf +�� +i +ciKxi +� +� +�� +� +∈DK +, ψ +� +HL +��� ≤ +� +Cψ +����� +� +i +ciKxi +����� +HK +(2.36) +which states that ψ ∈ dom(T ∗ +f ), which is condition (2.30). Now combine this with +(2.34), and we conclude that (2.32) is satisfied for ψ, i.e., that T ∗ +f ψ = ψ◦f holds. +□ +2.1. Basis approach. Let X, Y, K, L, f be as usual, and define Tf : HK → HL. +Since K is p.d. on X × X, the RKHS HK allows an ONB {hi}i∈N, hi ∈ HK; by +general theory, we get the pointwise formula: +K (x1, x2) = +� +i∈N +hi (x1) hi (x2) . +(2.37) +Then our condition in Theorem 2.6 is equivalent to the following: +(X ∋ x �−→ L (f (x) , y)) ∈ HK +(a) +⇕ +� +i∈N +|(Tf (hi)) (y)|2 < ∞. +(b) +Proof. (a) ⇒ (b) is detailed below; but the converse implication will follow by the +same argument. So by (a), Ly ∈ dom(T ∗ +f ) and therefore: +T ∗ +f (Ly) (·) ∈ HK. +(2.38) + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +10 +Since {hi} is an ONB in HK, +� +i +��� +� +hi, T ∗ +f (Ly) +� �� � +∈HK +���� +2 += +��T ∗ +f (Ly) +��2 +HK < ∞. +(2.39) +But from the LHS(2.39): +� +hi, T ∗ +f (Ly) +� +HK = ⟨Tf (hi) , Ly⟩HL = (Tf (hi)) (y) , +and (b) follows. +□ +Key Question: When is Ff,y (·) ∈ HK? The cleanest answer to the question +of what functions X +f +−−→ Y have the property that +Ff,y (x) = L (f (x) , y) is in HK +(2.40) +is this: +Theorem 2.8. Let K, L and f be given, then +Ff,y in (2.40) is in HK ⇐⇒ Ly ∈dom(T ∗ +f ), +(2.41) +where the operator Tf : HK → HL is given by +Tf (Kx) := L (f (x) , ·) . +Moreover, (2.41) holds for all y ∈ Y ⇐⇒ Tf is closable. +2.2. Dual pairs of operators. Consider a symmetric pair of operators with dense +domains: +HK +Tf +� +HL +T ∗ +f +� +(T = Tf, since it will depend on f) where +span {Kx}x∈X is dense in HK +(2.42) +and +span {Ly}y∈Y is dense in HL +(2.43) +such that +Kx ∈ dom(Tf), and +(2.44) +Ly ∈ dom(T ∗ +f ) +(2.45) +where “dom” denotes the respective operator domains. +Note. We note that +Tf (Kx) (·) = L (f (x) , ·) ∈ HL +is always well defined, with dense domain, but the secret is T ∗ +f . +Also note that (2.45) is the condition in Theorem 2.6. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +11 +Let f : X → Y be as before, and the two p.d. kernels K and L are fixed. We +then introduce the corresponding (densely defined) operator Tf : HK → HL by +setting +Tf (Kx) = L (f (x) , ·) ∈ HL. +(2.46) +Notation and convention. Kx (·) is the kernel function in HK as usual: +Kx (t) = K (x, t) , ∀t ∈ X and similarly, +(2.47) +Ly (u) = L (y, u) , ∀u ∈ Y. +(2.48) +Lemma 2.9. When Ly ∈ dom(T ∗ +f ) then +� +T ∗ +f (Ly) +� +(x) = L (f (x) , y) on X, +(2.49) +equivalently, +T ∗ +f (Ly) (·) = L (f (·) , y) on X. +(2.50) +Proof of (2.49). The conclusion (2.49) is equivalent to the following assertion: +� +Tf (Kx) (·) +� +�� +� +L(f(x),·) +, Ly +� +HL +� += +� +Kx, +∈HK +� +�� +� +L (f (·) , y) +� +�� +� +T ∗ +f (Ly) +� +HK +� +L (f (x) , y) +The conclusion (2.49) follows since the respective kernel functions span dense sub- +spaces. +□ +Recall, +the function x �−→ Ff,y (x) = L (f (x) , y) ∈ HK +⇕ +Ly ∈ dom(T ∗ +f ). +Assume that Ly ∈ dom(T ∗ +f ) , then apply the condition for functions in HK to +Ff,y (·), so ∀n, ∀ (xi)n +1, ∀ (ci)n +1, ci ∈ R: +����� +� +i +ciFf,y (xi) +����� +2 += +����� +� +i +ciL (f (xi) , y) +����� +2 +≤ +������ +�� +i +ciKxi, T ∗ +f (Ly) +� +HK +������ +2 +≤ +Schwarz +����� +� +i +ciKxi +����� +2 +HK +��T ∗ +f (Ly) +��2 +HK += +� +i +� +j +cicjK (xi, xj) +��T ∗ +f (Ly) +��2 +HK . + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +12 +Lemma 2.10. The implication below is both directions: +X ∋ x �−→ L (f (x) , y) ∈ HK for ∀y +⇕ +the condition in (2.6) is satisfied. +Even if we fix y ∈ Y , then +Ly ∈ dom(T ∗ +f ) ⇐⇒ (x �−→ L (f (x) , y)) ∈ HK. +(2.51) +Proof sketch. By definition, Ly ∈ dom(T ∗ +f ), ∃Cy < ∞ ⇐⇒ +���⟨Tfϕ, Ly⟩HL +��� = |(Tf (ϕ)) (y)| ≤ Cy ∥ϕ∥HK +holds for ∀ϕ ∈ span {Kx}x∈X . But +Tf (Kx) (y) = L (f (x) , y) , and +(2.52) +��� +� +i +ciL (f (xi) , y) +� +�� +� +Ff,y(xi) +���2 = +����� +� � +i +ciKxi, T ∗ +f (Ly) +� +HK +����� +2 +(2.53) +≤ +��T ∗ +f (Ly) +��2 +HK +� +i +� +j +cicjK (xi, xj) +� +�� +� +<∞ +(2.54) +and so by the basic lemma for HK (see the proof of Theorem 2.6), we conclude +that functions Ff,y ∈ HK, i.e., (x �−→ L (f (x) , y)) ∈ HK. +Conclusion: the bi-implication ⇐⇒ in (2.51) is valid. +□ +2.3. Functions and Operators. In general if T : H1 → H2 is an operator with +dense domain D ⊂ H1, where Hi, i = 1, 2, are two Hilbert spaces, we know that +T is closable ⇐⇒ T ∗ is densely defined, i.e., iff dom(T ∗) is dense in H2 (see e.g., +[JT21]). So we apply this to T = Tf, H1 = HK, H2 = HL, and the condition in +Theorem 2.6 holds ⇐⇒ Ly ∈ dom(T ∗ +f ) ∀y ∈ Y . Since span {Ly}y∈Y is dense in +HL, the condition in Theorem 2.6 =⇒ Tf is closable. +Given K and L as above, introduce +Fub (K, L) = {f : Tf is closable} , and +(2.55) +Fb (K, L) = {f : Tf is bounded from HK into HL} . +(2.56) +In both cases, the operators T = Tf depends on the choice of function X +f +−−→ Y , +but the two conditions (2.55) and (2.56) are different: +(Tf (Kx)) (y) = L (f (x) , y) = +� +(Tf)∗ (Ly) +� +(x) , +(2.57) +for all x ∈ X, and y ∈ Y . See details below: +Some general comments about the operator Tf : HK → HL. As before, K and +L are fixed p.d. kernels, and f : X → Y is a function. We need to understand the +conclusion from Theorem 2.6, i..e, when is +(X ∋ x �−→ L (f (x) , y)) ∈ HK for all y ∈ Y ? +(2.58) +Answer: (2.58) ⇐⇒ Ly ∈ dom(T ∗ +f ). + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +13 +Note that then the function in (2.58) is T ∗ +f (Ly); see (2.57). +But note that, +starting with a function X +f +−−→ Y , there are requirements for having (2.57) yield a +well defined linear operator Tf with dense domain in HK, s.t. +Tf (Kx) (·) = L (f (x) , ·) . +(2.59) +The case when Tf is bounded is easy since then dom(T ∗ +f ) = HL. Notationally, +L (f (x) , ·) ∈ Lf(x) ∈ HL, but we must also verify the implicit kernel function for +all finite sums: +� +i +� +j +cicjK (xi, xj) = 0 =⇒ +� +i +� +j +cicjL (f (xi) , f (xj)) = 0. +2.4. The case when K = L. As demonstrated in Section 3 below, for applications +to multi-level NNs, the recursive constructions simplify when the same p.d. kernel +K is used at each level. Hence below, we specialize to the case when X = Y , and +K = L; see the setting in Theorems 2.6 and 2.8. +Theorem 2.11. Consider a positive definite kernel K on X × X, and the corre- +sponding RKHS HK, i.e., the Hilbert completion of {Kx}x∈X where Kx := K (·, x). +Fix a function X +f +−−→ X with the property (see Theorem 2.6) that +(X ∋ x �−→ K (f (x) , y)) ∈ HK for all y ∈ X. +(2.60) +Hence, the operator Tf : HK → HK defined by +Tf (K (·, y)) := K (f (·) , y) +(2.61) +is a densely defined operator from HK into HK, with domain +DK := span {Kx}x∈X . +(2.62) +(1) Then the closure of Tf (also denoted Tf) is well defined and normal, i.e., +the two operators Tf and T ∗ +f commute. +(2) In particular, Tf has a projection-valued spectral resolution, i.e., there is a +projection-valued measure Q (·) on BC (= the Borel subsets in C) such that +Tf = +� +spect(Tf ) +λ Q (dλ) : DK → HK. +(2.63) +Proof. Note that part (2) follows from (1) and the Spectral Theorem for normal +operators (in the Hilbert space HK.) +Part (1). When the operator T ∗ +f is introduced, we get the following commuta- +tivity: +K (·, x) +Tf +� +T ∗ +f +� +K (·, f (x)) ∈ DK +T ∗ +f +� +K (f (·) , x) +Tf +� K (f (·) , f (x)) ∈ HK +Figure 2.2. Commutativity of Tf and T ∗ +f . +which is the desired conclusion (1). +□ + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +14 +Given a function f : X → X as in Theorem 2.11. +Below we make use of +the corresponding projection valued measure Q(f) from Theorem 2.11 in order to +establish an assignment from pairs (x, y) of points in X, into systems of complex +measures µ(x,y) on the spectrum of Tf. In this assignment, n-fold composition- +iteration of the function f yields the nth moment of each of the measures µ(x,y). +Corollary 2.12. Let K, X and f be as specified in Theorem 2.11, and let Q = +Q(K,f) (·) be the corresponding projection valued measure in (2.63). Then for every +pair x, y ∈ X, we get a corresponding Borel measure +µ(f) +x,y (B) = ⟨Kx, Q (B) Ky⟩HK = (Q (B) (Ky)) (x) , +(2.64) +for all B ∈ BC. Inductively, setting +f ◦n = f ◦ · · · ◦ f +� +�� +� +n fold +we arrive at the following moment formula for the respective complex measures: +µ(f) +f ◦n(x),y (B) = +� +B +λn µ(f) +x,y (dλ) . +(2.65) +We now turn to the role of multipliers in the RKHS HK. +Definition 2.13. A scalar valued function ϕ on X is said to be a multiplier for +HK iff one of the two equivalent conditions hold: +(1) The multiplication operator Mϕ acting on HK via MϕF = ϕF (via point- +wise product) leaves HK invariant. +(2) We have the following identity for the adjoint operator: +M ∗ +ϕ (Kx) = ϕ (x) Kx for all x ∈ X +(2.66) +where Kx denotes the kernel function Kx = K (·, x). +Remark 2.14. The equivalence of (1) and (2) follows from the standard reference +on RKHSs; see e.g., [JT21]. +Theorem 2.15. Let K be a fixed p.d. +kernel on X × X, and let HK be the +corresponding RKHS. Let X +f +−−→ X be a function such that (2.60) holds, i.e., +(X ∋ x �−→ K (f (x) , y)) ∈ HK for all y ∈ X. +Then for every multiplier ϕ for HK, we have: +M(ϕ◦f)T ∗ +f = T ∗ +f Mϕ. +(2.67) +Proof. It is clear that the conclusion (2.67) has the following equivalent form: +TfM ∗ +(ϕ◦f) = M ∗ +ϕTf; +(2.68) +and below we shall prove (2.68). +Let f and ϕ be as specified in the theorem. We then have the following commu- +tative diagram: + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +15 +K (·, x) +M ∗ +(ϕ◦f) +� +Tf +� +ϕ (f (x)) K (·, x) +Tf +� +K (·, f (x)) +M ∗ +ϕ +� ϕ (f (x)) K (·, f (x)) +Figure 2.3. +Commutative diagram corresponding to (2.68). +In the verification of the assertions in Figure 2.3, we used the conclusions in +Theorems 2.6 and 2.8 above. +□ +3. Neural Network-activation functions from p.d. kernels +In the previous section we introduced the use of positive definite kernels, and +associated generating function for the NN algorithms. Below we make use of the +kernel analysis in design of the generating NN functions. +The next definition makes use of the iterative generation of feedforward functions +as in the literature, e.g., [CC21, CKM22, Han16]. The recursive steps used here in +the definition and Lemma 3.2 below serve as applications of our general framework +from Theorems 2.6 and 2.8 above. +Definition 3.1. Let K be a positive definite kernel on R. An l-layer feedforward +network with kernel K is a function of the form +x �→ y1 = K (A1x + b1, c1) �→ y2 = K (A2y1 + b2, c2) �→ · · · +· · · �→ yl = K (Alyl−1 + bl, cl) �→ yout = K (⟨al+1, yl⟩ + bl+1, cl+1) +where +• x ∈ Rn0; +• Aj ∈ Rnj×nj−1, bj, cj ∈ Rnj for j = 1, · · · , l; +• al+1 ∈ Rnl, bl+1, cl+1 ∈ R; +and for vectors x, y ∈ Rm, +K (x, y) := [K (x1, y1) , · · · , K (xm, ym)] . +Lemma 3.2. Let K (x, y) = min (x, y), and a, b, c, d be nonzero constants. Then +(1) K (ax + b, c) = aK +� +x, a−1 (c − b) +� ++ b; +(2) K (K (x, a) , b) = K (x, K (a, b)); +(3) K (dK (ax + b, c) + e, f) = daK +� +x, K +� +a−1 (c − b) , a−1 � +d−1 (f − e) − b +��� ++ +db + e. +Proof. +(1) K (ax + b, c) = +� +ax + b +x < a−1 (c − b) +c +x > a−1 (c − b) +(2) Assume a < b, then +K (K (x, a) , b) = +� +x +x < a +a +x ≥ a = K (x, a) . +The case a > b is similar. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +16 +(3) This follows from (1)–(2): +K (dK (ax + b, c) + e, f) += +dK +� +K (ax + b, c) , d−1 (f − e) +� ++ e += +dK +� +aK +� +x, a−1 (c − b) +� ++ b, d−1 (f − e) +� ++ e += +d +� +aK +� +K +� +x, a−1 (c − b) +� +, a−1 � +d−1 (f − e) − b +�� ++ b +� ++ e += +daK +� +K +� +x, a−1 (c − b) +� +, a−1 � +d−1 (f − e) − b +�� ++ db + e += +daK +� +x, K +� +a−1 (c − b) , a−1 � +d−1 (f − e) − b +��� ++ db + e +□ +In what follows, all the networks are restricted to be defined on compact sub- +sets Ω in Rd, e.g., Ω = [0, 1]d (hypercubes). This is in consideration of standard +normalizations in training neural networks. +In Theorems 3.3 and 3.4, we present in detail the particular relative Reproducing +Kernel Hilbert Spaces which have as their respective dipole system (see (3.5)) the +generalized ReLu functions illustrated here in Figures 2.1 and 3.1. +Here we specify the kernel K1 for Brownian motion W indexed by R. As a result, +the corresponding p.d. kernel on R × R is as follows: +K1 (x, y) = +� +|x| ∧ |y| = min (|x| , |y|) +if xy ≥ 0 (so same sign) +0 +if xy < 0, so opposite sign. +(3.1) +Proof. The connection between the kernel K1 and the Brownian motion {Wx}x∈R +is as follows: +K1 (x, y) = E ((Wx − W0) (Wy − W0)) +(3.2) +for all x, y ∈ R. The asserted formula (3.1) follows from this, combined with the +independence of increments for Brownian motion. +□ +Theorem 3.3. Let K1 be the p.d. kernel (3.1) on R × R, with the corresponding +RKHS +HK1 = +� +f : f ′ ∈ L2� +, +∥f∥2 +HK1 = +� +|f ′|2 dλ1. +On Ω = [0, 1]d, consider the p.d. kernel +Kd (x, y) = K1 (x1, y1) · · · K1 (xd, yd) , +so that +HKd = +� +f : ∇f ∈ L2� +, +∥f∥2 +HKd = +� +|∇f|2 dλd, +where λd denotes the d-dimensional Lebesgue measure. +Given f : Ω → R, and a fixed c ∈ R, set +F : Rd → R1, +F (x) = K1 (f (x) , c) . +Then, +F ∈ HKd ⇐⇒ +�� +f −1([0,c]) +|∇f|2 dλd < ∞. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +17 +Theorem 3.4. Let µ be a non-atomic σ-finite measure on (R, BR), and consider +Stieltjes measures dF on (R, BR) such that +dF ≪ µ +(3.3) +(absolutely continuous). Then the relative RKHS Hµ for the p.d. kernel +Kµ (A, B) = µ (A ∩ B) +(3.4) +consists functions F such that +F (b) − F (a) = +� +v(µ) +a,b , F +� +Hµ +(3.5) +� +R +���� +dF +dµ +���� +2 +dµ < ∞ +(3.6) +where the relative kernels v(µ) +a,b are as follows: +dv(µ) +a,b +dµ +(x) = χ[a,b](x), +see Figure 3.1. +𝑣!,# +$ +𝑎 +𝑏 +𝜇 𝑎,𝑏 +Figure 3.1. Illustration dipole functions that reproduce differ- +ences of values of functions in the space Hµ. Compare with Figure +2.1 above. +Proof. See [AJV14, JT18] and the details in the proof of Theorem 3.3. +□ +Remark 3.5. The positive definite kernel Kµ which is “responsible” for the relative +RKHS Hµ is defined on B × B, where B denotes the Borel σ-algebra of subsets +of R. Using [JT18], one checks that +Kµ (A, B) := µ (A ∩ B) for all A, B ∈ B. +(3.7) +We further note that Kµ is the covariance for the generalized µ-Brownian motion +{W (µ) +A }A∈B, i.e., subject to +E +� +W (µ) +A W (µ) +B +� += µ (A ∩ B) for all A, B∈ B. +(3.8) +The corresponding Ito-lemma for W (µ) is defined for differentiable functions f on +R via +f +� +W (µ) +A +� +− f (0) = +� +A +f ′ � +W (µ) +t +� +dW (µ) +t ++ 1 +2 +� +A +f ′′ � +W (µ) +t +� +µ (dt) . +(3.9) +In particular, the measure µ is the quadratic variation of W (µ) +t +. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +18 +4. Applications to fractal images +In recent decades, it has become evident that fractal features arise in diverse +datasets, in time series and in image analysis, to mention two. (See e.g., [KLW20, +KLLW21].) Perhaps the best known examples of fractal features include precise +symmetries of scales. Via a prescribed system of affine maps, they take the form +of self-similarity. And a special case, includes iterated function systems (IFS), and +maximal-entropy measures, also called IFS measures. The more familiar Cantor +constructions, e.g., scaling by 3 or scaling by 4, are examples of IFS measures. For +each of these cases, the RKHS framework we present in Section 3, serve as ideal +tools for such adapted NN algorithms. In particular, this may be illustrated with +large numbers of images, say 5000 generated images, each one is a fractal, either +2D or 3D, with random rotation, with zooming, and coloring; half of them have +scaling 3, the other half have scaling 4. This leads to training of a network serving +to classify the images by scaling factors. +In particular, the Cantor-type activation functions, or the cumulative functions of +Cantor-like measures (Figure 3.1), have vanishing derivatives over structured subin- +tervals of [0, 1]. This feature may lead to several benefits in neural networks. For +example, such functions can introduce sparsity and regularization into the network, +which improves its generalization performance and reduces the risk of overfitting. +Additionally, these functions can make the network more robust to noise and other +perturbations in the input data, which improves its performance on unseen data. +Furthermore, activation functions whose derivative is zero over subintervals allow +the network to learn more complex and non-linear patterns in the data. This can +improve the expressiveness and flexibility of the network, making it more accurate +and effective for a wider range of tasks. Additionally, these functions can make +the network easier to optimize and train, since the gradient of the activation func- +tion is well-structured, thus reduce the computational complexity and improve the +convergence rate of the training algorithm. +More generally, a neural network with a custom activation function (see e.g. +the dipoles in Figure 2.1) uses a non-standard activation function with adjustable +parameters that can be trained and optimized during the learning process. This +allows the network to learn more complex and non-linear relationships between the +input and output data, which can improve the accuracy of the network’s predictions. +The use of a custom activation function with trainable parameters can be useful +in a variety of applications, such as image recognition, natural language processing, +and time series forecasting. It can also be used to improve the performance of other +machine learning algorithms, such as decision trees and support vector machines +(see e.g., [CST01, PORSTS21, HSTHD11, HST10, RSSST06, STWCK05]). +Below we apply a custom activation function in a ConvNet to classify fractal +images. In this setting, the activation function should be designed to capture the +complex, self-similar patterns that are characteristic of the fractal images. The +network is trained on a dataset of fractal images with corresponding labels. It is +optimized using a gradient-based algorithm, such as stochastic gradient descent. +Once trained, the network will be used to classify new fractal images and predict +their classes with high accuracy. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +19 +In the example below, a dataset1 of 15,000 Cantor-like 3D images is generated +in Mathematica. Parameters of each image, such as zoom factor, viewing angle, +and scaling factor, are uniformly distributed. A sample of the images is shown in +Figure 4.1. +The images are split into three categories according to their scaling factors, +labeled as class “1”, “2” and “3”, respectively. The entire dataset is divided into +a training set (size = 10,000), validation set (size = 2,500) and test set (size = +2,500). The task is to train a ternary classifier using the training set, along with +the validation set (for model selection), whose performance is then tested on the +test images. +In the experiment, a small ConvNet is implemented in Keras. Its architecture is +shown in Figure 4.2. The loss and accuracy of the model are recorded for 20 epochs +(Figure 4.3). +In comparison with a standard Relu network (Figure 4.3a) of the same archi- +tecture, the use of Cantor-like activation is better at reducing overfitting (Figure +4.3b); it is expected that with a systematic hyperparameter tuning, such a network +has the potential to outperform Relu networks in certain applications. +Class 1 +Class 2 +Class 3 +Figure 4.1. +A random sample of the dataset of 3D Cantor im- +ages. +1Available at https://www.kaggle.com/dsv/4791103. + +OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +20 +Model: "Cantor-activation" +Layer (type) +Output Shape +Param # +input_1 (InputLayer) +(None, 128, 128, 3) +0 +rescaling (Rescaling) +(None, 128, 128, 3) +0 +conv2d (Conv2D) +(None, 126, 126, 16) +448 +max_pooling2d (MaxPooling2D) +(None, 63, 63, 16) +0 +conv2d_1 (Conv2D) +(None, 61, 61, 32) +4640 +max_pooling2d_1 (MaxPooling2D) +(None, 30, 30, 32) +0 +conv2d_2 (Conv2D) +(None, 28, 28, 64) +18496 +max_pooling2d_2 (MaxPooling2D) +(None, 14, 14, 64) +0 +conv2d_3 (Conv2D) +(None, 12, 12, 128) +73856 +flatten (Flatten) +(None, 18432) +0 +dense (Dense) +(None, 3) +55299 +Figure 4.2. +A ConvNet for fractal image classification. +(a) Relu activation +(b) Cantor-like activation +Figure 4.3. +Training loss and validation loss, illustrated with +use of a ConvNet. + +16 +training loss +validation loss +14 +12 +LD +SSI +0.6 +t0 +0.2 +0.0 +25 +5.D +7.5 +14.0 +12.5 +15.0 +17.5 +24.0 +Epoch&1DO +0.95 +0.90 +0.B5 +fed +Accur +0.BO +0.75 +0.70 +0.65 +training accuracy +validation accuracy +25 +5.D +7.5 +14.0 +12.5 +15.0 +17.5 +24.0 +Epachs175 +training loss +150 +validation loss +125 +SS +0.75 +0.50 +0.25 +0.0 +25 +5.D +7.5 +14.0 +12.5 +15.0 +17.5 +24.0 +EpachsLD +60 +0.B +80.7 +0.6 +training accuracy +0.5 +validation accuracy +25 +5.D +7.5 +14.0 +12.5 +15.0 +17.5 +20.0 +EpachsOPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS +21 +References +[AJ21] +Daniel Alpay and Palle E. 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Jorgensen) Department of Mathematics, The University of Iowa, Iowa +City, IA 52242-1419, U.S.A. +Email address: palle-jorgensen@uiowa.edu +(Myung-Sin Song) Department of Mathematics and Statistics, Southern Illinois +University Edwardsville, Edwardsville, IL 62026, USA +Email address: msong@siue.edu +(James F. Tian) Mathematical Reviews, 416 4th Street Ann Arbor, MI 48103-4816, +U.S.A. +Email address: jft@ams.org + diff --git a/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/load_file.txt b/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ff748e397e34076270322f6e9877e5125b0c8dc --- /dev/null +++ b/UtAzT4oBgHgl3EQfX_xg/content/tmp_files/load_file.txt @@ -0,0 +1,1041 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf,len=1040 +page_content='OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS PALLE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' JORGENSEN, MYUNG-SIN SONG, AND JAMES TIAN Dedicated to the memory of Professor Ka-Sing Lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In this paper we show how specific families of positive definite kernels serve as powerful tools in analyses of iteration algorithms for multiple layer feedforward Neural Network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Our focus is on particular kernels that adapt well to learning algorithms for data-sets/features which display intrinsic self-similarities at feedforward iterations of scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Neural networks (NN), and reproducing kernel Hilbert spaces (RKHS) 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Basis approach 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Dual pairs of operators 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Functions and Operators 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The case when K = L 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Neural Network-activation functions from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernels 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Applications to fractal images 18 References 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Introduction Recently many authors have offered diverse approaches to feedforward Neural Network (NN) algorithms [ZC23, AK23, HL23, DWZ+23], as well as optimization terms based on kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Here we establish some new results in operator theory, and we bring them to bear on the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The list of applications of feedforward NN models includes a variety of machine learning settings, and deep NN based on kernels [GK23, MK23, BSW23, SHO22, GKNV22, GPR+21, Kut20, GKP20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A common theme in feedforward NN models is specific prescribed iterations which entail (i) ReLu functions [JR23, OSZ22, CKM22, CCK22, JR22, CL22], (ii) substitution from prescribes systems of affine mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, (iii) each step is then linked to the next with a choice of an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In this paper we show that there are natural positive definite kernels associated with the three steps going into feedforward NN constructions, as well as to their iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 41A30, 46E22, 47B32, 68T07, 92B20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' algorithms, multipliers, spectral resolutions, normal operators, it- erated function systems, fractal measures, feedforward neural network, explicit kernels, ReLU, reproducing kernel Hilbert spaces, positive definite kernels, composition operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='01327v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='LG] 3 Jan 2023 OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 2 believe that this then yields a more direct tool for kernel-based feedforward NN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This advantage of our approach is based on two facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' First, we identify a direct notion of kernel iteration which accounts for traditional function theoretic feedforward NN steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Secondly, our approach offers a more direct and natural choices of kernels which govern approximations involved in deep NN models, for example graph NN constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' While positive definite kernels and their associated reproducing kernel Hilbert spaces have found diverse applications in pure and applied mathematics, we shall focus here on a new role of kernels in feedforward network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In more detail, the main purpose of our paper is a presentation of choices of particular families of positive definite kernels which serve as powerful tools in analyses of multiple layer feedforward Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In general, reproducing kernel constructions, and the corresponding RKHSs are powerful tools in diverse applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In the present framework of kernel neural networks (KNNs) , their role may be summarized as follows: Starting with the problem at hand, when we build our RKHS(µ) via IFS iterations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', via Cantor- like fractal limits), then the Cantor-like µ activation functions arise as relative reproducing dipole functions for RKHS(µ) as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Neural networks (NN), and reproducing kernel Hilbert spaces (RKHS) A main theme in our paper is a development of new tools for design of feedforward Neural Network constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For this purpose we point out the use of positive definite kernels, and associated generating function for the NN algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' These kernel based functions include the more familiar ReLu functions, see Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We stress that the particular RKHS constructs will be relative in the sense of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the inner product reproduces differences of function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Our approach to the use of kernels and functions for feedforward Neural Network (NN) algorithms, is based on a systematic study of two classes of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' They act as follows: (i) between prescribed kernel Hilbert spaces, and (ii) other operators acting at indexed levels in the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', operators acting at fixed levels, so within choices of kernel Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Case (i) includes a systematic study of composition operators (see Corollaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='9) in the context of kernel Hilbert spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' and case (ii), the study of multiplier operators and their adjoints, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We emphasize that the two classes of operations discussed below depend on choices of kernels at each level in particular NN-network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Together these families of operators allow for realizations of black box filter-entries in associated generalized multi-resolutions systems, including operators which consist of composition followed by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Specific 3D applications are presented in the subsequent sections, secs 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Inside the paper we shall work with Hilbert spaces of functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', reproducing kernel Hilbert spaces (RKHSs), L2 spaces, and Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' It will be assumed that these are Hilbert spaces of real valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Inner prod- ucts will be written ⟨·, ·⟩, and we shall use subscripts on ⟨·, ·⟩ to indicate the Hilbert space under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, in our use of differentiation, or differential operators, we shall mean weak derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', differentiation in the sense of dis- tributions, or making use of the natural duality for the spaces under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 3 Our restriction here to the real valued case is dictated by our present applications to feedforward Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' However, many of our general results in Section 2 below extend to complex RKHS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The latter in turn are important in the study of geometry and potential theory of complex domains, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [Eng96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The power of kernel machines derives in part from the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' First, kernel machines serve to map points in a low-dimensional data sets (typically non- linear) into higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The dimensionality of this linear “hyperspace” may be infinite but is designed for optimization and efficient encoding of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Hence the kernel method allows one to find coefficients of separating hyperplanes for the problem at hand via RKHS-inner products, one selected for each pair of high- dimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' While kernel machines of various types have been used for decades, it was with the invention of support vector machines (SVMs) that kernels have now taken center stage (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [CST01, PORSTS21, HSTHD11, HST10, RSSST06, STWCK05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' By now, SVMs are used in diverse applications, including in bioinformatics (for finding similarities between different protein sequences), ma- chine vision, and handwriting recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Deep neural networks (to be discussed in Sections 3 and 4 below) are made of layers of artificial neurons: input layer, an output layer, and multiple hidden layers in-between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Deeper the networks have more hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The parameters of the network represent the strengths of the connections between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Training a network yield determination of values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Once trained, the ANN represents a model for turning an input (say, an image) into an output (a label or category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The variety of uses of forward Neural Network algorithms, the recent literature is substantial and diverse, especially with regards to applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [ASA+23, MM23, KG22, CC21, MCA20, Han16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The following lemma is a basic result in the theory of RKHSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For details, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [JT22] and the papers cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Fix a p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel X × X K −−→ R (or C), let HK denote the corre- sponding RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then a function F on X is in HK if and only if there exists a constant CF < ∞, such that the following estimate holds for all n ∈ N, all (ξi)n i=1, ξi ∈ R (or C), and all (xi)n i=1, xi ∈ X: ����� n � i=1 ξiF (xi) ����� 2 ≤ CF n � i=1 n � j=1 ξiξjK (xi, xj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' With the construction K �→ HK (referring to a RKHS of a fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel K), we arrive at the following two conclusions: (1) For all x ∈ X, the function Kx := K (·, x) is in HK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' and (2) For all F ∈ HK, and x ∈ X, we have F (x) = ⟨F, K (·, x)⟩HK , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the values of functions F in HK are reproduced via the inner product ⟨·, ·⟩HK, and the kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In addition to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2), we shall also consider relative reproducing kernels, and relative RKHSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' As noted in [AJV14], the relative reproducing property takes the following form F (y) − F (x) = ⟨F, vx,y (·)⟩Hrel , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3) OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 4 now valid for all pairs of points x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' So this entails double-indexed kernel functions vx,y ∈ Hrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A particular class of Hrel spaces are considered in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' There the setting is X = R, and the relative kernel functions va,b take the form of activation functions for classes of feedforward-NN-algorithms, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A systematic study of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3) is undertaken in [AJV14] where it is shown that the setting of relative reproducing is characterized by conditionally negative definite functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We now recall the RKHS for the standard 1-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [JT22, AJ21, JT21, JT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=') Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' When K is the Brownian motion kernel on R≥ × R≥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', K (x, y) = x ∧ y = |x| + |y| − |x − y| 2 , x, y ∈ R≥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4) the corresponding RKHS HK is the Hilbert space of absolutely continuous functions f on R such that the derivative f ′ = df/dx is in L2 (R), and f (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, ∥f∥2 HK = � R≥ |f ′ (x)|2 dx, for all f ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The key observation is that, if x > 0, the function R≥ ∋ y �−→ Fx (y) := K (y, x) = � y if y ≤ x x if y > x (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6) has weak derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Indeed, we have dFx dy = χ[0,x], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the indicator function of the interval [0, x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Hence if f is a function with f ′ ∈ L2 (R) and f (0) = 0, then f (x) = f (x) − f (0) = � x 0 f ′ (y) dy = � R F ′ x (y) f ′ (y) dy, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8) and the RHS in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8) is the inner product from the Hilbert space defined by the RHS in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The corresponding implication follows from the general theory of RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Recall that the RKHS of a kernel is a Hilbert space completion of the functions y �−→ K (x, y) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='9) as x varies over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, for K (x, y) = x ∧ y, ⟨K (·, x1) , K (·, x2)⟩HK = K (x1, x2) = x1 ∧ x2, and we can compute as follows: � R � d dy K (·, x1) � � d dy K (·, x2) � dy = � R χ[0,x1] (y) χ[0,x2] (y) dy = λ ([0, x1] ∩ [0, x2]) = x1 ∧ x2 = K (x1, x2) where λ = dy denotes the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 5 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Note that the functions va,b (called dipoles) in HK which satisfy f (b) − f (a) = ⟨f, va,b⟩ , for all f ∈ HK (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8)) are as follows: va,b (x) = � � � � � 0 if x < a x − a if a ≤ x < b b − a if x > b, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Also compare with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1, and the iterations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 𝑥 − 𝑎 𝑏 − 𝑎 𝑎 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=',#(⋅) 𝑏 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The generating dipole function {va,b} indexed by pairs a, b such that a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Compare with Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Induced metrics For a general p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel K on X × X, there is an induced metric on X, dK : X × X → R+ defined as (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [AJ21]) dK (x, y) = ∥K (·, x) − K (·, y)∥2 HK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='10) In particular, dK (x, y) = K (x, x) + K (y, y) − 2ℜ {K (x, y)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Note that d1/2 K is also a metric on X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For K (x, y) = x ∧ y on R × R as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4), ∥K (·, s) − K (·, t)∥2 HK = ��(· ∧ s)′ − (· ∧ t)′��2 L2 = ��χ[0,s] − χ[0,t] ��2 L2 = |s − t| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The results below deal with a general framework of pairs of sets X and Y , each equipped with a positive definite kernel, K resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', L, K on X, and L on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' With view to realization of feedforward Neural Network-functions, we will present an explicit framework (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='20)) which allows us to pass from (nonlinear) functions f : X → Y to linear operators Tf acting between the respective RKHSs HK and HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This will be a representation in the sense that composition of functions will map into products of the corresponding linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Some care must be exercised as the linear operators Tf will in general be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Nonetheless, we shall show that the operators still fall in a class where spectral resolutions are available, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 6 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Consider p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernels X × X K −−→ R and Y × Y L −−→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let f : X → Y be Lipschitz continuous with respect to the induced metrics dK, dL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', dL (f (x) , f (y)) ≤ cfdK (x, y) , x, y ∈ X, for some constant cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Define the operator Tf : HK → HL by Tf (Kx) (y) = L (f (x) , y) and extend it by linearity and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then, for any fixed c ∈ Y , the function Ff : X → R, Ff (x) := L (f (x) , c) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11) is in the RKHS HK if, and only if Lc ∈ dom(T ∗ f ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='12) the domain of the adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, Ff,y (x) := L (f (x) , y) ∈ HK, ∀y ∈ Y ⇕ Tf is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (See also Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let the setting be as in the statement of the theorem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', X f −−→ Y assumed continuous with respect to the two metrics, dK on X and dL on Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' so in particular, for pairs of points x1, x2 ∈ X, we have dK (x1, x2) = ∥K (·, x1) − K (·, x2)∥2 HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='13) = K (x1, x1) + K (x2, x2) − 2K (x1, x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='14) We further fix a point c ∈ Y , and set F = Ff,c, specified as follows: F (x) = L (f (x) , c) , for all x ∈ X, so F : X → R+ ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Now, for every N, and every subset SN = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' , xN) ⊂ X, consider the following matrix operations (in N dimensions): F �� N := � �� F (x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' F (xN) � �� � �� � column vectors , and QN := ��F �� N ⟩⟨ F �� N �� � �� � matrix of a rank-1 operator , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='15) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the rank-1 operator on RN written in Dirac’s notation, defined as QN (ξ) = ⟨FN, ξ⟩ FN (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='16) for all ξ ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Set KN := (K (xi, xj))N i,j=1 = � �� K (x1, x1) · · K (x1, xN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' K (xN, x1) · · K (xN, xN) � �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='17) a sample matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 7 For the convex cone of all positive definite N × N matrices, we introduce the following familiar ordering, K ≪C K′ iff (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=') ∃C < ∞ such that ξT KNξ ≤ CξT K′ Nξ for all ξ ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='18) Now an application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 above shows that the assertion in the theorem is equivalent to the existence of a finite constant C (independent of SN = (xi)N i=1) satisfying QN ≪C KN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the estimate ����� � i ξL (f (xi) , c) ����� 2 ≤ C � i � j ξiK (xi, xj) ξj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='19) holds for all N, all SN = {xi}N i=1, and all ξ ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We get this from the assumption on f in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' See details below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ Summary of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6: Start with K p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' on X × X, L p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' on Y × Y , and f : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We introduce the metrics dK on X, and dL on Y , and we consider f continuous, or Lipchitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' To get the desired conclusion (x �−→ L (f (x) , y)) ∈ HK, we must introduce an operator Tf : HK → HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The right choice is Ly ∈ dom(T ∗ f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' See details below: Fixing two kernels K and L, assumed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' on X × X, and on Y × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Pass to the corresponding RKHS HK and HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Find conditions on functions X f −−→ Y with the property that, for ∀y ∈ Y , then the induced function (X ∋ x �−→ L (f (x) , y)) � �� � Ff,y(·) as a function on X ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='20) The argument stressed below is via dual operators (bounded) HK Tf � HL T ∗ f � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' but the unbounded case is also interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Some remarks on the definition of the operator Tf : HK → HL in the case when no additional assumptions are placed on X f −−→ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We define Tf (Kx) (y) = L (f (x) , y) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' and so we extend Tf to linear combinations: DK := � � i ciKxi � �� � � function on X Tf −−−−−→ � i ciL (f (xi) , ·) � �� � function on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='21) But to make sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='21) so it is well defined, we must be careful with equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 8 If f : X → Y is a general function, the (generalized) operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='21) may be non-closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' However, we can still define the adjoint T ∗ f , but its domain might be “small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Set DK := span {Kx}x∈X , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='22) then (Definition) a vector ψ ∈ HL is in dom(T ∗ f ) iff ∃Cψ < ∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' ���⟨Tfϕ, ψ⟩HL ��� ≤ Cψ ∥ϕ∥HK , ∀ϕ ∈ DK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='23) We then set T ∗ f ψ = the solution to ⟨Tfϕ, ψ⟩HL = � ϕ, T ∗ f ψ � HK , ∀ϕ ∈ DK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='24) Let K, L, f be as specified, and assume for some y ∈ Y , that we have Ly ∈ dom(T ∗ f ), then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='24) for Ly ∈ HL T ∗ f −−→ HK, ϕ = Kx, ψ = Ly yields x �−→ T ∗ f (Ly) (x) = L (f (x) , y) ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' So the conclusion in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 that (x �−→ L (f (x) , y)) ∈ HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='25) holds iff Ly ∈ dom(T ∗ f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In this case there are no difficulties with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='21) and we get a dual pair Tf and T ∗ f , ⟨Tfϕ, ψ⟩HL = � ϕ, T ∗ f ψ � HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='26) for ∀ϕ ∈ DK, and ψ ∈ DL, HK Tf � HL T ∗ f � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Setting ϕ = Kx, and ψ = Ly, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='26) implies T ∗ f (Ly) (x) = L (f (x) , y) = (Tf (Kx)) (y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='27) But the previous condition Ly ∈ dom(T ∗ f ) (compare (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='26)) amounts to the asser- tion that T ∗ f (Ly) ∈ HK, and by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='27), this is then the conclusion for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='7 (composition operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let X, Y , K and L be as specified above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' in particular, K is a fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel on X × X, and the RKHS HK is a Hilbert space of scalar valued functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Similarly, HL is a Hilbert space of scalar valued functions on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Both HK and HL satisfy the defining axioms for RKHSs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' As noted, every function f, X f −−→ Y , induces a linear operator HK Tf −−→ HL, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='28) with dense domain DK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' see the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For the adjoint operator T ∗ f , HL T ∗ f −−→ HK, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='29) we have the following: For a function ψ in HL, the two characterizations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31) hold: ψ ∈ dom(T ∗ f ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30) OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 9 ⇕ ψ ◦ f ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31) In the affirmative, T ∗ f (ψ) = ψ ◦ f : X → R, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='32) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', T ∗ f is the composition operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30)⇒(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30), we then apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='23), and get Cψ < ∞ with: ����� � i ci � T ∗ f (ψ) � (xi) ����� 2 ≤ Cψ � i � j cicjK (xi, xj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='33) But T ∗ f (ψ) (xi) = � Kxi, T ∗ f (ψ) � HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='34) = ⟨Tf (Kxi) , ψ⟩HL = ⟨L (f (xi) , ·) , ψ⟩HL = ψ (f (xi)) , where we used the RKHS property for HL in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Substitution into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='33) yields ����� � i ciψ (f (xi)) ����� 2 ≤ Cψ � i � j cicjK (xi, xj) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='35) and, so by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 applied to F = ψ ◦ f, conclusion in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31)⇒(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Assume (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='31), we then reverse the above reasoning to get ��� � Tf �� i ciKxi � � �� � ∈DK , ψ � HL ��� ≤ � Cψ ����� � i ciKxi ����� HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='36) which states that ψ ∈ dom(T ∗ f ), which is condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Now combine this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='34), and we conclude that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='32) is satisfied for ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', that T ∗ f ψ = ψ◦f holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Basis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let X, Y, K, L, f be as usual, and define Tf : HK → HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Since K is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' on X × X, the RKHS HK allows an ONB {hi}i∈N, hi ∈ HK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' by general theory, we get the pointwise formula: K (x1, x2) = � i∈N hi (x1) hi (x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='37) Then our condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 is equivalent to the following: (X ∋ x �−→ L (f (x) , y)) ∈ HK (a) ⇕ � i∈N |(Tf (hi)) (y)|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (b) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (a) ⇒ (b) is detailed below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' but the converse implication will follow by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' So by (a), Ly ∈ dom(T ∗ f ) and therefore: T ∗ f (Ly) (·) ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='38) OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 10 Since {hi} is an ONB in HK, � i ��� � hi, T ∗ f (Ly) � �� � ∈HK ���� 2 = ��T ∗ f (Ly) ��2 HK < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='39) But from the LHS(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='39): � hi, T ∗ f (Ly) � HK = ⟨Tf (hi) , Ly⟩HL = (Tf (hi)) (y) , and (b) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ Key Question: When is Ff,y (·) ∈ HK?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The cleanest answer to the question of what functions X f −−→ Y have the property that Ff,y (x) = L (f (x) , y) is in HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='40) is this: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K, L and f be given, then Ff,y in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='40) is in HK ⇐⇒ Ly ∈dom(T ∗ f ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='41) where the operator Tf : HK → HL is given by Tf (Kx) := L (f (x) , ·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Moreover, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='41) holds for all y ∈ Y ⇐⇒ Tf is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Dual pairs of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Consider a symmetric pair of operators with dense domains: HK Tf � HL T ∗ f � (T = Tf, since it will depend on f) where span {Kx}x∈X is dense in HK (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='42) and span {Ly}y∈Y is dense in HL (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='43) such that Kx ∈ dom(Tf), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='44) Ly ∈ dom(T ∗ f ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='45) where “dom” denotes the respective operator domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We note that Tf (Kx) (·) = L (f (x) , ·) ∈ HL is always well defined, with dense domain, but the secret is T ∗ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Also note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='45) is the condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 11 Let f : X → Y be as before, and the two p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernels K and L are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We then introduce the corresponding (densely defined) operator Tf : HK → HL by setting Tf (Kx) = L (f (x) , ·) ∈ HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='46) Notation and convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Kx (·) is the kernel function in HK as usual: Kx (t) = K (x, t) , ∀t ∈ X and similarly, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='47) Ly (u) = L (y, u) , ∀u ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='48) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' When Ly ∈ dom(T ∗ f ) then � T ∗ f (Ly) � (x) = L (f (x) , y) on X, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='49) equivalently, T ∗ f (Ly) (·) = L (f (·) , y) on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='50) Proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The conclusion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='49) is equivalent to the following assertion: � Tf (Kx) (·) � �� � L(f(x),·) , Ly � HL � = � Kx, ∈HK � �� � L (f (·) , y) � �� � T ∗ f (Ly) � HK � L (f (x) , y) The conclusion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='49) follows since the respective kernel functions span dense sub- spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ Recall, the function x �−→ Ff,y (x) = L (f (x) , y) ∈ HK ⇕ Ly ∈ dom(T ∗ f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Assume that Ly ∈ dom(T ∗ f ) , then apply the condition for functions in HK to Ff,y (·), so ∀n, ∀ (xi)n 1, ∀ (ci)n 1, ci ∈ R: ����� � i ciFf,y (xi) ����� 2 = ����� � i ciL (f (xi) , y) ����� 2 ≤ ������ �� i ciKxi, T ∗ f (Ly) � HK ������ 2 ≤ Schwarz ����� � i ciKxi ����� 2 HK ��T ∗ f (Ly) ��2 HK = � i � j cicjK (xi, xj) ��T ∗ f (Ly) ��2 HK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 12 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The implication below is both directions: X ∋ x �−→ L (f (x) , y) ∈ HK for ∀y ⇕ the condition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Even if we fix y ∈ Y , then Ly ∈ dom(T ∗ f ) ⇐⇒ (x �−→ L (f (x) , y)) ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='51) Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' By definition, Ly ∈ dom(T ∗ f ), ∃Cy < ∞ ⇐⇒ ���⟨Tfϕ, Ly⟩HL ��� = |(Tf (ϕ)) (y)| ≤ Cy ∥ϕ∥HK holds for ∀ϕ ∈ span {Kx}x∈X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' But Tf (Kx) (y) = L (f (x) , y) , and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='52) ��� � i ciL (f (xi) , y) � �� � Ff,y(xi) ���2 = ����� � � i ciKxi, T ∗ f (Ly) � HK ����� 2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='53) ≤ ��T ∗ f (Ly) ��2 HK � i � j cicjK (xi, xj) � �� � <∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='54) and so by the basic lemma for HK (see the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6), we conclude that functions Ff,y ∈ HK, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', (x �−→ L (f (x) , y)) ∈ HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Conclusion: the bi-implication ⇐⇒ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='51) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Functions and Operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In general if T : H1 → H2 is an operator with dense domain D ⊂ H1, where Hi, i = 1, 2, are two Hilbert spaces, we know that T is closable ⇐⇒ T ∗ is densely defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', iff dom(T ∗) is dense in H2 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [JT21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' So we apply this to T = Tf, H1 = HK, H2 = HL, and the condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 holds ⇐⇒ Ly ∈ dom(T ∗ f ) ∀y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Since span {Ly}y∈Y is dense in HL, the condition in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 =⇒ Tf is closable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Given K and L as above, introduce Fub (K, L) = {f : Tf is closable} , and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='55) Fb (K, L) = {f : Tf is bounded from HK into HL} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='56) In both cases, the operators T = Tf depends on the choice of function X f −−→ Y , but the two conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='55) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='56) are different: (Tf (Kx)) (y) = L (f (x) , y) = � (Tf)∗ (Ly) � (x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='57) for all x ∈ X, and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' See details below: Some general comments about the operator Tf : HK → HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' As before, K and L are fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernels, and f : X → Y is a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We need to understand the conclusion from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='.e, when is (X ∋ x �−→ L (f (x) , y)) ∈ HK for all y ∈ Y ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='58) Answer: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='58) ⇐⇒ Ly ∈ dom(T ∗ f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 13 Note that then the function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='58) is T ∗ f (Ly);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' But note that, starting with a function X f −−→ Y , there are requirements for having (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='57) yield a well defined linear operator Tf with dense domain in HK, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Tf (Kx) (·) = L (f (x) , ·) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='59) The case when Tf is bounded is easy since then dom(T ∗ f ) = HL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Notationally, L (f (x) , ·) ∈ Lf(x) ∈ HL, but we must also verify the implicit kernel function for all finite sums: � i � j cicjK (xi, xj) = 0 =⇒ � i � j cicjL (f (xi) , f (xj)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The case when K = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' As demonstrated in Section 3 below, for applications to multi-level NNs, the recursive constructions simplify when the same p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel K is used at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Hence below, we specialize to the case when X = Y , and K = L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' see the setting in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Consider a positive definite kernel K on X × X, and the corre- sponding RKHS HK, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the Hilbert completion of {Kx}x∈X where Kx := K (·, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Fix a function X f −−→ X with the property (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6) that (X ∋ x �−→ K (f (x) , y)) ∈ HK for all y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='60) Hence, the operator Tf : HK → HK defined by Tf (K (·, y)) := K (f (·) , y) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='61) is a densely defined operator from HK into HK, with domain DK := span {Kx}x∈X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='62) (1) Then the closure of Tf (also denoted Tf) is well defined and normal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', the two operators Tf and T ∗ f commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2) In particular, Tf has a projection-valued spectral resolution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', there is a projection-valued measure Q (·) on BC (= the Borel subsets in C) such that Tf = � spect(Tf ) λ Q (dλ) : DK → HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='63) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Note that part (2) follows from (1) and the Spectral Theorem for normal operators (in the Hilbert space HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=') Part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' When the operator T ∗ f is introduced, we get the following commuta- tivity: K (·, x) Tf � T ∗ f � K (·, f (x)) ∈ DK T ∗ f � K (f (·) , x) Tf � K (f (·) , f (x)) ∈ HK Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Commutativity of Tf and T ∗ f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' which is the desired conclusion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 14 Given a function f : X → X as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Below we make use of the corresponding projection valued measure Q(f) from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11 in order to establish an assignment from pairs (x, y) of points in X, into systems of complex measures µ(x,y) on the spectrum of Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In this assignment, n-fold composition- iteration of the function f yields the nth moment of each of the measures µ(x,y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K, X and f be as specified in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='11, and let Q = Q(K,f) (·) be the corresponding projection valued measure in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then for every pair x, y ∈ X, we get a corresponding Borel measure µ(f) x,y (B) = ⟨Kx, Q (B) Ky⟩HK = (Q (B) (Ky)) (x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='64) for all B ∈ BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Inductively, setting f ◦n = f ◦ · · · ◦ f � �� � n fold we arrive at the following moment formula for the respective complex measures: µ(f) f ◦n(x),y (B) = � B λn µ(f) x,y (dλ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='65) We now turn to the role of multipliers in the RKHS HK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A scalar valued function ϕ on X is said to be a multiplier for HK iff one of the two equivalent conditions hold: (1) The multiplication operator Mϕ acting on HK via MϕF = ϕF (via point- wise product) leaves HK invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2) We have the following identity for the adjoint operator: M ∗ ϕ (Kx) = ϕ (x) Kx for all x ∈ X (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='66) where Kx denotes the kernel function Kx = K (·, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The equivalence of (1) and (2) follows from the standard reference on RKHSs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [JT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K be a fixed p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel on X × X, and let HK be the corresponding RKHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let X f −−→ X be a function such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='60) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', (X ∋ x �−→ K (f (x) , y)) ∈ HK for all y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then for every multiplier ϕ for HK, we have: M(ϕ◦f)T ∗ f = T ∗ f Mϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='67) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' It is clear that the conclusion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='67) has the following equivalent form: TfM ∗ (ϕ◦f) = M ∗ ϕTf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='68) and below we shall prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let f and ϕ be as specified in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' We then have the following commu- tative diagram: OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 15 K (·, x) M ∗ (ϕ◦f) � Tf � ϕ (f (x)) K (·, x) Tf � K (·, f (x)) M ∗ ϕ � ϕ (f (x)) K (·, f (x)) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Commutative diagram corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In the verification of the assertions in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3, we used the conclusions in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Neural Network-activation functions from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernels In the previous section we introduced the use of positive definite kernels, and associated generating function for the NN algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Below we make use of the kernel analysis in design of the generating NN functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The next definition makes use of the iterative generation of feedforward functions as in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [CC21, CKM22, Han16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The recursive steps used here in the definition and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2 below serve as applications of our general framework from Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K be a positive definite kernel on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' An l-layer feedforward network with kernel K is a function of the form x �→ y1 = K (A1x + b1, c1) �→ y2 = K (A2y1 + b2, c2) �→ · · · · · �→ yl = K (Alyl−1 + bl, cl) �→ yout = K (⟨al+1, yl⟩ + bl+1, cl+1) where x ∈ Rn0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Aj ∈ Rnj×nj−1, bj, cj ∈ Rnj for j = 1, · · · , l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' al+1 ∈ Rnl, bl+1, cl+1 ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' and for vectors x, y ∈ Rm, K (x, y) := [K (x1, y1) , · · · , K (xm, ym)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K (x, y) = min (x, y), and a, b, c, d be nonzero constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then (1) K (ax + b, c) = aK � x, a−1 (c − b) � + b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (2) K (K (x, a) , b) = K (x, K (a, b));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (3) K (dK (ax + b, c) + e, f) = daK � x, K � a−1 (c − b) , a−1 � d−1 (f − e) − b ��� + db + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (1) K (ax + b, c) = � ax + b x < a−1 (c − b) c x > a−1 (c − b) (2) Assume a < b, then K (K (x, a) , b) = � x x < a a x ≥ a = K (x, a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The case a > b is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' KERNELS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' AND FEEDFORWARD NEURAL NETWORKS 16 (3) This follows from (1)–(2): K (dK (ax + b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' c) + e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' f) = dK � K (ax + b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' c) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' d−1 (f − e) � + e = dK � aK � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 (c − b) � + b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' d−1 (f − e) � + e = d � aK � K � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 (c − b) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 � d−1 (f − e) − b �� + b � + e = daK � K � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 (c − b) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 � d−1 (f − e) − b �� + db + e = daK � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' K � a−1 (c − b) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' a−1 � d−1 (f − e) − b ��� + db + e □ In what follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' all the networks are restricted to be defined on compact sub- sets Ω in Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', Ω = [0, 1]d (hypercubes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This is in consideration of standard normalizations in training neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4, we present in detail the particular relative Reproducing Kernel Hilbert Spaces which have as their respective dipole system (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5)) the generalized ReLu functions illustrated here in Figures 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Here we specify the kernel K1 for Brownian motion W indexed by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' As a result, the corresponding p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel on R × R is as follows: K1 (x, y) = � |x| ∧ |y| = min (|x| , |y|) if xy ≥ 0 (so same sign) 0 if xy < 0, so opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The connection between the kernel K1 and the Brownian motion {Wx}x∈R is as follows: K1 (x, y) = E ((Wx − W0) (Wy − W0)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2) for all x, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The asserted formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1) follows from this, combined with the independence of increments for Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let K1 be the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1) on R × R, with the corresponding RKHS HK1 = � f : f ′ ∈ L2� , ∥f∥2 HK1 = � |f ′|2 dλ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' On Ω = [0, 1]d, consider the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel Kd (x, y) = K1 (x1, y1) · · · K1 (xd, yd) , so that HKd = � f : ∇f ∈ L2� , ∥f∥2 HKd = � |∇f|2 dλd, where λd denotes the d-dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Given f : Ω → R, and a fixed c ∈ R, set F : Rd → R1, F (x) = K1 (f (x) , c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then, F ∈ HKd ⇐⇒ �� f −1([0,c]) |∇f|2 dλd < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 17 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Let µ be a non-atomic σ-finite measure on (R, BR), and consider Stieltjes measures dF on (R, BR) such that dF ≪ µ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3) (absolutely continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Then the relative RKHS Hµ for the p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' kernel Kµ (A, B) = µ (A ∩ B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='4) consists functions F such that F (b) − F (a) = � v(µ) a,b , F � Hµ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5) � R ���� dF dµ ���� 2 dµ < ∞ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6) where the relative kernels v(µ) a,b are as follows: dv(µ) a,b dµ (x) = χ[a,b](x), see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 𝑣!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=',# $ 𝑎 𝑏 𝜇 𝑎,𝑏 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Illustration dipole functions that reproduce differ- ences of values of functions in the space Hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Compare with Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' See [AJV14, JT18] and the details in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The positive definite kernel Kµ which is “responsible” for the relative RKHS Hµ is defined on B × B, where B denotes the Borel σ-algebra of subsets of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Using [JT18], one checks that Kµ (A, B) := µ (A ∩ B) for all A, B ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='7) We further note that Kµ is the covariance for the generalized µ-Brownian motion {W (µ) A }A∈B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', subject to E � W (µ) A W (µ) B � = µ (A ∩ B) for all A, B∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='8) The corresponding Ito-lemma for W (µ) is defined for differentiable functions f on R via f � W (µ) A � − f (0) = � A f ′ � W (µ) t � dW (µ) t + 1 2 � A f ′′ � W (µ) t � µ (dt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='9) In particular, the measure µ is the quadratic variation of W (µ) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Applications to fractal images In recent decades, it has become evident that fractal features arise in diverse datasets, in time series and in image analysis, to mention two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [KLW20, KLLW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=') Perhaps the best known examples of fractal features include precise symmetries of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Via a prescribed system of affine maps, they take the form of self-similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' And a special case, includes iterated function systems (IFS), and maximal-entropy measures, also called IFS measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The more familiar Cantor constructions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', scaling by 3 or scaling by 4, are examples of IFS measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For each of these cases, the RKHS framework we present in Section 3, serve as ideal tools for such adapted NN algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In particular, this may be illustrated with large numbers of images, say 5000 generated images, each one is a fractal, either 2D or 3D, with random rotation, with zooming, and coloring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' half of them have scaling 3, the other half have scaling 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This leads to training of a network serving to classify the images by scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In particular, the Cantor-type activation functions, or the cumulative functions of Cantor-like measures (Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1), have vanishing derivatives over structured subin- tervals of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This feature may lead to several benefits in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' For example, such functions can introduce sparsity and regularization into the network, which improves its generalization performance and reduces the risk of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Additionally, these functions can make the network more robust to noise and other perturbations in the input data, which improves its performance on unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Furthermore, activation functions whose derivative is zero over subintervals allow the network to learn more complex and non-linear patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This can improve the expressiveness and flexibility of the network, making it more accurate and effective for a wider range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Additionally, these functions can make the network easier to optimize and train, since the gradient of the activation func- tion is well-structured, thus reduce the computational complexity and improve the convergence rate of the training algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' More generally, a neural network with a custom activation function (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' the dipoles in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1) uses a non-standard activation function with adjustable parameters that can be trained and optimized during the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' This allows the network to learn more complex and non-linear relationships between the input and output data, which can improve the accuracy of the network’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The use of a custom activation function with trainable parameters can be useful in a variety of applications, such as image recognition, natural language processing, and time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' It can also be used to improve the performance of other machine learning algorithms, such as decision trees and support vector machines (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=', [CST01, PORSTS21, HSTHD11, HST10, RSSST06, STWCK05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Below we apply a custom activation function in a ConvNet to classify fractal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In this setting, the activation function should be designed to capture the complex, self-similar patterns that are characteristic of the fractal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The network is trained on a dataset of fractal images with corresponding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' It is optimized using a gradient-based algorithm, such as stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Once trained, the network will be used to classify new fractal images and predict their classes with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 19 In the example below, a dataset1 of 15,000 Cantor-like 3D images is generated in Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Parameters of each image, such as zoom factor, viewing angle, and scaling factor, are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A sample of the images is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The images are split into three categories according to their scaling factors, labeled as class “1”, “2” and “3”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The entire dataset is divided into a training set (size = 10,000), validation set (size = 2,500) and test set (size = 2,500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The task is to train a ternary classifier using the training set, along with the validation set (for model selection), whose performance is then tested on the test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In the experiment, a small ConvNet is implemented in Keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Its architecture is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' The loss and accuracy of the model are recorded for 20 epochs (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' In comparison with a standard Relu network (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3a) of the same archi- tecture, the use of Cantor-like activation is better at reducing overfitting (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' it is expected that with a systematic hyperparameter tuning, such a network has the potential to outperform Relu networks in certain applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Class 1 Class 2 Class 3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A random sample of the dataset of 3D Cantor im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 1Available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='com/dsv/4791103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' OPERATOR THEORY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' KERNELS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' AND FEEDFORWARD NEURAL NETWORKS 20 Model: "Cantor-activation" Layer (type) Output Shape Param # input_1 (InputLayer) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 3) 0 rescaling (Rescaling) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 3) 0 conv2d (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 16) 448 max_pooling2d (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 16) 0 conv2d_1 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 32) 4640 max_pooling2d_1 (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 32) 0 conv2d_2 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 64) 18496 max_pooling2d_2 (MaxPooling2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 64) 0 conv2d_3 (Conv2D) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 128) 73856 flatten (Flatten) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 18432) 0 dense (Dense) (None,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 3) 55299 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A ConvNet for fractal image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' (a) Relu activation (b) Cantor-like activation Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Training loss and validation loss, illustrated with use of a ConvNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 16 training loss validation loss 14 12 LD SSI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='6 t0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 Epoch&1DO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='B5 fed Accur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='BO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='65 training accuracy validation accuracy 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='0 EpachsOPERATOR THEORY, KERNELS, AND FEEDFORWARD NEURAL NETWORKS 21 References [AJ21] Daniel Alpay and Palle E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Jorgensen, New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry, Opuscula Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 41 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 3, 283–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' MR 4302453 [AJV14] Daniel Alpay, Palle Jorgensen, and Dan Volok, Relative reproducing kernel Hilbert spaces, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 142 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 11, 3889–3895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' MR 3251728 [AK23] George A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Anastassiou and Seda Karateke, Richards’s curve induced Banach space valued ordinary and fractional neural network approximation, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Cienc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Exactas Fís.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' A Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' RACSAM 117 (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 1, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' MR 4505203 [ASA+23] Mohd Rashid Admon, Norazak Senu, Ali Ahmadian, Zanariah Abdul Majid, and So- heil Salahshour, A new efficient algorithm based on feedforward neural network for solving differential equations of fractional order, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Nonlinear Sci.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 440 (2023), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' 127671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' MR 4505410 (Palle E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Jorgensen) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Email address: palle-jorgensen@uiowa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='edu (Myung-Sin Song) Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026, USA Email address: msong@siue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='edu (James F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Tian) Mathematical Reviews, 416 4th Street Ann Arbor, MI 48103-4816, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content=' Email address: jft@ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} +page_content='org' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf'} diff --git a/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/2301.00155v1.pdf.txt b/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/2301.00155v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4728aa375b444c5f0ed267d63dd5dfa7563a76d --- /dev/null +++ b/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/2301.00155v1.pdf.txt @@ -0,0 +1,883 @@ +Kibble-Zurek scaling in one-dimensional localization transitions +Xuan Bu,1, ∗ Liang-Jun Zhai,2, 3, ∗ and Shuai Yin1, † +1School of Physics, Sun Yat-Sen University, Guangzhou 510275, China +2The school of mathematics and physics, Jiangsu University of Technology, Changzhou 213001, China +3Department of Physics, Nanjing University, Nanjing 210093, China +(Dated: January 3, 2023) +In this work, we explore the driven dynamics of the one-dimensional (1D) localization transitions. +By linearly changing the strength of disorder potential, we calculate the evolution of the localization +length ξ and the inverse participation ratio (IPR) in a disordered Aubry-Andr´e (AA) model, and +investigate the dependence of these quantities on the driving rate. At first, we focus on the limit +in the absence of the quasiperiodic potential. We find that the driven dynamics from both ground +state and excited state can be described by the Kibble-Zurek scaling (KZS). Then, the driven +dynamics near the critical point of the AA model is studied. Here, since both the disorder and the +quasiperiodic potential are relevant directions, the KZS should include both scaling variables. Our +present work not only extends our understanding of the localization transitions but also generalize +the application of the KZS. +I. +INTRODUCTION +The physics of phase transitions between localized +and metallic phases in disordered systems have attracted +long-term attentions since the pioneering work of An- +derson [1–6]. +As a result of the destructive interfer- +ence of scattered waves, the wave function can be lo- +calized at some isolated sites. +Theoretically, it was +shown that for one- and two-dimensional disordered sys- +tems, the localization transition happens for infinitesimal +disorder strength, whereas for higher-dimensional sys- +tems, the localization transition happens for finite dis- +order strength [2, 3]. Moreover, universality classes of +Anderson transition have been categorized [7–11]. +In +addition, besides the disordered systems, it was shown +that the localization can also happen in quasiperiodic +systems [12–33]. +For instance, it was shown that the +Aubry-Andr´e (AA) model hosts a localization transition +at finite strength of quasiperiodic potential [12, 13, 20– +26]. +Experimentally, the localization transition has +been observed in various platforms [34–39], such as cold +atomic systems [34, 35], quantum optics [36, 37], acoustic +waves [38], and electronic systems [39]. +On the other hand, great progresses have been made +in controlling quantum matter with high precision in the +last decades, inspiring the investigations on the nonequi- +librium dynamics of quantum systems [40–43]. In par- +ticular, the driven dynamics across a critical point has +aroused wide concern due to its potential application in +adiabatic quantum computations [44]. A general theory +describing the driven critical dynamics is the celebrated +Kibble-Zurek scaling (KZS) [45–53]. By linearly chang- +ing the distance to the critical point, the KZS states that +the whole driven process can be divided into different +stages. +In the initial stage, the system evolves adia- +batically along the equilibrium state. Then, the system +∗ These authors contribute equally to this work. +† yinsh6@mail.sysu.edu.cn +enters an impulse region, in which the evolution of the +system lags behind the external driving as a result of the +critical slowing down. A full finite-time scaling form with +the driving rate being a typical scaling variable has been +proposed in characterizing the nonequilibrium dynamics +in the whole process [54–56]. This full scaling form has +been verified in both classical and quantum phase tran- +sitions [47, 57–61]. +Recently, the nonequilibrium dynamics in the localiza- +tion transition have also attracted increasing attentions, +which have extended our understanding of localization +transitions and universality far from equilibrium [62–72]. +For instance, in disordered systems, dynamical phase +transition characterized by the peaks in the Loschmidt +echo after a sudden quench was studied [66]. In addition, +the KZS has been investigated in the localization transi- +tions in quasiperiodic AA model and its non-Hermitian +variant for changing the quasiperiodic potential to cross +the critical point [70–72]. However, there is still unknown +whether the KZS is applicable for changing the disorder +strength. +In this work, we study the driven dynamics of localiza- +tion transitions in one-dimensional (1D) disordered sys- +tems. We illustrate the dynamic scaling in a disordered +AA model and focus on two cases. In the first case, there +is no quasiperiodic potential and this model recovers the +usual Anderson model. In the second case, the system +is located near the AA critical point. For both cases, we +change the disorder coefficient across the transition point +and calculate the evolution of the localization length ξ +and the inverse participation ratio (IPR). For the Ander- +son model, we find that the evolution of these quantities +satisfy the usual KZS from both ground state and high- +est excited state; whereas for the disordered AA model, +since the quasiperiodic potential is another relevant di- +rection, the full scaling form should also include the con- +tribution from this term. In particular, in the overlap +region between the critical regions of the AA model and +the Anderson transition, we show that the dynamic scal- +ing behaviors can be described by both the AA critical +arXiv:2301.00155v1 [cond-mat.stat-mech] 31 Dec 2022 + +2 +exponents and the critical exponents of the Anderson lo- +calization. +The rest of the paper is arranged as follows. The 1D +disordered AA model and the characteristic quantities +are introduced in Sec. II. In Sec. III A, the driven dy- +namics in the Anderson model is studied. Then, we ex- +plore the driven dynamics near the AA critical point in +Sec. III B. A summary is given in Sec. IV. +II. +MODEL AND STATIC SCALING +PROPERTIES +The +Hamiltonian +of +the +disordered +AA +model +reads [73] +H = −J +L +� +j +(c† +jcj+1 + h.c.) +(1) ++(2J + δ) +L +� +j +cos [2π(γj + φ)]c† +jcj ++ε +L +� +j +wjc† +jcj, +in which c† +j(cj) is the creation (annihilation) operator of +the hard-core boson at site j, and J is the hopping ampli- +tude between the nearest-neighboring sites and is chosen +as the unity of energy, (2J +δ) measures the amplitude of +the quasiperiodic potential, γ is an irrational number, φ +is the phase of the potential with a uniform distribution +in [0, 1), wj provides the quenched disorder distributed +uniformly in the interval of [−1, 1], and ε is the coeffi- +cient of the disorder. To satisfy the periodic boundary +condition, γ has to be approximated by a rational num- +ber Fn/Fn+1 where Fn+1 = L and Fn are the Fibonacci +numbers [23, 71]. +The phase diagram of model (1) is shown in Fig. 1. +For δ = −2J, Eq. (1) recovers the Anderson model in +which all states are localized for any finite ε. Thus its +critical point of the localization transition is at ε = 0. +In the critical region, the localization length ξ, defined +as [70, 71, 73] +ξ = +� +� +� +� +L +� +j +[(j − jc)2]Pj, +(2) +with Pj being the probability of the wave function at site +j, and jc ≡ � +j jPj being the localization center, diverges +as +ξ ∝ ε−ν, +(3) +in which ν = 2/3 [73, 74]. Another quantity to character- +ize the localization transition is the inverse participation +ratio (IPR), which is defined as [75, 76] +IPR = +L +� +j=1 +|Ψ(j)|4, +(4) +FIG. 1. +Sketch of the phase diagram of the disorder AA +model. When δ = −2J (denoted by the yellow point), this +model recovers the Anderson model. The dark blue region (re- +gion A) denotes the critical region of localization transition of +the disordered AA model. The light green region (region B) +denotes the critical region of the Anderson localization transi- +tion. Near the critical point of δ = 0 and ε = 0, these critical +regions overlap with each other. +where Ψ(j) is the wavefunction. For a localized state, +the wave function is localized on some isolated sites, and +IPR ∝ L0, whereas IPR ∝ L−1 for the delocalized states. +Close to the critical point, IPR scales with ε as +IPR ∝ εs, +(5) +with the critical exponent being s = 2/3 [73]. In addition, +the dynamic exponent z for the Anderson model is z = +2 [74]. +For ε = 0, Eq. (1) recovers the AA model. It was shown +that all the eigenstates of are localized when δ > 0, and +all the eigenstates are delocalized when δ < 0. In the +critical region, the localization length ξ satisfies +ξ ∝ δ−νδ, +(6) +with νδ = 1 [70, 74]. And the IPR obeys +IPR ∝ δsδ, +(7) +with sδ ≈ 0.333 [73]. Besides, the dynamic exponent z +for the AA critical point is zδ ≈ 2.37 [70]. +Moreover, previously we showed that the disorder ε +also provides a relevant direction in the AA critical point. +For δ = 0, the localization length ξ obeys +ξ ∝ ε−νε, +(8) +with νε = 0.46(1) [73]. Note that this exponent is re- +markably different from ν and νδ. In addition, the IPR +obeys +IPR ∝ εsδνε/νδ. +(9) + +A +B +S=-2J3 +III. +THE KZS IN THE LOCALIZATION +TRANSITION +A. +KZS for the Anderson model +Here, we consider the driven dynamics of the Anderson +model with δ = −2J in Eq. (1). At first, we show the +detailed driven process. +Initially, the system is in the +localization phase for a specific realization of wj with +coefficient ε0 > 0. Then ε is decreased according to +ε = ε0 − Rt, +(10) +to cross the critical point, and wj keep invariant. Then +wi is resampled for another process with same initial ε0. +At last, the quantities are averaged for many realization +of samples to make the evolution curves smooth. +The KZS states that when ε > R1/νr with r = z +1/ν, +the system can evolve adiabatically since the state has +enough time to adjust to the change in the Hamiltonian; +in contrast, when ε < R1/νr, the system enter the impulse +region and the system stop evolving as a result of the crit- +ical slowing down. However, investigations showed that +the assumption that the system does not evolve in the +impulse region is oversimplified. To improve it, a finite- +time scaling theory has been proposed and demonstrates +that the external driving provides a typical time scale +of ζ ∝ R−z/r [54–56]. In the impulse region, ζ controls +the dynamic scaling behaviors and macroscopic quanti- +ties can be scaled with ζ. For instance, for large enough +system size, the full scaling form of the localization length +ξ around the critical point reads [71] +ξ(ε, R) = R−1/rf1(εR−1/rν), +(11) +in which f1 is the scaling function. +When ε +> +R−1/rν, the evolution is in the adiabatic stage, in which +f1(εR−1/rν) ∼ (εR−1/rν)−ν. +Accordingly, ξ satisfies +Eq. (3) and does not depend on the driving rate R. +In contrast, near the critical point, when ε < R−1/rν, +f1(εR−1/rν) tends to a constant and ξ ∝ R−1/r, demon- +strating that the divergence of ξ at the critical point has +been truncated by the external driving and ξ decreases +as R increases. +Similarly, the driven dynamics of the IPR around the +critical point satisfies +IPR(ε, R) = Rs/rνf2(εR−1/rν). +(12) +When ε > R−1/rν, f2(εR−1/rν) ∼ (εR−1/rν)s and +Eq. (12) recovers Eq. (5). In contrast, near the critical +point, when ε < R−1/rν, IPR ∝ Rs/r. +To verify the scaling functions of Eq. (11) and (12), we +numerically solve the Schrodinger equation for model (1), +and calculate the dependence of ξ and IPR on ε for var- +ious driving rate R. The finite difference method in the +time direction is used, and the time interval is chosen as +10−3. The lattice size is chosen as L = 500, which is large +enough to ignore the finite-size effect. ε0 is set as ε0 = 2, +which is far enough from the critical point at ε = 0. +� � �� +0.0 +0.5 +5 +10 +15 +20 +25 +� +� + 3 + 4 + 5 + 6 + 7 + 8 + 9 +R=10-3× +(a1) +(b1) +(b2) +� � +� � +0 +2 +4 +0 +2 +4 +� R1/r +� R-1/r� +(a2) +� � �� +0.0 +0.5 +0.05 +0.10 +0.15 +IPR +� +� � +� � +0 +2 +4 +0.2 +0.4 +0.6 +0.8 +IPRR-s/r� +� R-1/r� +FIG. 2. +Driven dynamics in the Anderson model with the +initial state being the ground state. The curves of ξ versus ε +before (a1) and after (a2) rescaled for different R. The curves +of IPR versus ε before (b1) and after (b2) rescaled for different +R. The arrows in (a1) and (b1) point the quench direction. +� � �� +0.0 +0.5 +5 +10 +15 +20 +� +� + 3 + 4 + 5 + 6 + 7 + 8 + 9 +R=10-3× +(a1) +(b1) +(b2) +� � +� � +0 +2 +4 +2 +4 +� R1/r +� R-1/r� +(a2) +� � �� +0.0 +0.5 +0.1 +IPR +� +� � +� � +0 +2 +4 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +IPRR-s/r� +� R-1/r� +FIG. 3. +Driven dynamics in the Anderson model with the +initial state being the highest excited state. The curves of +ξ versus ε before (a1) and after (a2) rescaled for different R. +The curves of IPR versus ε before (b1) and after (b2) rescaled +for different R. The arrows in (a1) and (b1) point the quench +direction. +First, the initial state is chosen as the ground state of +model (1) for ε = ε0. Figure 2 (a1) shows the evolution +of the localization length ξ for different R. Initially, one +finds that ξ almost does not depend on R, indicating the +system evolves adiabatically in this stage. Then when ε +approaches to the critical point, the curves for different +R begin to separate from each other, indicating that the +system enters the impulse region. After rescaling ξ and + +4 +� � �� +� � �� +0.0 +0.2 +0.4 +2 +4 +6 +8 +� +� + 3 + 4 + 5 + 6 + 7 + 8 + 9 +� � +0 +5 +1 +2 +� R1/r� +� R-1/� � r� +R=10-3× +� � �� +� � �� +0.0 +0.2 +0.4 +0.2 +0.4 +0.6 +IPR +� +(b1) +(b2) +(a1) +(a2) +� � +0 +5 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +IPR R-s� /� � r� +� R-1/� � r� +FIG. 4. +Driven dynamics near the AA critical point with +fixed δR−1/rενδ = 0.3. The curves of ξ versus ε before (a1) +and after (a2) rescaled for different R. +The curves of IPR +versus ε before (b1) and after (b2) rescaled for different R. +The arrows in (a1) and (b1) point the quench direction. +ε as ξR1/r and εR−1/νr, respectively, we find that the +rescaled curves collapse onto each other near the critical +point, as shown in Fig. +2 (a2). These results confirm +Eq. (11). In particular, exactly at the critical point, i.e., +ε = 0, Fig. 2 (a2) demonstrates ξ ∝ R−1/r. +Similarly, Fig. 2 (b1) shows the evolution of IPR for +different R. After an initial adiabatic stage, in which the +evolution of IPR is almost independent of R, hysteresis +effect of IPR appears near the critical point and the IPR +increases as R increases. After rescaling IPR and ε as +IPRR−s/νr and εR−1/νr, respectively, we find that the +rescaled curves match with each other near the critical +point, as shown in Fig. +2 (b2). These results confirm +Eq. (12). In particular, exactly at the critical point, i.e., +ε = 0, Fig. 2 (b2) demonstrates IPR ∝ Rs/νr. These +results clearly demonstrates that the KZS is applicable +in the localization transition of the Anderson model. +Moreover, different from the usual quantum phase +transition which happens only in the ground state, here +the Anderson localization happens in all eigenstates. It +is interesting to explore the driven dynamics with the +initial state being the excited state. To this end, we cal- +culate the dynamics of ξ and IPR with the initial state +being the highest excited state and show the results in +Fig. 3. +After rescaling the curves by R, we find that +the rescaled curves collapse onto each other, as shown in +Fig. 3, verifying Eqs. (11) and (12) and demonstrating +that the KZS is still applicable in the driven dynamics +from the excited states. +� � �� +� � �� +0.0 +0.2 +0.4 +2 +4 +6 +8 +10 +� +� + 3 + 4 + 5 + 6 + 7 + 8 + 9 +� � +0 +5 +1 +2 +3 +� R1/r� +� R-1/� � r� +R=10-3× +� � �� +� � �� +0.0 +0.2 +0.4 +0.2 +0.4 +0.6 +IPR +� +(b1) +(b2) +(a1) +(a2) +� � +0 +5 +0.2 +0.4 +0.6 +0.8 +1.0 +IPR R-s� /� � r� +� R-1/� � r� +FIG. 5. Driven dynamics near the AA critical point with fixed +δR−1/rενδ = −0.3. The curves of ξ versus ε before (a1) and +after (a2) rescaled for different R. The curves of IPR versus ε +before (b1) and after (b2) rescaled for different R. The arrows +in (a1) and (b1) point the quench direction. +� � �� +� � �� +0.0 +0.2 +0.4 +0 +2 +4 +6 +8 +10 +� +� + 3 + 4 + 5 + 6 + 7 + 8 + 9 +� � +0 +5 +0 +1 +2 +� R1/r +� R-1/� r +R=10-3× +� � �� +� � �� +0.0 +0.2 +0.4 +0.2 +0.4 +0.6 +IPR +� +(b1) +(b2) +(a1) +(a2) +� � +0 +5 +0.5 +1.0 +1.5 +IPR R-s/� r +� R-1/� r +FIG. 6. Driven dynamics near the AA critical point with fixed +δ = −0.1. The curves of ξ versus ε before (a1) and after (a2) +rescaled for different R. The curves of IPR versus ε before +(b1) and after (b2) rescaled for different R. The arrows in +(a1) and (b1) point the quench direction. +B. +KZS for the disordered AA model +In this section, we consider the driven dynamics near +the AA critical point with small δ in model (1) by chang- +ing the coefficient of the disorder term. Note that dif- +ferent from the Anderson model, there are two relevant +directions near the critical point of the disordered AA +model. One direction is the quasiperiodic potential, rep- +resented by δ, the other is the disorder term, represented +by ε. Thus, in the full scaling form, both two relevant +terms should be included. + +5 +In analogy to the analyses in Sec. III A, the evolution +of the localization length ξ should satisfy +ξ(ε, δ, R) = R−1/rεf3(εR−1/rενε, δR−1/rενδ), +(13) +in which rε = zδ + 1/νε. For R → 0 and δ = 0, f3 ∼ +(εR−1/rενε)−νε and Eq. (13) restores Eq. (8). For R → 0 +and ε = 0, f3 ∼ (δR−1/rενδ)−νδ and Eq. (13) restores +Eq. (6). +Similarly, under external driving, the IPR should sat- +isfy +IPR(ε, δ, R) = Rsδ/rενδf4(εR−1/rενε, δR−1/rενδ).(14) +For R → 0 and δ = 0, f4 ∼ (εR−1/rενε)sδνε/νδ and +Eq. (14) recovers Eq. (9). +For R → 0 and ε = 0, +f3 ∼ (δR−1/rενδ)sδ and Eq. (14) restores Eq. (7). +Equations (13) and (14) should be applicable for any +values of δ and ε near the critical point of the AA model. +Particularly, for δ < 0, there is an overlap critical region +between the critical region of the AA critical point and +the critical region of the Anderson localization, as illus- +trated in Fig. 1. Therefore, in this overlap region, the +driven critical dynamics of should simultaneously satisfy +Eqs. (11) and (13) for ξ and Eqs. (12) and (14) for the +IPR. +We at first examine Eqs. (13) and (14) for δ > 0 with +the initial state being the ground state. +For a fixed +δR−1/rενδ, we calculate the evolution of ξ and IPR for +various driving rate R. +After rescaling the evolution +curves with R, we find that the rescaled curves match +with each other, as shown in Fig. 4, confirming Eqs. (13) +and (14). +For δ < 0, Fig. 5 shows the evolution of ξ and IPR +with various driving rate R for a fixed δR−1/rενδ. After +rescaling the curves by R with the critical exponents of +the AA critical point, we find that the curves collapse +onto each other, as shown in Fig. 5, confirming Eqs. (13) +and (14). Moreover, Fig. 6 shows the evolution of ξ and +IPR for various driving rate R with a fixed δ, which is +near the AA critical point. +After rescaling the curves +by R with the critical exponents of the Anderson model, +we find that the curves also collapse onto each other, as +shown in Fig. 6, obeying Eqs. (11) and (12). Thus, we +confirm that for δ < 0 the driven critical dynamics can +simultaneously be described by Eqs. (11) and (13) for ξ +and Eqs. (12) and (14) for the IPR. +Here we remark on the results. (a) Although here we +only show the results with the initial state being the +ground state, it is expected that these scaling analyses +are also applicable for the excited states, similar to the +results shown in Sec. III A. (b) In Ref. [70], the driven dy- +namics in the AA model without the disorder term was +studied for changing the quasiperiodic potential. Here +we change the disorder strength to cross the AA critical +point. Comparing these two cases, we find that although +the scaling forms of the KZS are similar, the dimensions +of the driving rate are different in two cases. Combin- +ing these results, we find that the KZS can apply in the +localization transitions for different driving dynamics. +IV. +SUMMARY +In summary, we have studied the driven dynamics in +the localization transitions in 1D disordered AA model. +By changing the disorder coefficient to cross the criti- +cal point, we calculate the dynamics of the localization +length ξ and the IPR. For both the critical point of the +Anderson model and the AA model, we have verified +that the KZS is applicable in characterizing the driven +dynamics. Moreover, we have also generalized the KZS +to describe the driven dynamics from the excited states. +In addition, in the overlap critical region near the AA +critical point, we have found that the driven dynamics +can be simultaneously described by the KZS with both +the critical exponents of the AA model and the critical +exponents of the Anderson model. As one possible gen- +eralization, one can also investigate the driven dynamics +in the many-body localization transition [30–33, 77–81]. +ACKNOWLEDGMENTS +B. X. and S. Y. is supported by the National Natural +science Foundation of China (Grant No. 12075324), the +Science and Technology Projects in Guangzhou (Grant +No. +202102020367) and the Fundamental Research +Funds for Central Universities, Sun Yat-Sen University +(Grant No. 22qntd3005). L.-J. 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Matter +Phys. 6, 383 (2015). + diff --git a/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/load_file.txt b/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..262293e4b57d1f4e0c944ac186a3cff1380acebc --- /dev/null +++ b/VtAyT4oBgHgl3EQfV_ek/content/tmp_files/load_file.txt @@ -0,0 +1,940 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf,len=939 +page_content='Kibble-Zurek scaling in one-dimensional localization transitions Xuan Bu,1, ∗ Liang-Jun Zhai,2, 3, ∗ and Shuai Yin1, † 1School of Physics, Sun Yat-Sen University, Guangzhou 510275, China 2The school of mathematics and physics, Jiangsu University of Technology, Changzhou 213001, China 3Department of Physics, Nanjing University, Nanjing 210093, China (Dated: January 3, 2023) In this work, we explore the driven dynamics of the one-dimensional (1D) localization transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' By linearly changing the strength of disorder potential, we calculate the evolution of the localization length ξ and the inverse participation ratio (IPR) in a disordered Aubry-Andr´e (AA) model, and investigate the dependence of these quantities on the driving rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' At first, we focus on the limit in the absence of the quasiperiodic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' We find that the driven dynamics from both ground state and excited state can be described by the Kibble-Zurek scaling (KZS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then, the driven dynamics near the critical point of the AA model is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Here, since both the disorder and the quasiperiodic potential are relevant directions, the KZS should include both scaling variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Our present work not only extends our understanding of the localization transitions but also generalize the application of the KZS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' INTRODUCTION The physics of phase transitions between localized and metallic phases in disordered systems have attracted long-term attentions since the pioneering work of An- derson [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' As a result of the destructive interfer- ence of scattered waves, the wave function can be lo- calized at some isolated sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Theoretically, it was shown that for one- and two-dimensional disordered sys- tems, the localization transition happens for infinitesimal disorder strength, whereas for higher-dimensional sys- tems, the localization transition happens for finite dis- order strength [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Moreover, universality classes of Anderson transition have been categorized [7–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In addition, besides the disordered systems, it was shown that the localization can also happen in quasiperiodic systems [12–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For instance, it was shown that the Aubry-Andr´e (AA) model hosts a localization transition at finite strength of quasiperiodic potential [12, 13, 20– 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Experimentally, the localization transition has been observed in various platforms [34–39], such as cold atomic systems [34, 35], quantum optics [36, 37], acoustic waves [38], and electronic systems [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' On the other hand, great progresses have been made in controlling quantum matter with high precision in the last decades, inspiring the investigations on the nonequi- librium dynamics of quantum systems [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In par- ticular, the driven dynamics across a critical point has aroused wide concern due to its potential application in adiabatic quantum computations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' A general theory describing the driven critical dynamics is the celebrated Kibble-Zurek scaling (KZS) [45–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' By linearly chang- ing the distance to the critical point, the KZS states that the whole driven process can be divided into different stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the initial stage, the system evolves adia- batically along the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then, the system ∗ These authors contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' † yinsh6@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='cn enters an impulse region, in which the evolution of the system lags behind the external driving as a result of the critical slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' A full finite-time scaling form with the driving rate being a typical scaling variable has been proposed in characterizing the nonequilibrium dynamics in the whole process [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' This full scaling form has been verified in both classical and quantum phase tran- sitions [47, 57–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Recently, the nonequilibrium dynamics in the localiza- tion transition have also attracted increasing attentions, which have extended our understanding of localization transitions and universality far from equilibrium [62–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For instance, in disordered systems, dynamical phase transition characterized by the peaks in the Loschmidt echo after a sudden quench was studied [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In addition, the KZS has been investigated in the localization transi- tions in quasiperiodic AA model and its non-Hermitian variant for changing the quasiperiodic potential to cross the critical point [70–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' However, there is still unknown whether the KZS is applicable for changing the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In this work, we study the driven dynamics of localiza- tion transitions in one-dimensional (1D) disordered sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' We illustrate the dynamic scaling in a disordered AA model and focus on two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the first case, there is no quasiperiodic potential and this model recovers the usual Anderson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the second case, the system is located near the AA critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For both cases, we change the disorder coefficient across the transition point and calculate the evolution of the localization length ξ and the inverse participation ratio (IPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For the Ander- son model, we find that the evolution of these quantities satisfy the usual KZS from both ground state and high- est excited state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' whereas for the disordered AA model, since the quasiperiodic potential is another relevant di- rection, the full scaling form should also include the con- tribution from this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In particular, in the overlap region between the critical regions of the AA model and the Anderson transition, we show that the dynamic scal- ing behaviors can be described by both the AA critical arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='00155v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='stat-mech] 31 Dec 2022 2 exponents and the critical exponents of the Anderson lo- calization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The rest of the paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The 1D disordered AA model and the characteristic quantities are introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' III A, the driven dy- namics in the Anderson model is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then, we ex- plore the driven dynamics near the AA critical point in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' A summary is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' MODEL AND STATIC SCALING PROPERTIES The Hamiltonian of the disordered AA model reads [73] H = −J L � j (c† jcj+1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=') (1) +(2J + δ) L � j cos [2π(γj + φ)]c† jcj +ε L � j wjc† jcj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' in which c† j(cj) is the creation (annihilation) operator of the hard-core boson at site j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' and J is the hopping ampli- tude between the nearest-neighboring sites and is chosen as the unity of energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (2J +δ) measures the amplitude of the quasiperiodic potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' γ is an irrational number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' φ is the phase of the potential with a uniform distribution in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' wj provides the quenched disorder distributed uniformly in the interval of [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' and ε is the coeffi- cient of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' To satisfy the periodic boundary condition, γ has to be approximated by a rational num- ber Fn/Fn+1 where Fn+1 = L and Fn are the Fibonacci numbers [23, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The phase diagram of model (1) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For δ = −2J, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (1) recovers the Anderson model in which all states are localized for any finite ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Thus its critical point of the localization transition is at ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the critical region, the localization length ξ, defined as [70, 71, 73] ξ = � � � � L � j [(j − jc)2]Pj, (2) with Pj being the probability of the wave function at site j, and jc ≡ � j jPj being the localization center, diverges as ξ ∝ ε−ν, (3) in which ν = 2/3 [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Another quantity to character- ize the localization transition is the inverse participation ratio (IPR), which is defined as [75, 76] IPR = L � j=1 |Ψ(j)|4, (4) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Sketch of the phase diagram of the disorder AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' When δ = −2J (denoted by the yellow point), this model recovers the Anderson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The dark blue region (re- gion A) denotes the critical region of localization transition of the disordered AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The light green region (region B) denotes the critical region of the Anderson localization transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Near the critical point of δ = 0 and ε = 0, these critical regions overlap with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' where Ψ(j) is the wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For a localized state, the wave function is localized on some isolated sites, and IPR ∝ L0, whereas IPR ∝ L−1 for the delocalized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Close to the critical point, IPR scales with ε as IPR ∝ εs, (5) with the critical exponent being s = 2/3 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In addition, the dynamic exponent z for the Anderson model is z = 2 [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For ε = 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (1) recovers the AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' It was shown that all the eigenstates of are localized when δ > 0, and all the eigenstates are delocalized when δ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the critical region, the localization length ξ satisfies ξ ∝ δ−νδ, (6) with νδ = 1 [70, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' And the IPR obeys IPR ∝ δsδ, (7) with sδ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='333 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Besides, the dynamic exponent z for the AA critical point is zδ ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='37 [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Moreover, previously we showed that the disorder ε also provides a relevant direction in the AA critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For δ = 0, the localization length ξ obeys ξ ∝ ε−νε, (8) with νε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='46(1) [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Note that this exponent is re- markably different from ν and νδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In addition, the IPR obeys IPR ∝ εsδνε/νδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (9) A B S=-2J3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' THE KZS IN THE LOCALIZATION TRANSITION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' KZS for the Anderson model Here, we consider the driven dynamics of the Anderson model with δ = −2J in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' At first, we show the detailed driven process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Initially, the system is in the localization phase for a specific realization of wj with coefficient ε0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then ε is decreased according to ε = ε0 − Rt, (10) to cross the critical point, and wj keep invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then wi is resampled for another process with same initial ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' At last, the quantities are averaged for many realization of samples to make the evolution curves smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The KZS states that when ε > R1/νr with r = z +1/ν, the system can evolve adiabatically since the state has enough time to adjust to the change in the Hamiltonian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' in contrast, when ε < R1/νr, the system enter the impulse region and the system stop evolving as a result of the crit- ical slowing down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' However, investigations showed that the assumption that the system does not evolve in the impulse region is oversimplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' To improve it, a finite- time scaling theory has been proposed and demonstrates that the external driving provides a typical time scale of ζ ∝ R−z/r [54–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In the impulse region, ζ controls the dynamic scaling behaviors and macroscopic quanti- ties can be scaled with ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For instance, for large enough system size, the full scaling form of the localization length ξ around the critical point reads [71] ξ(ε, R) = R−1/rf1(εR−1/rν), (11) in which f1 is the scaling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' When ε > R−1/rν, the evolution is in the adiabatic stage, in which f1(εR−1/rν) ∼ (εR−1/rν)−ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Accordingly, ξ satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (3) and does not depend on the driving rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In contrast, near the critical point, when ε < R−1/rν, f1(εR−1/rν) tends to a constant and ξ ∝ R−1/r, demon- strating that the divergence of ξ at the critical point has been truncated by the external driving and ξ decreases as R increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Similarly, the driven dynamics of the IPR around the critical point satisfies IPR(ε, R) = Rs/rνf2(εR−1/rν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (12) When ε > R−1/rν, f2(εR−1/rν) ∼ (εR−1/rν)s and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (12) recovers Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In contrast, near the critical point, when ε < R−1/rν, IPR ∝ Rs/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' To verify the scaling functions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11) and (12), we numerically solve the Schrodinger equation for model (1), and calculate the dependence of ξ and IPR on ε for var- ious driving rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The finite difference method in the time direction is used, and the time interval is chosen as 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The lattice size is chosen as L = 500, which is large enough to ignore the finite-size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' ε0 is set as ε0 = 2, which is far enough from the critical point at ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 5 10 15 20 25 � � 3 4 5 6 7 8 9 R=10-3× (a1) (b1) (b2) � � � � 0 2 4 0 2 4 � R1/r � R-1/r� (a2) � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='15 IPR � � � � � 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='8 IPRR-s/r� � R-1/r� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Driven dynamics in the Anderson model with the initial state being the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of ξ versus ε before (a1) and after (a2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of IPR versus ε before (b1) and after (b2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The arrows in (a1) and (b1) point the quench direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 5 10 15 20 � � 3 4 5 6 7 8 9 R=10-3× (a1) (b1) (b2) � � � � 0 2 4 2 4 � R1/r � R-1/r� (a2) � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='1 IPR � � � � � 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='7 IPRR-s/r� � R-1/r� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Driven dynamics in the Anderson model with the initial state being the highest excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of ξ versus ε before (a1) and after (a2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of IPR versus ε before (b1) and after (b2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The arrows in (a1) and (b1) point the quench direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' First, the initial state is chosen as the ground state of model (1) for ε = ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Figure 2 (a1) shows the evolution of the localization length ξ for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Initially, one finds that ξ almost does not depend on R, indicating the system evolves adiabatically in this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Then when ε approaches to the critical point, the curves for different R begin to separate from each other, indicating that the system enters the impulse region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling ξ and 4 � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 2 4 6 8 � � 3 4 5 6 7 8 9 � � 0 5 1 2 � R1/r� � R-1/� � r� R=10-3× � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 IPR � (b1) (b2) (a1) (a2) � � 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 IPR R-s� /� � r� � R-1/� � r� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Driven dynamics near the AA critical point with fixed δR−1/rενδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of ξ versus ε before (a1) and after (a2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of IPR versus ε before (b1) and after (b2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The arrows in (a1) and (b1) point the quench direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' ε as ξR1/r and εR−1/νr, respectively, we find that the rescaled curves collapse onto each other near the critical point, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2 (a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' These results confirm Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In particular, exactly at the critical point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=', ε = 0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2 (a2) demonstrates ξ ∝ R−1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Similarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2 (b1) shows the evolution of IPR for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After an initial adiabatic stage, in which the evolution of IPR is almost independent of R, hysteresis effect of IPR appears near the critical point and the IPR increases as R increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling IPR and ε as IPRR−s/νr and εR−1/νr, respectively, we find that the rescaled curves match with each other near the critical point, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2 (b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' These results confirm Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In particular, exactly at the critical point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=', ε = 0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2 (b2) demonstrates IPR ∝ Rs/νr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' These results clearly demonstrates that the KZS is applicable in the localization transition of the Anderson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Moreover, different from the usual quantum phase transition which happens only in the ground state, here the Anderson localization happens in all eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' It is interesting to explore the driven dynamics with the initial state being the excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' To this end, we cal- culate the dynamics of ξ and IPR with the initial state being the highest excited state and show the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling the curves by R, we find that the rescaled curves collapse onto each other, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 3, verifying Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11) and (12) and demonstrating that the KZS is still applicable in the driven dynamics from the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 2 4 6 8 10 � � 3 4 5 6 7 8 9 � � 0 5 1 2 3 � R1/r� � R-1/� � r� R=10-3× � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 IPR � (b1) (b2) (a1) (a2) � � 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 IPR R-s� /� � r� � R-1/� � r� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Driven dynamics near the AA critical point with fixed δR−1/rενδ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of ξ versus ε before (a1) and after (a2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of IPR versus ε before (b1) and after (b2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The arrows in (a1) and (b1) point the quench direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0 2 4 6 8 10 � � 3 4 5 6 7 8 9 � � 0 5 0 1 2 � R1/r � R-1/� r R=10-3× � � �� � � �� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='6 IPR � (b1) (b2) (a1) (a2) � � 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='5 IPR R-s/� r � R-1/� r FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Driven dynamics near the AA critical point with fixed δ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of ξ versus ε before (a1) and after (a2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The curves of IPR versus ε before (b1) and after (b2) rescaled for different R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' The arrows in (a1) and (b1) point the quench direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' KZS for the disordered AA model In this section, we consider the driven dynamics near the AA critical point with small δ in model (1) by chang- ing the coefficient of the disorder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Note that dif- ferent from the Anderson model, there are two relevant directions near the critical point of the disordered AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' One direction is the quasiperiodic potential, rep- resented by δ, the other is the disorder term, represented by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Thus, in the full scaling form, both two relevant terms should be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 5 In analogy to the analyses in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' III A, the evolution of the localization length ξ should satisfy ξ(ε, δ, R) = R−1/rεf3(εR−1/rενε, δR−1/rενδ), (13) in which rε = zδ + 1/νε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For R → 0 and δ = 0, f3 ∼ (εR−1/rενε)−νε and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (13) restores Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For R → 0 and ε = 0, f3 ∼ (δR−1/rενδ)−νδ and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (13) restores Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Similarly, under external driving, the IPR should sat- isfy IPR(ε, δ, R) = Rsδ/rενδf4(εR−1/rενε, δR−1/rενδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (14) For R → 0 and δ = 0, f4 ∼ (εR−1/rενε)sδνε/νδ and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (14) recovers Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For R → 0 and ε = 0, f3 ∼ (δR−1/rενδ)sδ and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (14) restores Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Equations (13) and (14) should be applicable for any values of δ and ε near the critical point of the AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Particularly, for δ < 0, there is an overlap critical region between the critical region of the AA critical point and the critical region of the Anderson localization, as illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Therefore, in this overlap region, the driven critical dynamics of should simultaneously satisfy Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11) and (13) for ξ and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (12) and (14) for the IPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' We at first examine Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (13) and (14) for δ > 0 with the initial state being the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For a fixed δR−1/rενδ, we calculate the evolution of ξ and IPR for various driving rate R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling the evolution curves with R, we find that the rescaled curves match with each other, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 4, confirming Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For δ < 0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 5 shows the evolution of ξ and IPR with various driving rate R for a fixed δR−1/rενδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling the curves by R with the critical exponents of the AA critical point, we find that the curves collapse onto each other, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 5, confirming Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (13) and (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Moreover, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 6 shows the evolution of ξ and IPR for various driving rate R with a fixed δ, which is near the AA critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' After rescaling the curves by R with the critical exponents of the Anderson model, we find that the curves also collapse onto each other, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 6, obeying Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Thus, we confirm that for δ < 0 the driven critical dynamics can simultaneously be described by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (11) and (13) for ξ and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (12) and (14) for the IPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Here we remark on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (a) Although here we only show the results with the initial state being the ground state, it is expected that these scaling analyses are also applicable for the excited states, similar to the results shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' (b) In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [70], the driven dy- namics in the AA model without the disorder term was studied for changing the quasiperiodic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Here we change the disorder strength to cross the AA critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Comparing these two cases, we find that although the scaling forms of the KZS are similar, the dimensions of the driving rate are different in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Combin- ing these results, we find that the KZS can apply in the localization transitions for different driving dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' SUMMARY In summary, we have studied the driven dynamics in the localization transitions in 1D disordered AA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' By changing the disorder coefficient to cross the criti- cal point, we calculate the dynamics of the localization length ξ and the IPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' For both the critical point of the Anderson model and the AA model, we have verified that the KZS is applicable in characterizing the driven dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Moreover, we have also generalized the KZS to describe the driven dynamics from the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' In addition, in the overlap critical region near the AA critical point, we have found that the driven dynamics can be simultaneously described by the KZS with both the critical exponents of the AA model and the critical exponents of the Anderson model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' As one possible gen- eralization, one can also investigate the driven dynamics in the many-body localization transition [30–33, 77–81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' ACKNOWLEDGMENTS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' is supported by the National Natural science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 12075324), the Science and Technology Projects in Guangzhou (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 202102020367) and the Fundamental Research Funds for Central Universities, Sun Yat-Sen University (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 22qntd3005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Zhai is supported by the National Natural science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 11704161) and China Postdoctoral Science Foun- dation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 2021M691535), and Zhongwu Youth Innovation Talent Support Plan of Jiangsu University of technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Anderson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 109, 1492 (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Thouless, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 13, 93 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [3] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Evers and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Mirlin, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 80, 1355 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [4] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Hatano and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Nelson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' B 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' A 95, 062118 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Longhi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 122, 237601 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [22] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Tang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Zhang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Zhang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' A 101, 063612 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Jiang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' B 95, 184201 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' [65] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Romito, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Lobo, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} +page_content=' Recati, Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtAyT4oBgHgl3EQfV_ek/content/2301.00155v1.pdf'} 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diff --git a/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/2301.03624v1.pdf.txt b/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/2301.03624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..08d502f958b0950b46d419f955d98231b1c0c727 --- /dev/null +++ b/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/2301.03624v1.pdf.txt @@ -0,0 +1,3132 @@ +Prepared for submission to JCAP +EDGES of the dark forest: A new +absorption window into the composite +dark matter and large scale structure +Anoma Ganguly, Rishi Khatri, Tuhin S. Roy +Department of Theoretical Physics, Tata Institute of Fundamental Research, +Homi Bhabha Road, Mumbai 400005, India +E-mail: anoma@theory.tifr.res.in, khatri@theory.tifr.res.in, tuhin@theory.tifr.res.in +Abstract. We propose a new method to hunt for dark matter using dark forest/absorption +features across the whole electromagnetic spectrum from radio to gamma rays, especially +in the bands where there is a desert i.e. regions where no strong lines from baryons are +expected. Such novel signatures can arise for dark matter models with a composite nature +and internal electromagnetic transitions. The photons from a background source can interact +with the dark matter resulting in an absorption signal in the source spectrum. In case of a +compact source, such as a quasar, such interactions in the dark matter halos can produce +a series of closely spaced absorption lines, which we call the dark forest. +We show that +the dark forest feature is a sensitive probe of the dark matter self-interactions and the halo +mass function, especially at the low mass end. Moreover, the absorption of CMB photons by +dark matter gives rise to a global absorption signal in the CMB spectrum. For dark matter +transition energies in the range 2.5 × 10−4 eV − 5 × 103 eV, such absorption features result +in spectral distortions of the CMB in the COBE/FIRAS band of 60-600 GHz. If the dark +matter transition frequency is ∼156 GHz, we show that the absorption of CMB photons by +dark matter can provide an explanation for the anomalous absorption feature detected by +the EDGES collaboration in 50-100 MHz range. +arXiv:2301.03624v1 [astro-ph.CO] 9 Jan 2023 + +Contents +1 +Introduction +1 +2 +A theoretical framework for the composite dark sector +4 +3 +Experimental signatures +9 +3.1 +Absorption lines in the spectrum of a compact source +10 +3.1.1 +A dark line: absorption by a single dark matter halo +12 +3.1.2 +Dark forest: absorption by multiple dark matter halos +15 +3.1.3 +Detectability of dark forest +20 +3.2 +Global absorption signal in the CMB spectrum +21 +3.2.1 +The EDGES anomaly +24 +3.2.2 +General predictions for the shape of the dark absorption feature +25 +3.2.3 +General predictions for the dark absorption feature in different parts +of the CMB spectrum +27 +4 +Observational constraints +27 +4.1 +Early Universe constraints +27 +4.2 +Late Universe constraints from Milky Way +28 +5 +Conclusions +30 +A Dark matter model +32 +B Dark matter halo density and temperature profile +33 +C Dark forest +34 +D Convergence of distribution functions +37 +E Radiative transfer equation in an expanding Universe +37 +F Limits on CMB spectral distortions from COBE/FIRAS +39 +G Milky Way model for constraining the radiative coupling from CMB anisotropy +maps +39 +1 +Introduction +Dark matter, although making up a major fraction of the matter content in our Universe, +continues to remain a mystery. Owing to its elusive nature, the different experiments search- +ing for it [1, 2] have till now only succeeded in placing stringent constraints on its possible +interactions beyond the usual gravitational force. The allowed parameter space for possible +dark matter candidates is huge, with masses ranging from 10−22 eV ultralight bosons to 1043 +GeV compact objects. Thus a more clear picture of its true nature can only start emerging +if we find new methods to look for dark matter in experiments. The current searches for +– 1 – + +dark matter broadly fall into three categories, direct detection: where detectors search for +nuclear/electronic recoil due to dark matter, indirect detection: where experiments look for +emission signals of different standard model particles produced from dark matter annihila- +tion, decay, etc., or collider searches: where high energy accelerators try to produce dark +matter by colliding standard model particles. The absence of a distinct signature of dark +matter in any of these searches so far not only hints at its far more nuanced nature, but also +calls for new detection strategies. In this work, we propose a novel method to look for dark +matter in the absorption lines of a background source. The main advantage of absorption +lines is their ability to probe very weak interactions between dark matter and photons, which +is possible if the background source is sufficiently bright. +Absorption lines are a generic feature of a class of models where dark matter is a compos- +ite particle with a discrete energy spectrum. The presence of a small electromagnetic coupling +can allow the transitions between different dark matter energy states via emission/absorption +of a photon. As a specific example, we will consider dark matter to be a composite parti- +cle made of two elementary particles of the dark sector. A strong dark attractive potential +between the constituents makes dark matter stable on cosmological scales. As a whole dark +matter stays electromagnetically neutral, while the constituents carry a millicharge. This +simple model allows us to describe dark matter as a bound state with weak electromagnetic +transitions similar to a hydrogen atom. The generic signatures of such models will include +both absorption as well as emission lines in experiments. While a lot of work in the past +has been on dark matter induced emission lines [3–6] and electromagnetic signals in colliders +[7, 8], only a few touch upon the absorption signatures of dark matter [9, 10]. In addition +to electromagnetic/radiative transitions, the transitions between different energy states can +happen via inelastic scattering between dark matter particles. This has been studied in the +context of small scale structure problems like the core-cusp problem and the missing satellite +problem [11–27]. +In this work, we focus on the less studied and more promising signature of composite +dark matter: the absorption of light. The absorption line from dark matter inside a single +galaxy cluster at gamma ray frequencies was studied in [9, 10]. However, there is no a priori +reason to be confined to the gamma-ray band. In particular, the detection of an absorption +line unidentifiable with the known transitions in baryonic atoms or molecules in any part of +the electromagnetic spectrum is a tell-tale signature of such dark matter models. In addition +to a single absorption line, we can even have a collection of absorption lines or dark forest +similar to the Ly-α or 21 cm forest generated by the neutral hydrogen atoms. The dark +forest arises due to absorption of light by dark matter halos along the line of sight (LoS) to +a quasar. We show that the dark forest opens a new window to the large scale structure as +it traces the evolution of dark matter temperature and distribution inside the dark matter +halos through the cosmic history. +We make a detailed study of the evolution of dark forest from redshift of 7 to 0 for dark +matter transitions at radiowave frequency of 156 GHz. Interestingly, we find that the amount +of absorption by a dark matter halo of a given mass is sensitive to the presence of dark matter +self-interactions. Moreover, the density of absorption lines has a strong dependence on the +smallest dark matter sub-structures present in the Universe. In general, the dark forest can +appear in any part of the electromagnetic spectrum including radio, microwave, infrared, +optical, X-ray, and gamma ray bands. In particular, the detection of absorption lines in the +spectrum of a bright quasar at z ∼ 6 at frequencies < 200 MHz, below the 21 cm forest of +neutral hydrogen, will be a smoking gun signature for this model. This may be possible with +– 2 – + +the upgraded Giant Meterwave Telescope (uGMRT) [28, 29] and the Square Kilometer Array +(SKA) [30] which have lowest frequency bands in 50-350 MHz and 125-250 MHz respectively. +When the isotropic cosmic microwave background (CMB) acts as a background source, +the absorption of CMB photons by inelastic composite dark matter gives rise to a global +absorption feature in the CMB spectrum. The origin of the global absorption signal from +transitions in dark matter is similar to the global absorption feature caused by the hyper-fine +transitions in neutral hydrogen during the Dark Ages [31–34]. After dark matter decouples +from the electron-baryon plasma, it cools as (1 + z)2, with the temperature soon becoming +much lower than the CMB temperature. At very high redshifts, the strong inelastic collisions +between dark matter particles bring the two dark matter energy levels in kinetic equilibrium +with the dark matter temperature. The dark matter temperature being much lower than +the CMB temperature implies that the dark matter particles in the ground state can absorb +the CMB photons and generate an absorption signal in the CMB spectrum. As the Universe +cools and the number density of dark matter particles gets diluted, the radiative transitions +due to CMB photons take over the dark matter collisional transitions, bringing the level +population in equilibrium with the CMB temperature and the signal vanishes. An important +difference from 21 cm cosmology is the role of bremsstrahlung, which is important before +recombination when the Universe has ample number of electrons and protons. This process +can erase spectral distortions in the low frequency tail of the CMB before recombination +and establish an almost the perfect black body spectrum in the Rayleigh-Jeans tail. Thus +the high redshift (low frequency) edge of the absorption signal is entirely determined by +bremsstrahlung, while the low redshift (high frequency) edge of the signal depends on the +ratio of collisional to radiative coupling of dark matter. +Tantalizingly, we find that the absorption of CMB photons by dark matter with a transi- +tion frequency of 156 GHz at redshifts ∼ 3000−1000 can produce a global absorption feature +that matches the amplitude, width and location in frequency measured by the Experiment to +Detect the Global Epoch of reionization Signal (EDGES) [35, 36]. The EDGES collaboration +reported a strong absorption feature which is almost double in amplitude compared to the +maximum absorption expected from 21cm absorption by hydrogen in the standard model of +cosmology. However a recent experiment Shaped Antenna measurement of the background +Radio Spectrum (SARAS 3) [37] disfavors the EDGES absorption profile being cosmological +in origin. A recent paper from the EDGES collaboration [38] does a Bayesian analysis jointly +constraining the receiver calibration, foregrounds, and the measured signal reaffirming the +presence of an absorption feature. Several other groups such as Large aperture Experiment +to detect the Dark Ages (LEDA) [39–41], Probing Radio Intensity at high Z from Marion +(PRIZM) [42], Radio Experiment for the Analysis of Cosmic Hydrogen (REACH) [43], Sonda +Cosmol´ogica de las Islas para la Detecci´on de Hidr´ogeno Neutro (SCI-HI) [44], and Cosmic +Twilight Polarimeter (CTP) [45] are working on validating these claims. Even if the exact +profile measured by EDGES turns out to be due to systematics, an anomalously large ampli- +tude of the absorption signal would still require beyond standard model physics, if confirmed +by other experiments. +We begin with a discussion of the theoretical framework of composite dark matter +in the section 2. +We make a detailed study of the unique absorption signatures of such +dark matter models in section 3, where we discuss the physics of dark forest in subsection +3.1, and the global absorption feature in the CMB spectrum in subsection 3.2. We then +proceed towards the implications of different astrophysical and cosmological experiments on +the allowed parameter space for our dark matter model in section 4. We conclude our results +– 3 – + +in section 5. +We use the Planck 2018 [46] cosmological parameters (Hubble constant: H0 = 100 h = +67.66 km s−1 Mpc−1, Ωm = 0.3111, and Ωb = 0.049). We also use the publicly available +codes Recfast++ [47, 48], Colossus [49], and FeynCalc [50–52] in our analysis. +2 +A theoretical framework for the composite dark sector +In this section, we set up a theoretical framework for the underlying physics of composite +dark sector whose principal modes of observation are absorption features in the spectrum of +a background source. Specifically, we provide a proof of principle scenario that can accom- +modate a dark sector which leaves its signature through photon absorption lines. +Before we build such a model, we note the following set of considerations. Even though +each of these conditions need not be fulfilled strictly, stating them is useful. This not only +allows us to stay general but also provides hints at the emerging model for the dark sector. +(i) When the dark sector states make transitions among themselves, photons get absorbed/ +emitted. This immediately suggests that the dark sector consists of multi-level states +with mass gap(s). Further, if there exists a symmetry that makes at least the lowest lying +state cosmologically stable, the conservation of the associated quantum number dictates +that all such transitioning states must have identical quantum numbers/charges under +the same symmetry. +(ii) In order to emit/absorb a photon when a transition takes place, there must be contact +operators between the dark sector states and electromagnetic current. Such a coupling can +arise if dark matter possesses an electromagnetic charge. However, different observations +such as CMB [53, 54], virialization in dark matter halos, elliptical shape of galaxies [55], +bullet cluster [56], etc. strongly constrain the electromagnetic charge dark matter can +possess. To evade such constraints, we consider the transitioning states to be neutral +composits with electrically charged constituents. Such states are bound states analogous +to an atom of the visible sector. Unfortunately, the electric charge of the constituents +will lead to non-zero electromagnetic moment [10, 57] of the low lying dark state that can +give strong signals in the CMB, direct, and indirect dark matter searches. Such bounds +weaken when the constituents have very small electric charge or millicharge. +(iii) The ionization of dark matter, if accessible, would provide much stronger signals in ex- +periments compared to the absorption/emission lines. Ionized dark matter in the early +Universe will be subject to radiation pressure similar to the baryons which can modify the +CMB acoustic peaks, imprint acoustic oscillation features in the dark matter power spec- +trum, and erase structure on small scales. At late times, the Coulomb scattering between +ionized dark matter particles inside a halo can give rise to cored central density profiles. +This suggests that scenarios where the transition energy and binding/ionization energy +are similar (Etransition ≃ Ebinding), the signals from ionization are a far better probe. In +this work we concentrate on scenarios where the lines are the primary signatures of dark +matter. Therefore the model we construct should have an added feature that +Etransition ≪ Ebinding. +(2.1) +(iv) One can use data from CMB experiments to put a strong constraint on the ionization of +dark matter (see subsection 4.1 for details), +Ebinding ≳ 1.6 keV. +(2.2) +– 4 – + +All this information makes it quite clear that, +• The presence of a small electric charge of the constituents makes electromagnetism respon- +sible only for a small fraction of the binding energy of these states. +• The dark sector cannot be a simple scaled version of hydrogen-like atom. +While the first conclusion is obvious, the second demands a more careful explanation. For +a hydrogen-like dark matter with mass Mbound state, satisfying the condition in (iii) implies +that the transitions in dark matter cannot be Lyman-α like. Even though the hyper-fine +splitting (∆Ehf ≪ Ebinding) in these systems perfectly satisfies (iii), we find the simplicity +of the set-up over-constraining. For example, a hydrogen-like system satisfies the following +relation, +Mbound state ∼ Ebinding +�Ebinding +∆Ehf +� +=⇒ Mbound state ≫ Ebinding. +Thus, the desired hierarchy in ∆Ehf and Ebinding gets translated into a hierarchically large +Mbound state. If such a state represents dark matter, a large mass would result in small dark +matter number density which reduces the signal of absorption. The task therefore is to build +a framework which is calculable to a large extent and can accommodate the hierarchy in (iii) +while keeping the bound state mass within reach. +In this work, we propose a scenario where the leading order results get quantum cor- +rected due to additional light degrees of freedom (d.o.f.) in the dark sector. This is reminis- +cent of the proton-neutron mass difference occurring in nature, where the photons in the loop +cancel the tree level contribution from the mass difference between the up and down quarks. +Summarizing, we require the dark sector to consist of composite states that are massive and +cosmologically stable, and couple strongly to the light d.o.f. of the dark sector. +As a starting point, we take the dark states to be bound states of mass mχ where the +attractive potential between the constituents is dominantly due to dark-gluon exchanges. To +be specific, we take the underlying physics to be a non-abelian SU(N) gauge theory (we will +refer to it as dark color) with a suitably designed matter content and associated charges. The +strength of the gauge coupling constant roughly at the mass of the bound state i.e. αN(mχ) +determines most the features of the tower of dark states. For instance, for αN(mχ) ∼ O(1), +the bound state is relativistic and generically all energy splittings in the spectrum are of the +same order as the binding energy i.e. Etransition ∼ Ebinding, which does not satisfy condition +(iii). Therefore, we need a separate mass scale, MQ in the theory which would ultimately yield +mχ to be different from the scale (ΛN) at which gauge theory becomes strongly coupled. A +mild hierarchy between MQ and ΛN would allow the theory to be perturbative αN(mχ) ≪ 1, +resulting in non-relativistic dark bound states where the transition energy is much smaller +than the binding energy. +An additional flavor symmetry can play a constructive role, partly in providing stability +to the dark state and partly in giving rise to naturally light pseudo Nambu Goldstone bosons +(pNGBs) which couple strongly to the dark states. +The details of the model we employ +are provided in table 1, where we list the particle content in the ultraviolet (UV) and their +quantum numbers under the gauge as well as global symmetries. The dark quarks making +up the matter content follow the mass hierarchy: mq < ΛN < MQ. The mass term for the +dark quarks in Weyl representation is given by, +L ⊃ MQ QDQc +D + mqδab qDaqc +Db + h.c., +(2.3) +– 5 – + +SU(N) +SU(2)D +L +SU(2)D +R +U(1)D +U(1)em +qD +N +2 +1 +0 ++ϵ +qc +D +¯N +1 +¯2 +0 +−ϵ +QD +N +1 +1 ++1 ++ϵ +Qc +D +¯N +1 +1 +−1 +−ϵ +Table 1: The dark quarks in Weyl representation and their charges under gauge and global +symmetries. +where a and b are the flavor indices which take values 1, 2 and the hermitian conjugate is +abbreviated as h.c.. +The mass term for the light quarks in eq.(2.3) ensures that the condensate ⟨qDqc +D⟩ is +flavor diagonal signaling that the global SU(2)D +L × SU(2)D +R × U(1)D +L × U(1)D +R gets sponta- +neously broken into a diagonal SU(2)D +V × U(1)D, with an additional axial U(1)A broken by +the dark color itself. The axial SU(2)D +A is explicitly broken by the light quark mass mq in +eq. (2.3). After symmetry breaking, there are no massless NGB modes but there exist three +light pNGBs or dark pions πD whose mass mπD is controlled by light dark quark mass mq. +We take these non relativistic heavy-light bound states (made of heavy quark QD and +light quark qD) to form the dark tower of states protected by the darkness number (quantum +number) of the global U(1)D symmetry listed in table 1. Further, the dark states with light +quarks as constituents also carry the flavor charges of the light quark. This implies that the +chiral effective Lagrangian generated by the dark color will automatically include the dark +state - pNGB interactions. +The model proposed here may appear complicated, but it should be noted that the +physics of the dark sector described here mimics aspects of the physics of the visible sector. +In nature the strong interactions of Quantum chromodynamics (QCD) play a similar role +in producing heavy-light bound states such as D-mesons or B-mesons. +The spontaneous +breaking of the approximate chiral symmetry associated with the light quarks of the visible +sector gives rise to pions (pNGBs of the visible sector) with substantial interactions with the +heavy light mesons. Following the formalism of the chiral Lagrangian for the heavy flavor +[58–60], one can write down the form of interaction between the dark states and the dark +pions, and estimate the hyper-fine splitting, transition rates, etc. In appendix A, we chalk +out some aspects of this formalism in the context of our model. In the main body of this +work we simply outline the main features of the model: +• The dark pions are pNGBs lying the coset SU(2)D +L × SU(2)D +R +� +SU(2)D +V . One can use the +usual exponential parametrization to express πD as, +ΣD ≡ exp +�2iπD +fD +� +where πD ≡ πa +D ˜ta, and ΣD +SU(2)D +L ×SU(2)D +R +−−−−−−−−−−−→ LDΣDR† +D +(2.4) +The broken generators ˜ta belong to the coset, fD is the energy scale of the dark condensate +⟨qDqc +D⟩, and LD and RD are transformation operators for SU(2)D +L and SU(2)D +R respectively. +• We take ¯QD qD bound state (darkness number -1) as the candidate for dark states. Similar +to the matter anti-matter asymmetry in the visible sector, the asymmetry in the number +– 6 – + +of dark (darkness number -1) versus anti-dark states (darkness number +1) yields the +observed dark matter abundance. +• We designate the lowest lying pseudoscalar spin 0 state as χ and the spin 1 state as χ∗ +µ. +A convenient way to represent these four physical states collectively is using a matrix field +Xv which is an eigenstate of the velocity v of the dark bound state, +Xv ≡ P+ +� +χ∗ +µγµ − χγ5� +. +(2.5) +The projection operator P+ = 1 +2 +� +1 + /v +� +captures only the small fluctuations (≪ MQ) for +this Heavy Quark Effective Field theory (HQET) [61–64]. +• χ-χ∗ exhibits a nearly degenerate system. The mass difference between χ and χ∗, known +in the literature as the hyper-fine splitting, arises from operators shown in of appendix +A. This splitting is suppressed by the heavy quark mass MQ, but contains a number of +unknown parameters and in the limit MQ → ∞, χ and χ∗ become exactly degenerate. +• The spontaneous emission rate for χ∗ → χ γ process is computable within the paradigm of +the chiral Lagrangian for the heavy-light systems. The rate is also MQ suppressed like the +hyper-fine splitting but depends on a different set of unknown parameters. +• In order to derive the χ∗χ∗πD or χχ∗πD couplings, one needs to take into consideration +the symmetry properties of the heavy-light bound states. It is conventional to write the +interactions between the bound states and dark pions by defining the vectorial and axial +currents which contain the dark pions. These couplings also contribute non-trivially to the +hyper-fine splitting as well as the transition rates. +Even though the formalism of HQET seems to describe the different properties of the B +and D meson states, such as, the pattern of their couplings and mass gaps extremely well, +there is one crucial drawback. The physics of these states is described in terms of unknown +constants. While in the context of states in the visible sector, the existence of data allows +us to determine these constants (which in turn, makes it possible to predict several other +observables rather precisely), such a procedure is not practical for our dark bound states. +We also cannot just scale up QCD to predict these, since the number of colors, flavors, and +pattern of masses are not the same. Of course, lattice QCD can make definitive statements +about the size of splittings and photon transition rates, but such an exercise is beyond the +scope of this work. +For the purpose of phenomenology, it is sufficient to consider a simplified model where +we only keep a handful of states. Instead of using unknown parameters in the Lagrangian of +the theory, we parameterize the simplified model in terms of parameters which quantify the +energy splittings and transition rates of various physical processes. The underlying EFT is +depicted in terms of a simplified model in figure 1. The main features of this model relevant +for the estimation of absorption lines arising due to hyper-fine transitions in the dark sector +can be summarized as follows, +• Since the only relevant transitions happening in the dark states are the hyper-fine tran- +sitions, the physics is rather simple. We can disregard all the other higher energy states +and treat this as a two-state system with states 0 and 1. The state 0 corresponds to en- +ergy E0 = mχc2 and degeneracy factor g0, and state 1 corresponds to energy E1 = mχ∗c2 +– 7 – + +Figure 1: The left part shows the different energy scales of the theory. The right part is a +zoomed-in version of the hyper-fine splitting in the dark sector. +and degeneracy factor g1, where c is the speed of light in vacuum. The energy difference +between two states is given by, +∆Ehf = E1 − E0 = hν0 = kBT∗, +(2.6) +where h is the Planck’s constant, kB is the Boltzmann constant, ν0 is the transition fre- +quency, and T∗ is the transition temperature. +We will further assume ∆Ehf < mπD, which implies that as far as χ∗ ↔ χ transitions are +concerned, we can disregard the dark pions. In this work, for simplicity we take g1/g0 = 3. +• The population of dark matter particles in the two states is decided by the collisional and +radiative transition rates. The ratio of the number density of dark matter particles in the +ground state (n0) with respect to the excited state (n1) is parameterized by the excitation +temperature Tex, +n0 +n1 +≡ g0 +g1 +exp(T∗/Tex). +(2.7) +In our simplified model, the occupation number of 0 and 1 states gives the total dark +matter number density, +nχ = n0 + n1. +(2.8) +• The transition between the two states can happen via emission/absorption of a photon +which is parameterized in terms of Einstein A and B coefficients in the following way: +The number of radiative transitions per unit time per unit volume from level 0 to level 1 +is proportional to the Einstein coefficient B01, +dn0→1 +dt += n0B01 ¯J, +(2.9) +where ¯J is the mean intensity of incident light. The number of radiative transitions per unit +time per unit volume from level 1 to level 0 is a sum total of spontaneous emission which +proportional to the Einstein coefficient A10 and stimulated emission which is proportional +to B10, +dn1→0 +dt += n1(A10 + B10 ¯J). +(2.10) +– 8 – + +excited +states +m. +AEh +x +AN +m +TDThe Einstein coefficients A10, B01, and B10 are related to each other via the Einstein +relations which follow from the principle of detailed balance, +A10 = 2hν3 +0 +c2 B10, +g0B01 = g1B10. +(2.11) +In this work, we parameterize the Einstein coefficient for hyper-fine transitions in the dark +sector in terms of the Einstein coefficient for hyper-fine transitions in the hydrogen atom, +A10 = αA AHI +10, where AHI +10 = 2.85 × 10−15s−1. +(2.12) +We will set αA = 0.35 in our numerical computations (see subsection 4.2 for justification). +• The transition between the two states can also happen via inelastic collisions between +dark matter particles parameterized in terms of the collisional excitation and de-excitation +coefficients C01 and C10 respectively. The number of collisional transitions per unit volume +from level i to level j is given by, +dni→j +dt += niCij, +(2.13) +where i ̸= j and i, j run from 0 to 1. For a thermal velocity (Maxwell Boltzmann) distribu- +tion of dark matter particles at temperature Tχ, the two collisional coefficients are related +as, +Cij(Tχ) = gj +gi +exp(−T∗/Tχ) Cji(Tχ). +(2.14) +In case of a quasar, we simply consider the implications for two extreme scenarios, one +where inelastic collisions are completely absent (collisionless dark matter) and the other +where the inelastic collisions are strong (collisional dark matter) (see eq.(3.1) in subsection +3.1). In case of CMB as a background source, we use intermediate collisional cross-section +parameters as a function of dark matter temperature (see eq.(3.10) of subsection 3.2 for +details). +3 +Experimental signatures +The composite dark matter particle can make an electromagnetic transition from the ground +state to an excited state by absorbing a photon. Such transitions can give rise to unique +experimental signatures in the form of absorption lines in the spectrum of a bright background +source. In particular, the detection of a new absorption line, not identifiable with a known +atomic or molecular transition in any part of the electromagnetic spectrum, would be a +smoking gun signature for such dark matter models. +The high dark matter density in structures like dark matter halos and dwarf galaxies +makes these sites ideal targets that can generate such absorption signals. In particular, when +one such object lies along the line of sight (LoS) to a compact source, the absorption of light +by the composite dark matter particles inside these objects produces an absorption line in +the source spectrum. The shape of the line is characterized by the density and velocity dis- +tribution profile of dark matter particles inside the absorber. In reality, we will have multiple +– 9 – + +Figure 2: Schematic diagram of the line of sight intersecting an absorber at an impact +parameter p. +such absorbers along the LoS to a distant quasar resulting in a series of absorption lines at +different frequency locations in the observer’s frame. The frequency location of the absorp- +tion lines is decided by the transition frequency and the redshift of the absorber. We will +study the absorption lines in the spectrum of a compact source for a single absorber by taking +an example of a dwarf galaxy and a general dark matter halo in subsection 3.1.1. We then +proceed towards the case of multiple absorbers along the LoS to a quasar in subsubsection +3.1.2. +When the source is isotropic i.e. the CMB, the composite dark matter particles absorb +the CMB radiation giving rise to a broad global absorption feature in the CMB spectrum. +We study such a dark global absorption feature in subsection 3.2. +3.1 +Absorption lines in the spectrum of a compact source +When the LoS to the compact source passes through an absorber, such as a dark matter +halo, the composite dark matter particles inside these structures can absorb the incident +light, imprinting an absorption feature in the source spectrum. Similar to absorption, we +can also have emission lines from dark matter imposed on the average spectrum of a galaxy +inside the dark matter halo. +In the rest frame of a point-like absorber situated at redshift z0, the absorption/emission +happens at the transition frequency ν0. Due to the expansion of the Universe, the absorp- +tion/emission line is observed today at a frequency ν = ν0/(1 + z0). However, complication +arises for absorbers of finite size and non-trivial density and velocity profiles in different +possible astrophysical scenarios. +• Dark matter density profile: For an extended absorber intersected by the LoS to the +source at an impact parameter p (as shown in figure 2), the cumulative net absorption (true +absorption minus stimulated emission) gets contribution from all the particles present along +the LoS, which is denoted by s. Lets consider a line element ds (in figure 2) along the LoS. +The true absorption (stimulated emission) is proportional to the number density of dark +matter particles in the ground (excited) state, which in turn is proportional to the total +number density of dark matter particles nχ(r) = ρ(r)/mχ, where ρ(r) is the dark matter +density at a distance r from the center. +• Excitation temperature: The population of dark matter particles in the two states at +a radius r inside the dark matter halo is determined by the excitation temperature Tex +– 10 – + +Line of sight +s +ds +S=0 +Absorberdefined in eq. (2.7). The excitation temperature is determined by two processes, namely, +the radiative transitions due to the CMB photons which try to bring the two levels in +kinetic equilibrium with the CMB temperature (Tγ(z0)), and the collisional transitions +due to inelastic collisions between dark matter particles inside the halo, which try to bring +the two levels in kinetic equilibrium with the temperature of the halo (Th(r)). In this work +we will study two extreme scenarios for dark matter (DM) inelastic self-interactions, +Tex(r) = +� +Tγ (z0) +collisionless DM, +Th (r) +collisional DM. +(3.1) +The general scenario would lie somewhere between these two extremes. We will indeed find +that the absorption lines are sensitive to the collisional nature of dark matter. +• Doppler broadening: The non-trivial velocity profile of the dark matter particles along +the LoS gives rise to the Doppler broadening of the absorption line around ν0 in the halo’s +rest frame. This broadening is characterized by the line profile, +φ (νh, r, p) = +1 +√π∆νD (r) exp +� +−(νh − ν0 (1 + vLoS(r, p)/c))2 +∆νD (r)2 +� +, +where ∆νD (r) = ν0 (1 + vLoS(r, p)/c) +c +� +2kBTh (r) +mχ +, +(3.2) +νh being the absorption frequency in the halo’s rest frame, and vLoS being the peculiar +velocity of the dark matter halo along the LoS. The effect of vLoS is simply to shift the +frequency location of the line in observer’s frame. In this work, we will not be calculating +the two-point correlations but only the one-point statistics of the dark matter forest. So +we will ignore the halo peculiar velocity and set vLoS = 0. +In the presence of absorption, the flux density measured by the observer falls exponentially +with the column density along the LoS. Conventionally, the observed absorption line is quan- +tified by the optical depth τν which is defined as, +τν = log +�F 0 +ν +Fν +� +, +(3.3) +where Fν and F 0 +ν are the flux densities of the source in the presence and absence of absorption +respectively. +For a halo intersected at an impact parameter p (as shown in figure 2), the optical depth +profile in the halo’s rest frame is given by [65], +τ(νh, p) = +� do +−ds +g1 +g0 +αAAHI +10 c2 +8πν2 +0 +ρ (r) +mχ +φ (νh, r, p) +� +1 − e− +T∗ +Tex(r) +1 + (g1/g0) e− +T∗ +Tex(r) +� +ds, +(3.4) +where r2 = p2 + s2. We set the origin s = 0 to be the position where the impact parameter +intersects the LoS. We integrate the LoS from the source to the observer. +The distance +between the source and the absorber is denoted by ds and the distance between the observer +and the absorber is denoted by do. In the frame of the observer on Earth, the optical depth +profile is obtained by mapping νh → νh/(1 + z0). +– 11 – + +In the rest of the analysis we choose the following dark matter model parameters: +mχ = 1 MeV, ν0 = 156.2 GHz, and αA = 0.35 (see section 4 for justification) for our study. +Our main results are however quite general and we leave the full exploration of the parameter +space for future work. +3.1.1 +A dark line: absorption by a single dark matter halo +The dark matter halos are gravitationally bound structures which form the building block +of the non-linear matter distribution. We want to study how the different properties of dark +matter halos influence the absorption profile generated by them. +A dark matter halo is characterized by its mass parameter (Mh), a length parameter +(virial radius rvir), a temperature (Th), and a dark matter density profile ρ(r). Some of these +parameters are related (see appendix B for exact expressions). We assume a NFW density +profile [16] for the dark matter halo. The Doppler line profile for the halo is decided by the +halo temperature Th (see eq. (B.3) in appendix B for definition). +We proceed towards calculating the optical depth or the absorption profile generated by +a given halo mass Mh using eq.(3.4). The optical depth depends crucially on the dark matter +density profile, the impact parameter (p expressed in rvir units), and the Doppler broadening +due to random motion of the dark matter particles parameterized by the effective temperature +(Th) of the dark matter halo. We present our results in figure 3. We plot the optical depth +profiles for a given halo mass (106M⊙/h) at different redshifts (z = 6, 4 and 2) intersected +at p = 0.1 rvir in the top two panels. In the bottom two panels, we plot the optical depth +profiles at a given redshift (z = 5) for different halo masses (in M⊙/h units) intersected at +p = 0.5 rvir. We make the following observations: +(i) There is stronger absorption in collisional DM compared to collisionless DM. +(ii) As we go to higher redshifts, the total absorption by a halo, which is equal to the area +under the optical depth profile, grows (top two panels of figure 3). +(iii) The width of the absorption profile increases with halo mass and redshift. +(iv) In collisionless DM case, the peak amplitude of the absorption profile increases with +the halo mass (third panel of figure 3). +(v) In collisional DM case, the peak amplitude of the absorption profile decreases with the +halo mass (fourth panel of figure 3). +We explain these findings using figure 4 where we compare the halo temperature at the virial +radius rvir for different halo masses with the CMB temperature in the first panel. We also +compare the dark matter number density profile and the halo temperature profile at different +redshifts in the last two panels. Corresponding to the above observations, the explanations +are as follows: +(i) For halos of mass ≲ 108M⊙/h, Th < Tγ (first panel of figure 4). Therefore the excitation +temperature (defined in eq.(3.1)) for collisional DM is less than that for collisionless +DM, resulting in stronger absorption in the case of collisional DM. +(ii) As we go to higher redshifts, the number density of dark matter particles increases +which results in stronger absorption (second panel of figure 4). +– 12 – + +−6 +−4 +−2 +0 +2 +4 +6 +∆ f (MHz) +0.00 +0.05 +0.10 +0.15 +0.20 +Optical depth +Collisionless DM +z = 6 +z = 4 +z = 2 +−6 +−4 +−2 +0 +2 +4 +6 +∆ f (MHz) +0.0 +0.5 +1.0 +1.5 +2.0 +Optical depth +z = 2 +z = 4 +z = 6 +Collisional DM +−20 +−10 +0 +10 +20 +∆ f (MHz) +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +0.035 +Optical depth +Mh in M⊙/h +Collisionless DM +Mh = 108 +Mh = 107 +Mh = 106 +−20 +−10 +0 +10 +20 +∆ f (MHz) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Optical depth +Mh in M⊙/h +Collisional DM +Mh = 108 +Mh = 107 +Mh = 106 +Figure 3: Top: The optical depth profiles for 106M⊙/h halo at impact parameters 0.1 rvir in +the halo’s rest frame. Bottom: The optical depth profiles for different halo masses intersected +at 0.5 rvir at z = 5 in the halo’s rest frame. The solid lines refer to collisionless DM and dashed +lines refer to collisional DM. +(iii) The width of the optical depth profile in the halo’s rest frame ∝ √Th. +The halo +temperature, Th increases with both redshift and halo mass (third panel of figure 4). +(iv) In collisionless case, Tex = Tγ is independent of the halo mass. The number density +of dark matter particles increases with the halo mass resulting in stronger absorption +profiles for collisionless DM. +(v) In the collisional case, Tex = Th increases with the halo mass. Even though the total +dark matter number density increases with the halo mass, a higher value of Tex implies +less dark matter particles in the ground state, which combined with broadening of the +profile results in smaller peak amplitude of the absorption profile for higher halo masses. +Note that in this analysis we have assumed the dark matter density and velocity distribution +profiles to be the same in both collisionless and collisional cases. +The presence of dark +– 13 – + +0 +2 +4 +6 +Redshift +100 +101 +102 +Th(rvir) (K) +Mh = 109 +CMB +Mh = 108 +Mh = 107 +Mh in M⊙/h +0 +1 +2 +3 +4 +Radius (in rvir units) +100 +102 +104 +DM number density(cm−3) +z = 6 +z = 4 +z = 2 +0 +1 +2 +3 +4 +Radius (in rvir units) +0.2 +0.4 +0.6 +0.8 +1.0 +Th (K) +z = 6 +z = 4 +z = 2 +Figure 4: From left: The evolution of halo temperature at rvir as a function of redshift +for halo masses 107, 108, and 109 M⊙/h, the dark matter number density profile and the +halo temperature profile for 106 M⊙/h halo as a function of radius (in rvir units) at different +redshifts z = 6, 4 and 2. +−0.010 +−0.005 +0.000 +0.005 +0.010 +∆ f (MHz) +0.0 +0.5 +1.0 +1.5 +2.0 +Optical depth +2.0 rs +1.0 rs +0.5 rs +Figure 5: Optical depth profile for Leo T subhalo for impact parameters p = 0.5 rs, 1.0 rs +and 2.0 rs for collisionless DM represented by solid lines and collisional DM represented by +dashed lines. +matter self-interactions does not change the Maxwellian velocity profile as discussed in [66]. +In principle, strong dark matter collisions may modify the density profile of dark matter +halos which can modify the shape of the absorption profiles. +As a detour, to showcase the possibility of hunting for dark matter absorption signatures +in satellite galaxies of Milky Way, we take the example of absorption line generated by the +dark matter subhalo that hosts the Leo-T dwarf galaxy. +Leo-T dwarf galaxy: +Low mass dwarf galaxies are excellent venues to study dark matter +since they have low star formation activity and weak electromagnetic emission. Some of +the best current constraints on emission signatures of dark matter come from the dwarf +satellite galaxies of Milky Way. [67–72]. Thus we also expect that the absorption of light +from a background source by composite dark matter particles in dwarf galaxies would provide +strong tests for such dark matter models. We consider one such MW satellite galaxy, namely, +– 14 – + +Leo-T. We model its dark matter density profile using a Burkert profile from [73]. We assume +the velocity distribution of dark matter to be Maxwellian with a velocity dispersion ( +� +⟨v2⟩) +equal to that of hydrogen ∼ 6.9 km/s [74]. The temperature of the halo is defined by the +relation, kBTh = 1 +3mχ⟨v2⟩. For mχ = 1 MeV, we find Th ∼ 2.1 K. In figure 5, we show the +absorption profiles of Leo-T intersected at impact parameters 0.5 rs, 1.0 rs and 2.0 rs, where +rs is the scale radius of the halo. We note that since Th ∼ Tγ for Leo-T, the absorption +profiles in collisional and collisionless cases are similar. However the small difference in Tex +still gives a noticeably stronger absorption in case of collisional dark matter compared to +collisionless dark matter. +3.1.2 +Dark forest: absorption by multiple dark matter halos +If the LoS to the source passes through multiple halos (located at different redshifts z0), +each intersection gives rise to an absorption profile at ν = νh/(1 + z0) to an observer on +Earth. +Collectively, a large number of absorption lines coming from the same transition +in dark matter at different redshifts, and hence separated in frequency, are called forest in +spectroscopy. In this section we describe the procedure to simulate a dark forest and discuss +its qualitative and quantitative aspects. +The simulation consists of discretized frequency bins in a given frequency range with +the bin width adjusted such that each bin has an identical probability of net absorption. In +a pseudo experiment, we then simulate the absorption line by generating a random number +to select the bin where absorption occurs. We plot the observed dip in intensity in terms of +the relative transmission e−τν. We summarize the algorithm for generating the dark forest +spectra below [65, 75, 76]: +• We begin by selecting the frequency range of simulation. For an instrument sensitive in +νmin to νmax range, the absorption lines correspond to halos in zmax = ν0/νmin − 1 to +zmin = ν0/νmax − 1 redshift range. +• We find the equiprobable bin width ∆ν at a given ν by relating it to the probability of +finding a halo in redshift bin ∆z centered at z = ν0/ν − 1. This probability is equal to the +fraction of the area on the sky covered by halos of all masses in ∆z redshift bin. Thus the +probability of intersecting a halo in a frequency range ν to ν + ∆ν is given by, +∆Nh = ∆ν dNh +dν += ∆z dNh +dz += ∆z c (1 + z)2 +H (z) +� Mmax +Mmin +dMh +dn +dMh +(z)A (Mh, z) , +where +A (Mh, z) = πrmax (Mh, z)2 . +(3.5) +The halo mass function dn/dMh in comoving units is taken from [77], Mmin and Mmax +denote the minimum and maximum halo mass at a given redshift respectively, and rmax is +the physical radius of the halo at which the dark matter number density is equal to the +mean dark matter number density in the Universe. We choose the bin width ∆ν at each ν +such that the probability of absorption ∆Nh = 0.1. +• We generate a random number from a uniform distribution in [0, 1] in each frequency bin. +The bin is selected for absorption if the random number is ≤ 0.1. +• The absorption profile is characterized by the halo’s redshift z0, mass Mh, and impact +parameter p. For the selected bin, we choose Mh from the probability distribution function +– 15 – + +of the area fraction occupied by halos of mass Mh at redshift z0, +p (Mh, z0) ∝ +dn +dMh +A (Mh, z0) . +(3.6) +We choose the impact parameter from a uniform distribution over the cross-sectional area +of the halo A(Mh, z). +• We then generate the absorption profile in the halo’s rest frame using eq.(3.4) and map it +to the observer’s frame by transforming νh → νh/(1+z0). +We simulate the synthetic dark forest spectra for 100 different LoS in the redshift range 7 to 0 +(see appendix C for one such sample spectra). We quantify the information in the dark forest +by studying the distribution functions of the peaks (τpeak) and widths (b2) of the absorption +lines. The width is defined in terms of b2 given by [78], +b2 = −2 +� c +ν0 +�2 +τ 2 +ττ ′′ − τ ′2 = −2 +� c +ν0 +�2 τpeak +τ ′′ +peak +, +(3.7) +where τ ′ = dτ/dν, τ ′′ = d2τ/dν2. +By combining the spectra for 100 different LoS, we calculate the mean count N and +standard deviation in ∼ 30 bins to get the distribution functions for τpeak and b2. Note that +these distribution functions are unnormalized. The shape of the distribution functions at very +low values is an artifact of the maximum impact parameter at which a halo is intersected +(∼ 4.5rvir in our case). So we place a cutoff on τpeak and b2 at the lower end and plot the +distribution functions only above this cutoff, where we are not affected by the choice of the +maximum impact parameter. We present our results in figures 6 and 7. +We observe: +(i) The distribution functions for both τpeak and b2 are higher for Mmin = 104M⊙/h +compared to Mmin = 106M⊙/h. +(ii) The tails of both τpeak and b2 distribution functions in both collisionless and collisional +case extend to larger values at lower redshifts (redshift range 2-1 compared to redshift +range 6-5). +(iii) The τpeak distribution function for collisional DM rises above the collisionless DM at +large values of τpeak. +(iv) In the collisionless DM case, the τpeak distribution function rises monotonically at lower +values of τpeak. For collisional DM, the curve rises at lower values of τpeak, reaches a +peak, falls, and again rises at τpeak ∼ 10−3. +(v) The b2 distribution function for collisionless and collisional DM almost coincide. +(vi) The tail of b2 distribution function for Mmin = 106M⊙/h extends to larger values +compared to Mmin = 104M⊙/h. +To explain the findings above, we make two plots in figure 8. In the first plot we compare +the number of halos intersected per unit redshift for three different values of Mmin and in +the second plot we compare the unnormalized probability of intersecting a halo of mass (Mh) +at redshifts 1, 4 and 7. Corresponding to the above observations, the explanations are as +follows: +– 16 – + +10−2 +10−1 +100 +τpeak +10−1 +100 +101 +102 +dN/dτpeak +Mmin = 106M⊙/h +z : 6 - 5 +Collisionless +Collisional +10−2 +10−1 +100 +τpeak +10−1 +100 +101 +102 +dN/dτpeak +Mmin = 104M⊙/h +z : 6 - 5 +Collisionless +Collisional +10−3 +10−2 +10−1 +100 +τpeak +10−2 +10−1 +100 +101 +102 +103 +dN/dτpeak +Mmin = 106M⊙/h +z : 2 - 1 +Collisionless +Collisional +10−3 +10−2 +10−1 +100 +τpeak +10−2 +10−1 +100 +101 +102 +103 +dN/dτpeak +Mmin = 104M⊙/h +z : 2 - 1 +Collisionless +Collisional +Figure 6: Distribution function for the optical depth peaks for minimum halo mass 106 M⊙/h +and 104 M⊙/h in redshift ranges 6-5 (top) and range 2-1 (bottom) respectively. +(i) The lower limit of the halo mass function Mmin decides the contribution of the low mass +end of the halo mass function to eq.(3.5). Thus the probability of intersecting a halo +is higher for Mmin = 104M⊙/h compared to Mmin = 106M⊙/h (first panel of figure 8). +(ii) As the matter overdensities grow, the collapse fraction increases and the higher mass +halos start contributing to the mass function at lower redshifts. In addition at lower +redshifts, a given redshift interval dz corresponds to a larger comoving distance interval +dη = dz/H(z) resulting in more number of halo intersections (second panel of figure 8). +This in turn gives rise to higher τpeak and b2 values at low redshifts as there is a greater +chance of intersecting more massive halos as well as intersecting halos close to the center +where dark matter density and halo temperature is high. Moreover, the line width for +a halo also increases at lower redshifts due to smaller Doppler shift (b2 ∝ 1/(1 + z)2). +(iii) At a given redshift, the probability of hitting low mass halos along the LoS is higher, +since the halo mass function falls exponentially at larger masses (second panel of figure +– 17 – + +100 +101 +102 +103 +b2 (km2/s2) +10−2 +10−1 +100 +101 +102 +dN/d(b2[km2/s2]) +Mmin = 106M⊙/h +z : 6 - 5 +Collisionless +Collisional +10−1 +100 +101 +102 +103 +b2 (km2/s2) +10−2 +10−1 +100 +101 +102 +dN/d(b2[km2/s2]) +Mmin = 104M⊙/h +z : 6 - 5 +Collisionless +Collisional +100 +101 +102 +103 +104 +105 +b2 (km2/s2) +10−1 +100 +101 +102 +103 +dN/d(b2[km2/s2]) +Mmin = 106M⊙/h +z : 2 - 1 +Collisionless +Collisional +10−1 +100 +101 +102 +103 +104 +105 +b2 (km2/s2) +10−1 +100 +101 +102 +103 +dN/d(b2[km2/s2]) +Mmin = 104M⊙/h +z : 2 - 1 +Collisionless +Collisional +Figure 7: Distribution function for line widths for minimum halo mass 106 M⊙/h and +104 M⊙/h in redshift ranges 6-5 (top) and 2-1 (bottom) respectively. +8). As explained before, the absorption is stronger for collisional DM compared to +collisionless DM in halos of masses ≲ 108M⊙/h (first panel of figure 4). +(iv) A new absorption peak is generated when the tails of two or more absorption profiles +overlap. This can be seen in figure 9 where we compare the dark forest spectrum for +collisional and collisionless DM. Due to stronger absorption in collisional DM case, the +new lines give rise to an extra feature in the τpeak distribution function for collisional +case compared to collisionless case at the low τpeak end. +(v) We consider the same velocity distribution profiles for DM inside the halo for both +collisionless and collisional DM. The small differences mostly arise when the absorption +profiles of two or more halos overlap and give rise to new lines which can have different +line widths in collisionless versus collisional case. +(vi) More massive halos have higher halo temperatures resulting in a larger line width +(b2 ∝ Th). +– 18 – + +0.0 +2.5 +5.0 +7.5 +10.0 +Redshift +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +dN/dz +Mmin = 104 +Mmin = 105 +Mmin = 106 +Mmin in M⊙/h +5.0 +7.5 +10.0 +12.5 +15.0 +Log10(Mh) +10−25 +10−21 +10−17 +10−13 +10−9 +10−5 +10−1 +(dn/dMh) × A(Mh) +z=1 +z=4 +z=7 +Figure 8: On the left we plot the number of halos intersected per unit redshift. On the +right we plot the unnormalized probability distribution for intersecting a halo of mass Mh +along line of sight at redshifts 1, 4 and 7. +5.10 +5.11 +5.12 +5.13 +5.14 +Redshift +25.400 +25.425 +25.450 +25.475 +25.500 +25.525 +25.550 +25.575 +25.600 +Frequency (GHz) +0.996 +0.998 +1.000 +Relative transmission +5.10 +5.11 +5.12 +5.13 +5.14 +Redshift +25.400 +25.425 +25.450 +25.475 +25.500 +25.525 +25.550 +25.575 +25.600 +Frequency (GHz) +0.996 +0.998 +1.000 +Relative transmission +Figure 9: Dark forest spectrum for collisionless DM (above) and collisional DM (below) for +Mmin = 106M⊙/h. The absorption lines in the regions of overlap of absorption profiles of +two halos are stronger in collisional case compared to collisionless case. Both the spectra are +for identical halo intersections (i.e. same simulation), the only difference being the collisional +property of dark matter. +We check the convergence of the distribution functions by increasing the line of sight direc- +tions from 100 to 1000 in appendix D. +– 19 – + +10−2 +101 +104 +107 +1010 +Frequency (GHz) +10−5 +10−4 +10−3 +10−2 +Resolution (∆ν/ν) +uGMRT +LOFAR +SKA1 +JWST +X-shooter +BOSS +XMM-Newton +Chandra +Athena +10−4 +10−3 +10−2 +10−1 +100 +101 +τpeak +10−2 +10−1 +100 +101 +102 +103 +dN/dτpeak +0.1 αA +mχ +αA +mχ +10 αA +mχ +Sensitivity +Figure 10: On the left we plot the spectral resolution of different spectroscopic experiments +over the electromagnetic spectrum. On the right we show the scaling of the τpeak distribution +function (for collisionless DM in redshift range 2 to 1 and Mmin = 104M⊙/h) with αA/mχ +(see eq.(2.12) for the definition of αA). The dashed black line shows the minimum sensitivity +(τpeak > 0.01) of an experiment. +3.1.3 +Detectability of dark forest +The dark forest is a collection of absorption lines, where each line is characterized by a +frequency, width and a peak amplitude. For our choice of ν0 = 156.2 GHz, the absorption +lines were generated at radiowave frequencies with a typical width ∆ν/ν ≈ 10−3. A different +ν0 would give rise to the dark forest in a different part of the electromagnetic spectrum. The +existence of a large number of spectroscopic experiments spanning different frequency ranges +already make the detection of new dark absorption lines an exciting possibility. +To name a few, experiments like the Square Kilometer Array (SKA1) [30], Low-Frequency +Array (LOFAR) [79], and the upgraded Giant Meterwave Radio Telescope (uGMRT) [28, 29] +operate at radiowave frequencies, the James Webb Space Telescope (JWST) [80] covers the +infra-red, Baryon Oscillation Spectroscopic Survey (BOSS) [81] and X-shooter [82] look for +quasars at redshifts ∼ 2 to 5 at the optical frequencies, while Chandra [83] and X-ray Multi +Mirror Mission (XMM-Newton) [84] operate at UV and X-ray frequencies. The frequency +coverage and spectral resolution for these experiments are shown in the first panel of figure +10. We can see that existing experiments operating in radio and optical frequencies already +have the required spectral resolution (∆ν/ν ≳ 10−3) to detect the new dark absorption lines +in the spectrum of bright quasars/blazars. A high signal to noise ratio can be achieved by +increasing the duration of observation or the integration time which would allow the detec- +tion of the weak absorption lines. Note that the peak amplitude of an absorption line scales +as αA/mχ as shown in eq. (3.4). Our choice of αA is at the boundary of being disallowed for +dark matter of mass 1 MeV by the CMB constraints (shown in the third panel of figure 13 +in section 4). However, note that our constraints are very conservative and a more careful +analysis would considerably weaken them. Also for a different value of ν0 outside the observ- +able band of CMB, higher values of the radiative coupling αA would be allowed. Thus, all +our results and plots can be readily scaled for a different values of αA. +– 20 – + +We show the scaling of the τpeak distribution function with αA/mχ obtained by averaging +over 100 different LoS directions for 0.01 and 10 times αA/mχ compared to αA/mχ chosen in +this work in the second panel of figure 10. We find that if an instrument can detect absorption +lines with τpeak ≥ 0.01, the full distribution function for 10 × αA/mχ can be probed with +the spectra of ∼ 100 quasars. The sensitivity decreases for weaker absorption lines and a +sufficiently large quasar sample is required to probe the tail of the distribution function. For +instance, in 0.1 × αA/mχ case, only ∼ 1 in 100 quasar spectra contributes to τpeak > 0.01 in +the tail of the distribution function. +3.2 +Global absorption signal in the CMB spectrum +The absorption of photons of a particular frequency also leave a tell-tale signature in the +sky-averaged spectrum. Such features are called global signals. The much studied 21 cm +global signal [31–34] in the CMB spectrum due to neutral hydrogen, for example, carries +within important information about the growth of structure and first stars. Not surprisingly, +we expect a similar global absorption in CMB in case of transition among the dark sector +states. The underlying physics of the global absorption feature is more or less similar to the +absorption along the LoS to a bright source. However, there are some crucial differences: +• In case of absorption along the LoS to a compact source, the observed signal is equal to +absorption minus stimulated emission. The effect of spontaneous emission is negligible as it +gets distributed along all directions in the 4π solid angle. However, spontaneous emission +is important in case of CMB because it is an isotropic source and the observed signal along +a given LoS gets contribution from spontaneous emission. +• We assume that dark matter is in kinetic equilibrium with the baryonic plasma and CMB +till zdec > 105 (see subsection 4.1 for details). As long as dark matter is kinematically +coupled to the CMB, its temperature is equal to the CMB temperature ∝ (1+z). Since the +dark matter is non-relativistic at decoupling, it cools faster than the CMB with temperature +evolving with redshift ∝ (1 + z)2. +Tχ(z) = +� +� +� +Tγ(z) +z ≥ zdec, +Tγ(zdec) +� +1+z +1+zdec +�2 +z < zdec. +(3.8) +• First consider the redshift at which dark matter starts absorbing CMB photons. Let z0 be +the redshift at which dark matter absorbs a photon of frequency ν0 as before. The optical +depth per unit redshift is given by, +dτχ +dz = − +g1/g0 +1 + (g1/g0) e− +T∗ +Tex(z0) +� +1 − e− +T∗ +Tex(z0) +� αAAHI +10 c4 nχ (z0) +8πν3 +0H (z) +�1 + z0 +1 + z +� +δ (z − z0) . +(3.9) +The optical depth τχ in eq.(3.9) is obtained by integrating the line profile along the LoS +in an expanding Universe using Sobolev approximation [31–34, 47, 48, 85]. The Sobolev +approximation is valid as long as the Doppler line width is negligible compared to the width +of the global absorption feature. +• In case of a single source we analyzed two extreme limits: collisionless DM and highly colli- +sional DM. The effect of inelastic collisions is however essential to have a global absorption +– 21 – + +signal. The physics of dark matter inelastic collisions is described in detail in eq. (2.13) of +section 2. The exact functional form of the collisional coefficients depends on the details +of the dark matter model. Even for a simple system of a hydrogen atom, collision cross- +sections have a complicated temperature dependence [86]. For simplicity, we will assume +the dark matter collision cross-sections qualitatively similar to the inelastic cross-sections +of hyper-fine transitions in hydrogen. We parameterize C10 as a power law in Tχ: +C10 = nχ⟨σv⟩, +where ⟨σv⟩ = +� +� +� +� +� +a1 +� +Tχ(z) +Tχ(zrec) +�β +z < zsat, +a1 +� +Tχ(zsat) +Tχ(zrec) +�β +z ≥ zsat, +(3.10) +where a1 is the value of ⟨σv⟩ at zrec = 1100 (the redshift of hydrogen recombination), zsat +is the saturation redshift parameter, and β is the power law index which is taken to be +positive. +• In the presence of both radiative transitions as well as inelastic collisions, the change in +the population of dark matter particles in the ground state (n0) can be calculated from +eq.(2.9), eq.(2.10), and eq.(2.13), +dn0 +dz − 3n0 +1 + z = − +1 +H(z)(1 + z) +� +n1C10 − n0C01 + n1A10 + (n1B10 − n0B01) ¯J +� +, +(3.11) +where ¯J is the CMB intensity at ν0. +• From eq. (3.11), we find that the level population of dark matter particles which is pa- +rameterized in terms of the excitation temperature Tex (see eq. (2.7)) is determined by +the competition between the collisional rate and the radiative transition rate. By differ- +entiating eq.(2.7) with respect to redshift and substituting eq.(3.11) into it, the evolution +equation for excitation temperature Tex becomes, +dTex +dz += T 2 +ex +T∗ +� 1 +n1 +dn1 +dz − 1 +n0 +dn0 +dz +� +. +(3.12) +We note that eq.(3.12) is quite general and applies to any two level system, not just the +spin flip transitions. In particular, we have not made any assumptions about the smallness +of T∗ with respect to other temperatures in the problem. +• It is customary to express the specific intensity at frequency ν i.e. +Iν in terms of the +brightness temperature as, +Tb = +c2 +2ν2kB +Iν. +(3.13) +• Prior to recombination, the collisions between the free electrons and ions create and destroy +photons by the bremsstrahlung process. Bremsstrahlung plays a vital role in preserving +the blackbody spectrum of CMB by erasing any distortion that may have originated in the +past. Even when it becomes unimportant in maintaining the CMB blackbody spectrum +over most of the frequency range at z ≤ 106, it is still important in the low energy Rayleigh- +Jeans tail of the CMB spectrum. The bremsstrahlung process tries to bring the brightness +– 22 – + +temperature in equilibrium with the gas/baryon temperature Tg, which is equal to the +CMB temperature (Tb → Tg = Tγ) until z ≈ 500 [87, 88]. Consequently, the quantity +x ≡ +hν (z) +kBTg (z) = +hν0 +kBTγ (z0) +(3.14) +remains invariant till z ≈ 500. The optical depth due to bremsstrahlung (τbr) per unit +redshift is given by [89, 90], +dτbr +dz (x) = − +cσT αnenB +(24π3)1/2 H (1 + z) +gbr(x, z) +�kBTg +mec2 +�−7/2 � h +mec +�3 �1 − e−x +x3 +� +, +(3.15) +where α is the fine structure constant, σT is the Thomson scattering cross-section, me is +the mass of the electron, nB(z) and ne(z) are the number densities of baryons and electrons +respectively, gbr(x) ≡ Z2 +i ni⟨gff(x)⟩/nB, where Zi is the charge of the ith ion having number +density ni, and ⟨gff(x)⟩ is the thermally averaged Gaunt factor which has been taken from +[91]. +• At high redshifts, the number density of dark matter particles is high which results in +stronger collisional transitions between the two dark matter states, compared to radiative +transitions due to CMB photons. Thus, initially Tex is in kinetic equilibrium with the dark +matter temperature which is much lower than the CMB temperature (Tex → Tχ ≪ Tγ) at +z < zdec. The dark matter particles absorb the CMB photons and a net flow of energy takes +place from CMB to dark matter resulting in an absorption feature in the CMB spectrum. +The absorption of CMB photons by dark matter at redshift z0 generates an absorption +line at x (defined in eq.(3.14)). If this line lies in the low frequency Rayleigh Jeans tail +(x ≪ 1) of the CMB spectrum, it gets erased by the bremsstrahlung emission at subsequent +times (z < z0). As the number density and the temperature of dark matter falls due to +expansion of the Universe, the collision rate falls and the radiative transitions involving the +CMB photons begin to dominate over the collisions. This brings Tex in kinetic equilibrium +with the CMB temperature (Tex → Tγ). When this happens there is no net emission or +absorption of the CMB photons by dark matter and the absorption signal vanishes. Thus +we expect to see a broad absorption feature in the CMB spectrum, starting from the time +dark matter decouples until a later time when the radiative transitions take over. +• The evolution of brightness temperature at ν = ν0/(1 + z0) incorporating the effect of +absorption by dark matter at z = z0 (from eq.(3.9)) and bremsstrahlung (from eq.(3.15)) +is given by, +dTb(ν) +dz +− Tb(ν) +1 + z = dτχ +dz +� +−Tb(ν) + hν +kB +1 +(ehν/kBTex(z) − 1) +� ++dτbr(x) +dz +(−Tb(ν) + Tg) . +(3.16) +The differential brightness temperature δTb observed at frequency ν = ν0/(1+z0) is defined +as the brightness temperature (obtained by solving eq. (3.16)) minus the CMB temperature +today i.e. in the observer’s frame, +δTb (ν, z = 0)observer’s frame ≡ Tb (ν, z = 0) − Tγ (z = 0) . +(3.17) +– 23 – + +6000 +3000 +2000 +1500 +1200 +1000 +Redshift +50 +100 +150 +Frequency (MHz) +−0.8 +−0.6 +−0.4 +−0.2 +0.0 +Brightness temperature (K) +99 % CI +EDGES best fit +Case 1 +Case 2 +Figure 11: Dark absorption features for two cases denoted by red and blue lines respectively. +The choice of model parameters is given in table 2. The grey band shows the 99% CI for the +EDGES data. C10 ∝ T 4 +χ in case 1 results in a narrower right edge compared to C10 ∝ T 2 +χ in +case 2. +Parameter +Case 1 +Case 2 +Mass (mχ (MeV)) +1 +1 +Binding energy (Ebinding (MeV)) +0.06 +0.06 +Transition frequency (ν0 (GHz)) +156.2 +156.2 +Transition temperature (T∗ (K)) +7.5 +7.5 +Radiative coupling (αA) +0.35 +0.35 +Collisional coupling +� +a1 +� +cm3s−1� +, β, zsat +� +2.18 × 10−22, 2, 2700 +3.49 × 10−23, 4, 2000 +Table 2: Two sets of parameters that can give the amplitude and width required by the +EDGES experiment. +3.2.1 +The EDGES anomaly +The EDGES collaboration [35] reported a strong absorption feature in the CMB spectrum. +This feature is almost twice in amplitude compared to the maximum possible signal expected +from the 21 cm transitions in neutral hydrogen during the cosmic dawn. In this work, we +propose that this anomalous signal is caused by the absorption of CMB photons by composite +dark matter. In particular, we show below that we can get an absorption feature which has +the same central frequency, amplitude, and width as the EDGES signal by a suitable plausible +choice of dark matter model parameters listed in table 2. +Although there are many free +model parameters that decide the final shape of the signal, we would like to emphasize a few +important points: +• The left (low frequency) edge of the signal is entirely decided by the bremsstrahlung pro- +cess which erases the absorption by dark matter at high redshifts. This also fixes the +– 24 – + +location of the maximum absorption which happens around the redshift of recombina- +tion. In particular, the rapid decrease in bremsstrahlung efficiency around recombination +provides a sharp edge to the signal at the low frequency end. +• The right (high frequency) edge of the signal is decided by the strength of inelastic collision +cross-section relative to the radiative coupling of dark matter. We find that it is relatively +easy to get a narrow absorption feature, at least close to the peak, which has the same +amplitude as EDGES as shown in figure 11. The shape of the high frequency right edge is +a strong function of the temperature dependence of the collision cross-section which can +be tuned by having the collision cross-section to depend weakly or strongly on the dark +matter temperature. +3.2.2 +General predictions for the shape of the dark absorption feature +To understand the role of different model parameters in determining the shape of the global +absorption signal, we vary each model parameter one by one keeping all the other parameters +fixed. We keep the fiducial parameters as given in column 1 of table 2, however the qualitative +behavior and results are valid for any ν0 in the Rayleigh Jeans part of the CMB spectrum at +recombination. The variation in the absorption feature of the CMB for different dark matter +model parameters is shown in figure 12. +• Mass: Given the abundance of dark matter, smaller mass of dark matter implies a higher +dark matter number density which increases the strength of the absorption signal. +• Binding energy: Dark matter with a higher binding energy decouples earlier (see eq.(4.1) +of section 4), resulting in a lower dark matter temperature Tχ in eq.(3.8). Since the dark +matter inelastic collisional coupling ∝ T β +χ , where β > 0 in eq.(3.10), smaller dark matter +temperature implies weaker collisional coupling. A weaker collisional coupling compared +to radiative coupling drives Tex → Tγ earlier, resulting in smaller amplitude of the signal +for higher binding energies. +• Transition frequency: When the transition frequency is varied, there is an overall shift +in the position in frequency of the absorption signal . Moreover the bremsstrahlung rate +eq.(3.15) is a sensitive function of the transition frequency. As we go to lower frequencies, +the bremsstrahlung is more efficient in erasing the absorption signal resulting in a smaller +amplitude. +• Inelastic collision cross-section: The high frequency or right edge of the signal is +decided by the relative strength of collisional versus radiative coupling. If we change the +power law index (β) of the collision cross-section while keeping the amplitude at zrec = 1100 +fixed, a higher β would mean stronger collisional coupling at z > zrec. Thus we see stronger +absorption as we increase β. +• Radiative coupling: A higher spontaneous emission rate compared to collisional transi- +tion rate couples Tex → Tγ earlier, shifting the absorption signal to higher redshifts. The +bremsstrahlung process is more efficient in erasing the signal at higher redshifts. Hence we +see a smaller amplitude of the absorption signal at higher values of αA (see eq.(2.12) for +the definition of αA). +– 25 – + +5000 +1200 +700 +500 +400 +330 +Redshift +100 +200 +300 +400 +500 +Frequency (MHz) +−2 +−1 +0 +Brightness temperature (K) +mχ(MeV) +0.01 +1 +10 +(a) Mass of dark matter +5000 +2000 +1200 +900 +700 +Redshift +50 +100 +150 +200 +250 +Frequency (MHz) +−1.5 +−1.0 +−0.5 +0.0 +Brightness temperature (K) +Ebinding(MeV) +0.01 +0.06 +0.31 +(b) Binding energy +5000 +2000 +1200 +900 +Redshift +50 +100 +150 +200 +Frequency (MHz) +−0.6 +−0.4 +−0.2 +0.0 +Brightness temperature (K) +ν0(GHz) +135.4 +156.2 +177.0 +(c) Transition frequency +5000 +2000 +1200 +900 +Redshift +50 +100 +150 +200 +Frequency (MHz) +−1.0 +−0.5 +0.0 +Brightness temperature (K) +β +1 +2 +3 +(d) Inelastic collision cross-section +5000 2000 +1200 900 +700 600 +Redshift +100 +200 +300 +Frequency (MHz) +−1.0 +−0.5 +0.0 +Brightness temperature (K) +αA +0.035 +0.35 +3.5 +(e) Radiative coupling +Figure 12: Role of different model parameters in deciding the shape of the global absorption +signal from dark matter. +We vary each parameter at a given time keeping all the other +parameters fixed at the case 1 choice given in table 2 (solid blue line in figures 11 and 12 are +identical). +– 26 – + +3.2.3 +General predictions for the dark absorption feature in different parts of +the CMB spectrum +Irrespective of EDGES, composite dark matter predicts an absorption feature in the CMB +spectrum. Our choice of transition frequency was motivated by the EDGES observation. +In general for a different transition frequency ν0 and absorption redshift z0, the absorption +feature will be appear in a different part of the CMB spectrum. The upcoming experiment +called the Array of Precision Spectrometers for the Epoch of RecombinAtion (APSERA) [92], +which aims to detect the recombination lines in the CMB spectrum will also be sensitive to +the dark absorption feature in the 2-6 GHz frequency range. +Any dark absorption feature originating at z0 ≳ 2×106 in the CMB cannot be observed. +This is because Compton scattering [93] along with the photon number changing processes +like bremsstrahlung and double Compton scattering [94–97] are efficient in erasing any de- +viations from the black body spectrum till z ∼ 2 × 106. As the bremsstrahlung and double +Compton scattering rates fall ∝ ν−2 with frequency, they decouple at z ∼ 2×106 for photons +having x ∼ 0.01 [94, 98–103]. With only Compton scattering efficient in z ∼ 2 × 106 − 105 +range, the equilibrium spectrum is the Bose-Einstein spectrum and the resulting deviations +from blackbody are created in the form of µ-type spectral distortions. If the absorption hap- +pens at z ≲ 105, we will have a broad absorption feature in the CMB spectrum. The COsmic +Background Explorer/Far-InfraRed Absolute Spectrophotometer (COBE/FIRAS) [104] ex- +periment strongly limits the CMB spectral distortions in 60−600 GHz band (see appendix F). +Thus any absorption happening at z0 < 2 × 106 corresponding to 60 < ν0(GHz) < 1.2 × 109 +range will be strongly constrained by COBE. +In addition, CMB photons having x ∼ 50 correspond to different energy states of hydro- +gen and helium in 10.2 − 50 eV range at recombination (z ∼ 1100). If dark matter absorbs +these photons, there would be fewer CMB photons that can excite and ionize hydrogen and +helium speeding up recombination. An early recombination would modify the position and +amplitude of angular peaks and the Silk-damping tail of the CMB anisotropy power spec- +trum which is strongly constrained by Planck [105]. We leave the detailed analysis of the +constraining power of different CMB observations on our dark matter model to future work. +4 +Observational constraints +A crucial step towards establishing the viability of our dark matter model involves utilizing +the current astrophysical and cosmological data to constrain the different model parameters. +We divide this section into two parts, first we look at the early Universe constraints coming +from CMB observations and then we move towards the late Universe constraints from the +Milky Way galaxy. Note that we do a conservative analysis to find the absolutely allowed +parameter space. The goal here is not to derive the best constraints on our model but rather +to show that a significant and interesting part of the parameter space is allowed and has +unique signatures in experiments as explained before. +4.1 +Early Universe constraints +At early times, the Universe is dominated by the cosmic microwave background radiation +and the precise measurements of the CMB allow us to strongly constrain any possible electro- +magnetic interaction dark matter had in the past. For instance, when the temperature of the +DM-SM plasma is higher than the binding energy of the composite dark matter particles, dark +matter exists in the form of electromagnetically charged free dark quarks which thermally +– 27 – + +100 +102 +104 +106 +Mass of dark matter (keV) +10−7 +10−5 +10−3 +10−1 +101 +103 +Binding energy (keV) +CMB +Milky Way +100 +102 +104 +106 +Mass of dark matter (keV) +10−25 +10−23 +10−21 +10−19 +10−17 +10−15 +⟨σv⟩ (cm2) +Bullet cluster +Milky Way +100 +102 +104 +106 +Mass of dark matter (keV) +10−5 +10−4 +10−3 +10−2 +10−1 +100 +101 +102 +αA +DM cooling +DM emission +Figure 13: We show the absolutely allowed parameter space for the different astrophysical +parameters (left: binding energy, center: collision cross-section, and right: radiative coupling) +for our composite dark matter model. The colored region in the plots is ruled out. We choose +the transition frequency ν0 = 156.2 GHz which is motivated to solve the EDGES anomaly. +couple the dark matter plasma to the baryon-photon fluid via Coulomb scattering and Comp- +ton scattering. Such electromagnetic couplings of dark matter to baryon-photon plasma are +strongly constrained by the CMB observations. The Planck experiment [105] is sensitive to +angular scales up to l ∼ 3000 or co-moving wave numbers less than k ≈ l/rLSS ≈ 0.22 Mpc−1 +which enter the horizon at redshift z∗ ≃ 1.3 × 105. These considerations demand that the +rate of Coulomb and Compton scattering must be less than the Hubble expansion rate at +redshift z∗. Using these conditions, we find that the electromagnetic charge ϵ of free dark +quarks must obey an upper bound, ϵ ≲ 10−7. +We can avoid these constraints on ϵ if dark matter recombines into a stable and neutral +composite state by z ≥ z∗. As the Universe expands and the CMB temperature falls, the +peak of the CMB blackbody spectrum shifts towards lower energies. There are fewer high +energy photons left in the Wein tail of the CMB spectrum that can ionize the composite dark +matter. Assuming the recombination history of hydrogen and dark matter to be similar, +Binding Energy +kBTγ(z = 1100) +���� +hydrogen +≈ 50 ≈ Binding Energy +kBTγ(z = z∗) +���� +dark matter +. +(4.1) +This imposes a lower bound on the binding energy of dark matter, Ebinding ≥ 1.6 keV, as +shown in the first panel of figure 13. In this work, we assume that dark matter recombines +before z∗ and kinematically decouples from the thermal plasma thereafter. We also note that +the photo-ionization cross-section of dark matter ∝ ϵ2. This together with the high binding +energy implies that dark matter will not be reionized in most astrophysical environments. +4.2 +Late Universe constraints from Milky Way +Consistency with the precise observations of Milky Way (MW) gives additional constraints +on the properties of dark matter at late times, in particular the dark matter self-interactions +and the strength of electromagnetic transition. We derive these constraints for two extreme +scenarios, one where the dark matter self-interaction cross-section is very weak such that it +remains collisionless in the MW halo. In the other case, we consider the inelastic collisions +of dark matter to be strong enough such that the states corresponding to electromagnetic +transitions are in kinetic equilibrium with the virial temperature of the MW halo. +– 28 – + +(i) Collisionless dark matter: If the timescale of dark matter collisions denoted by tcollision = +(nχ⟨σv⟩)−1 in our local neighborhood (ρχ ∼ 0.4 GeV cm−3) is longer than the age of the +Milky Way (tMW ≈ 13 billion years [106]), the dark matter stays collisionless. This puts +an upper limit on the dark matter collision cross-section (elastic + inelastic), +⟨σv⟩ < +1 +tMW +�mχ +ρχ +� +. +(4.2) +We show this constraint in the second panel of figure 13. We also show constraints from +Bullet cluster [56] (σ/mχ < 2 cm2g−1) assuming the relative velocity between the two +clusters v ≈ 4700 km/s. This indirectly puts constraints on dark matter inelastic collision +cross-section. +We note that our back of the envelope constraints from MW are very conservative. A +more careful analysis taking into account the non-negligible dark matter self-interactions +which maybe preferred by data, would relax these constraints [107, 108]. +(ii) Collisional dark matter: The observations in fact allow dark matter self-interactions +as long as they do not significantly disturb the dark matter halo profile, e.g. by collapsing +the halo into a disk [109]. We assume here the other extreme case, where dark matter +self-interactions are strong enough that the excitation temperature of the two states re- +sponsible for transitions are in kinetic equilibrium with the virial temperature of the MW +halo i.e. Tex = Tvir. +• The dark matter particles in the MW halo are in virial equilibrium at a temperature +Tvir ∝ mχ (calculated using the virial theorem) and have an energy ∼ kBTvir. For the +dark matter to not ionise into charged dark quarks, the energy exchange in each inelastic +collision (∼ kBTvir) must be less than the binding energy of dark matter particles. This +imposes a lower bound on the binding energy as shown in the first panel of figure 13. +We note that this bound is weaker compared to the CMB bound derived in eq.(4.1). +Therefore we consider the CMB bound in our calculations. +• In a collisional excitation, the kinetic energy of dark matter particles is converted into +its internal energy. The excited dark matter particles can then de-excite by sponta- +neously emitting a photon, converting the internal energy of dark matter into radiation. +This would lead to a gravitationally unstable dark matter halo, which cools and starts +collapsing into a disk similar to baryons. From GAIA observations [110] such a scenario +is ruled out. The cooling timescale tcooling for this process is given by the ratio of the +thermal energy density U in dark matter to the radiative cooling rate C, +tcooling ≈ U +C +≈ +3 +2nχkBTvir +C +, where +C = n1A10kBT∗. +(4.3) +The MW dark matter halo remains gravitationally stable if the cooling timescale is +longer than the age of MW [106]. This puts an upper limit on the radiative coupling of +dark matter (see eq.(2.12) for the definition of αA), +αA < 3 +2 +Tvir +T∗ +1 +tMWAHI +10 +� +1 + g0 +g1 +e +T∗ +Tvir +� +. +(4.4) +In this analysis, we have assumed that dark matter only cools via the dark hyper-fine +transitions. In principle, if the levels with higher energies can be collisionally excited +– 29 – + +and have a higher spontaneous emission rate, they will contribute to the cooling process. +As discussed in section 2, in this model we assume that the other transitions correspond +to very high energies which cannot be excited collisionally. We plot this limit on αA +in the third panel of figure 13 for transition frequency 156.2 GHz or T∗ = 7.5 K. This +choice of parameters was motivated by the EDGES observation. +• The collisionally excited dark mater particles in the MW halo can de-excite by spon- +taneously emitting a photon, creating a background radiation with a dipole anisotropy +owing to our off-center position 8.3 kpc [111] away from the MW halo center. The +experiments like WMAP and Planck [105, 112] measure the fluctuations in the sky +temperature in broad frequency bands in the frequency range of 10-500 GHz. +Considering j to be the integrated specific intensity and javg be the average integrated +specific intensity due to dark matter emission (see appendix G for details), the temper- +ature fluctuations along a given LoS in a particular frequency band (νmin to νmax with +a central frequency νc) is given by, +δTνc = +� |j − javg| +νmax − νmin +� +c2 +2kBν2c +. +(4.5) +We note that this dipole will be aligned with the galactic plane and thus obscured by +Galactic emission. Also this dipole will appear in only one channel that includes ν0 +and would be absent in other channels. Without doing a detailed analysis, we can just +require very conservatively that this dipole must be smaller than the cosmological dipole +δTνc < 3 mK, imposing an upper bound on αA. The constraints on αA from eq.(4.4) +and eq.(4.5) are shown in the third panel of figure 13. In our analysis, we have chosen +ν0 = 156.2 GHz which falls in the frequency band of Planck centered at νc = 143 GHz +with a bandwidth ∆ν/ν = 0.33. We note that CMB is the most precisely measured part +of the electromagnetic spectrum over the whole sky and constraints from other parts of +the spectrum (for other values of ν0) would be much weaker. +We note that these constraints only apply if the collision cross-sections for these transitions +are large enough such that the levels are in kinetic equilibrium at MW virial temperature. +These constraints would get relaxed significantly for weaker collisions. We also note that +the collision cross-sections usually have a strong temperature dependence which suggests +that dark matter collisions may be insignificant in MW but maybe important in other +dark matter halos. +5 +Conclusions +In this work, we explore the unique experimental signature of a class of composite dark +matter models having electromagnetic transitions: the absorption lines in the spectrum of a +background source. Such absorption signatures occur as distinct dark lines in the spectrum +of a quasar, and as a global absorption feature or spectral distortion in the spectrum of the +CMB. +• The absorption of light by multiple dark matter halos along the LoS to a quasar gives rise +to a dark forest, similar to the Lyman-α forest from neutral hydrogen. In contrast to the +Lyman-α or 21 cm forest, the dark forest is less prone to the uncertainties related to the +non-linear baryonic physics. +– 30 – + +• The dark forest is sensitive to the self-interactions of dark matter. The dark forest is also +sensitive to the minimum halo mass and therefore the primordial power spectrum on small +scales. +• Absorption of CMB photons by dark matter gives rise to µ-type spectral distortions at +z ≳ 105 and a global absorption feature in the CMB spectrum at z ≲ 105. +• The bremsstrahlung process plays an important role if the dark matter absorption fre- +quency falls in the Rayleigh-Jeans tail of the CMB spectrum. In this case, bremsstrahlung +determines the high redshift edge of the global absorption feature. The low redshift shape +of the global signal is determined by the relative strength of inelastic collisional coupling +with respect to the radiative coupling of dark matter. +• Our dark matter model (with a suitable choice of parameters) can reproduce the absorp- +tion feature that matches the frequency location, width, and amplitude but not the exact +flattened shape of the anomalous signal measured by the EDGES collaboration [35]. +• If the EDGES signal is indeed produced by dark matter absorption, then our model predicts +that a dark forest must exist in a frequency band ten times higher than the 21 cm forest +frequency band. +• A global absorption feature or spectral distortion due to dark matter is a general prediction +of our model which can even occur in a different frequency band of the CMB spectrum. +Such signals could be detected in future experiments such as Primordial Inflation Explorer +(PIXIE) [113] or APSERA [92]. +• The absorption of CMB photons in the UV band can affect the recombination history and +this part of the parameter space can be probed by the CMB anisotropy measurements. +We have ignored the clustering of halos in this work. One needs to perform N-body simu- +lations to take into account the two point correlations of the dark forest which we leave for +future work. +Our results open up a new and unexpected interesting window into the nature of dark +matter and a potential new probe of the cosmic web. Our work motivates the search for dark +forest and global absorption features across the full electromagnetic spectrum, especially in +the desert i.e. +the part of the electromagnetic spectrum where there are no strong lines +expected from baryons in the standard model of cosmology. With the current sensitivity of +spectroscopic surveys in different frequency bands, it should already be possible to probe the +dark forest at optical and radio-wave frequencies. +Acknowledgments +We acknowledge the computational facilities offered by the Department of Theoretical Physics +at TIFR, Mumbai. This work is supported by Max Planck Partner Group for cosmology of +Max Planck Institute for Astrophysics Garching at TIFR funded by Max-Planck-Gesellschaft. +We acknowledge the support from Department of Atomic Energy (DAE) Government of +India, under Project Identification No. RTI 4002. AG is thankful to Yidong Xu, Steven +Furlanetto, Girish Kulkarni, Shadab Alam, Nissim Kanekar, Aritra Kumar Gon and Shikhar +Mittal for useful discussions. +– 31 – + +A +Dark matter model +By construction, the dark sector model is described by an asymptotically free non-abelian +SU(N) gauge theory. At UV energies, the theory is described by weakly coupled dark quarks. +The necessary d.o.f. are given in table 1 in section 2. At low energies, SU(N) is strong and +spectra of the dark sector is given in terms dark color neutral bound states. The formulation +we discuss here closely follows the chiral Lagrangian techniques developed in particle physics +[58–60]. One essential difference though is that there is no analogous weak force present in the +dark sector which allows the heavy-light bound dark matter states to decay in comparison to +the heavy mesons of the visible sector. We begin this section with the spontaneous breaking +of the chiral symmetry of the dark sector. At low energies, the strong dark QCD generates +the flavor diagonal condensate ⟨qDa qc +Db⟩ = δab Λ3 +N. The flavor symmetry gets spontaneously +broken in the following fashion, +SU(2)D +L × SU(2)D +R → SU(2)D +V . +(A.1) +This symmetry breaking results in three pNGBs/dark pions πD parameterized in eq.(2.4) +in section 2. In order to discuss the physics of heavy-light bound states, it is convenient to +introduce a field ξD defined via, +ΣD(x) = ξ2 +D(x); with ξD(x) +SU(2)D +L ×SU(2)D +R +−−−−−−−−−−−→ LD ξD(x) V † +D(x) = VD(x) ξD(x) R† +D, +(A.2) +where VD(x) is the transformation operator for the unbroken vectorial symmetry SU(2)D +V . +We choose our dark matter candidate to be a heavy-light bound state. To do this, we +first redefine the heavy and light dark quark Dirac spinors in the following way, +QD ≡ +�QDw +Qc † +Dw +� +, +qD ≡ +� +ξ† +DqDw +ξD qc † +Dw +� +, +(A.3) +where QDw and qDw are the Weyl spinors for the heavy and light dark quarks respectively. +The dark matter candidate is defined by the matrix field X as, +Xa ≡ ¯QD qDa +SU(2)D +L ×SU(2)D +R +−−−−−−−−−−−→ VDab(x)Xb. +(A.4) +In order to construct the operators responsible for interactions between dark bound state and +dark pions, we define the vector current VDµ and the axial current ADµ which incorporate +the dark pions πD, +VDµ = 1 +2(ξ† +D∂µξD + ξD∂µξ† +D), +ADµ = 1 +2(ξ† +D∂µξD − ξD∂µξ† +D). +(A.5) +Since the dark state X is a non-relativistic heavy-light bound state, its velocity v is conserved. +Using the framework of HQET, we can project the small scale fluctuations [61, 62] and write +the Lagrangian for Xv invariant under the full flavor symmetry SU(2)D +L × SU(2)D +R, +LXv = i tr +� ¯ +X a +v vµDab +µ X b +v +� ++ ig1 tr +� ¯ +X a +v γµγ5 Aab +µ X b +v +� ++ 1 +4f2 +D ∂µΣab∂µΣ† +ba ++g2 f2 +DΛNmqδab Σab + g3 mqδab tr +� ¯ +X a +v X b +v +� +ξD ξD + h.c. + ... , +(A.6) +– 32 – + +where the covariant derivative is defined as Dµ = ∂µ + VDµ and the trace is taken over the +gamma matrices. +The Lagrangian in eq.(A.6) does not distinguish the spin 1 state χ∗ from the spin 0 +state χ. The terms that violate the spin symmetry occur as powers in 1/MQ and contribute +to the mass gap between χ and χ∗ resulting in the hyper-fine splitting of the dark sector. +The tree level operators that generate this mass gap are given by, +λ +MQ +tr +� ¯ +X a +v σµν X a +v σµν +� +; λ′ mq +MQ +tr +� ¯ +X a +v σµν X a +v σµν +� +ξDξD ... etc. +(A.7) +We can decompose the matrix field Xv into a pseudoscalar field and a vector field using +eq.(2.5) and compute the tree level hyper-fine splitting from eq.(A.7). Further, the axial +current operator in eq.(A.6) gives rise to χ∗χ∗πD and χχ∗πD couplings which also contribute +to the hyper-fine splitting at the loop level. +B +Dark matter halo density and temperature profile +The halo mass density is modeled as an NFW profile [16] with a concentration model taken +from [114, 115] to describe the halo up to 10 times the virial radius [116] of the halo. The +halo mass Mh is defined as the total matter enclosed within a sphere around the halo center +that encloses a fixed density equal to 200 times the critical density. Thus r200c is defined as, +M200c(r < r200c) +4πr3 +200c/3 += 200ρc(z) +(B.1) +where halo mass Mh = M200c = +� r200c +0 +d3r ρ(r). +For simplicity, we assume the dark matter to follow a Maxwell-Boltzmann velocity +distribution with a radially dependent rms width σ described by the power law profile in +[117], +ρ +σ3 (r) = 101.46 +�ρc(z) +v3 +vir +�� r +rvir +�−1.9 +, +(B.2) +where r is the radius of the halo in physical units, ρc is the critical density of the Universe, +ρ(r) is the NFW density profile of the halo. vvir and rvir are the virial velocity and virial +radius of the dark matter halo respectively. One can associate a temperature to the width +of the velocity distribution using the relation, σ2(r) = 3kBTh(r)/mχ, which gives, +Th(r) = mχ +3k +�101.46 +ρ(r) +�ρc(z) +v3 +vir +�� r +rvir +�−1.9�−2/3 +. +(B.3) +The random motion of dark matter particles along the LoS gives rise to a Doppler line profile +whose line width in units of speed is given by, +b(r) = c∆νD +ν0 += (1 + vLoS(r, p)/c) +� +2kBTh(r) +mχ +. +(B.4) +– 33 – + +C +Dark forest +We show the sample dark forest spectra in the redshift range 7-0 for collisionless and collisional +DM for three different choices of Mmin = 106, 105 and 104M⊙/h respectively keeping other +dark matter model parameters fixed at: mχ = 1 MeV, T∗ = 7.5 K, and αA = 0.35. +6.00 +6.25 +6.50 +6.75 +7.00 +Redshift +21 +22 +23 +Frequency (GHz) +0.9 +1.0 +Rel. transmission (e−τ) +5.00 +5.25 +5.50 +5.75 +6.00 +Redshift +24 +25 +26 +27 +Frequency (GHz) +4.00 +4.25 +4.50 +4.75 +5.00 +Redshift +28 +30 +32 +Frequency (GHz) +3.00 +3.25 +3.50 +3.75 +4.00 +Redshift +35.0 +37.5 +40.0 +Frequency (GHz) +0.9 +1.0 +Rel. transmission (e−τ) +2.00 +2.25 +2.50 +2.75 +3.00 +Redshift +45 +50 +55 +Frequency (GHz) +1.00 +1.25 +1.50 +1.75 +2.00 +Redshift +60 +70 +80 +Frequency (GHz) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Redshift +100 +120 +140 +160 +Frequency (GHz) +0.6 +0.8 +1.0 +Relative transmission +Figure 14: Dark forest spectrum for collisionless DM in for Mmin = 106M⊙/h. +– 34 – + +6.00 +6.25 +6.50 +6.75 +7.00 +Redshift +21 +22 +23 +Frequency (GHz) +0.6 +0.8 +1.0 +Rel. transmission (e−τ) +5.00 +5.25 +5.50 +5.75 +6.00 +Redshift +24 +25 +26 +27 +Frequency (GHz) +4.00 +4.25 +4.50 +4.75 +5.00 +Redshift +28 +30 +32 +Frequency (GHz) +3.00 +3.25 +3.50 +3.75 +4.00 +Redshift +35.0 +37.5 +40.0 +Frequency (GHz) +0.6 +0.8 +1.0 +Rel. transmission (e−τ) +2.00 +2.25 +2.50 +2.75 +3.00 +Redshift +45 +50 +55 +Frequency (GHz) +1.00 +1.25 +1.50 +1.75 +2.00 +Redshift +60 +70 +80 +Frequency (GHz) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Redshift +100 +120 +140 +160 +Frequency (GHz) +0.6 +0.8 +1.0 +Relative transmission +Figure 15: Dark forest spectrum for collisional DM for Mmin = 106M⊙/h. +– 35 – + +6.00 +6.25 +6.50 +6.75 +7.00 +Redshift +21 +22 +23 +Frequency (GHz) +0.5 +1.0 +Rel. transmission (e−τ) +5.00 +5.25 +5.50 +5.75 +6.00 +Redshift +24 +25 +26 +27 +Frequency (GHz) +4.00 +4.25 +4.50 +4.75 +5.00 +Redshift +28 +30 +32 +Frequency (GHz) +3.00 +3.25 +3.50 +3.75 +4.00 +Redshift +35.0 +37.5 +40.0 +Frequency (GHz) +0.8 +1.0 +Rel. transmission (e−τ) +2.00 +2.25 +2.50 +2.75 +3.00 +Redshift +45 +50 +55 +Frequency (GHz) +1.00 +1.25 +1.50 +1.75 +2.00 +Redshift +60 +70 +80 +Frequency (GHz) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Redshift +100 +120 +140 +160 +Frequency (GHz) +0.6 +0.8 +1.0 +Relative transmission (e−τ) +Figure 16: Dark forest spectrum for collisionless DM for Mmin = 104M⊙/h. +– 36 – + +6.00 +6.25 +6.50 +6.75 +7.00 +Redshift +21 +22 +23 +Frequency (GHz) +0.0 +0.5 +1.0 +Rel. transmission (e−τ) +5.00 +5.25 +5.50 +5.75 +6.00 +Redshift +24 +25 +26 +27 +Frequency (GHz) +4.00 +4.25 +4.50 +4.75 +5.00 +Redshift +28 +30 +32 +Frequency (GHz) +3.00 +3.25 +3.50 +3.75 +4.00 +Redshift +35.0 +37.5 +40.0 +Frequency (GHz) +0.5 +1.0 +Rel. transmission (e−τ) +2.00 +2.25 +2.50 +2.75 +3.00 +Redshift +45 +50 +55 +Frequency (GHz) +1.00 +1.25 +1.50 +1.75 +2.00 +Redshift +60 +70 +80 +Frequency (GHz) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Redshift +100 +120 +140 +160 +Frequency (GHz) +0.4 +0.6 +0.8 +1.0 +Relative transmission (e−τ) +Figure 17: Dark forest spectrum for collisional DM for Mmin = 104M⊙/h. +D +Convergence of distribution functions +We compare the τpeak and b2 distribution functions for collisionless DM in the redshift range +6-5 for 100 and 1000 LoS directions in figure 18. The mean count N in a given τpeak or b2 bin +follows a Poisson distribution with a standard deviation ∝ +√ +N. When the average is taken +over spectra from 1000 different LoS directions, we find that the tail becomes smoother and +the error bars on the distribution functions contract as expected. +E +Radiative transfer equation in an expanding Universe +Let uν(Ω) be the radiation energy density per unit frequency per unit solid angle. In terms +of the photon distribution function f(ν), +uν(Ω)dν dΩ = hνf(ν)(h/c)3ν2 dν dΩ. +(E.1) +In an expanding Universe, frequency redshifts as 1/a, where a as the scale factor, while f(ν) +remains conserved. From eq.(E.1), in the absence of a source or sink, uν(Ω)/ν3 or uν(Ω)a3 +– 37 – + +10−2 +10−1 +100 +τpeak +10−3 +10−2 +10−1 +100 +101 +102 +dN/dτpeak +Mmin = 106M⊙/h +z : 6 - 5 +nseed = 100 +nseed = 1000 +100 +101 +102 +103 +b2 (km2/s2) +10−3 +10−2 +10−1 +100 +101 +102 +dN/d(b2[km2/s2]) +Mmin = 106M⊙/h +z : 6 - 5 +nseed = 100 +nseed = 1000 +Figure 18: The distribution functions for peak optical depth (left) and line width (right) +for collisionless DM for Mmin = 106M⊙/h in the redshift range 6-5 for 100 and 1000 LoS +directions. +remains constant with respect to time. +d(uνa3) +dt += 0, +(E.2) +a3 ∂uν +∂t + a3 +�dν +dt +�∂uν +∂ν + 3a2 ˙auν = 0, +(E.3) +a3 ∂uν +∂t − νa2 ˙a∂uν +∂ν + 3a2 ˙auν = 0, +(E.4) +∂uν +∂t − ν ˙a +a +∂uν +∂ν + 3 ˙a +auν = 0. +(E.5) +In the presence of a source having emission coefficient jν and absorption coefficient αν, the +radiative transfer equation becomes [118], +∂uν +∂t − ν ˙a +a +∂uν +∂ν + 3 ˙a +auν = −cανuν + jν. +(E.6) +The specific intensity Iν, defined as the energy per unit area per unit time per unit frequency +per unit solid angle is related to the energy density per unit frequency per unit solid angle +as uν = Iν/c. The specific intensity in temperature units is given by the following relation, +cuν = Iν = 2ν2 +c2 kBTb, +(E.7) +where Tb is the brightness temperature at a fixed frequency ν. We can use eq.(E.7) to cast +eq.(E.6) in terms of the brightness temperature [89, 119], +dTb +dt + ˙a +aTb = cαν +� +− Tb + +c2 +2ν2kB +jν +αν +� +. +(E.8) +If the level population defined in eq.(2.7) is in kinetic equilibrium at temperature Tex, by using +the Einstein relations we can find the ratio between the emission and absorption coefficients +– 38 – + +100 +200 +300 +400 +500 +600 +Frequency (GHz) +10−5 +10−4 +10−3 +10−2 +10−1 +∆Iν/Iν +Figure 19: +1-σ limits on the fractional deviation from the CMB blackbody spectrum +(∆Iν/Iν) as a function of frequency from COBE/FIRAS data [104, 120]. +which is given by [89], +jν +αν += 2hν3 +c2 +1 +(ehν/kBTex − 1). +(E.9) +This relation following from the principle of detailed balance depends only on the excitation +temperature Tex of the two levels. +F +Limits on CMB spectral distortions from COBE/FIRAS +We show the limits on spectral distortions of CMB from a perfect blackbody from COBE/FIRAS +[104, 120] in the 60-600 GHz band in figure 19. +G +Milky Way model for constraining the radiative coupling from CMB +anisotropy maps +We assume that dark matter in MW is distributed according to the following NFW [16] +density profile, +ρχ(r) = ρ0 +1 +r +r0 +� +1 + ( r +r0 ) +�2 , +(G.1) +with ρ0 (106 M⊙ kpc−3) = 3.486 and r0 (kpc) = 26.77 [79] and has a virial temperature Tvir. +The emissivity due to dark matter emission from MW along the LoS is given by [89], +jν = hν +4π φ(ν)n1(r)αAAHI +10, +(G.2) +where ν is the frequency of the emitted photon and φ(ν) is the Doppler line profile centered +at ν0 with a line width, +∆νD = ν0 +c +� +2kBTvir +mχ +. +(G.3) +– 39 – + +For a band covering a frequency range from νmin to νmax, the integrated specific intensity +along a particular LoS is given by, +j = +� νmax +νmin +dν jν. +(G.4) +– 40 – + +References +[1] Kimberly K. 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URL: +https://lambda.gsfc.nasa.gov/product/cobe/firas_monopole_get.html. +– 48 – + diff --git a/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/load_file.txt b/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6a4e3474e178ca82ecd6a256ade87732f08f9164 --- /dev/null +++ b/WNE2T4oBgHgl3EQfDgYF/content/tmp_files/load_file.txt @@ -0,0 +1,2370 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf,len=2369 +page_content='Prepared for submission to JCAP EDGES of the dark forest: A new absorption window into the composite dark matter and large scale structure Anoma Ganguly, Rishi Khatri, Tuhin S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Roy Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail: anoma@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='in, khatri@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='in, tuhin@theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='in Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We propose a new method to hunt for dark matter using dark forest/absorption features across the whole electromagnetic spectrum from radio to gamma rays, especially in the bands where there is a desert i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' regions where no strong lines from baryons are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such novel signatures can arise for dark matter models with a composite nature and internal electromagnetic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The photons from a background source can interact with the dark matter resulting in an absorption signal in the source spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In case of a compact source, such as a quasar, such interactions in the dark matter halos can produce a series of closely spaced absorption lines, which we call the dark forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We show that the dark forest feature is a sensitive probe of the dark matter self-interactions and the halo mass function, especially at the low mass end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Moreover, the absorption of CMB photons by dark matter gives rise to a global absorption signal in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For dark matter transition energies in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 × 10−4 eV − 5 × 103 eV, such absorption features result in spectral distortions of the CMB in the COBE/FIRAS band of 60-600 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If the dark matter transition frequency is ∼156 GHz, we show that the absorption of CMB photons by dark matter can provide an explanation for the anomalous absorption feature detected by the EDGES collaboration in 50-100 MHz range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='03624v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='CO] 9 Jan 2023 Contents 1 Introduction 1 2 A theoretical framework for the composite dark sector 4 3 Experimental signatures 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 Absorption lines in the spectrum of a compact source 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 A dark line: absorption by a single dark matter halo 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Dark forest: absorption by multiple dark matter halos 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 Detectability of dark forest 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Global absorption signal in the CMB spectrum 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 The EDGES anomaly 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 General predictions for the shape of the dark absorption feature 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 General predictions for the dark absorption feature in different parts of the CMB spectrum 27 4 Observational constraints 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 Early Universe constraints 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Late Universe constraints from Milky Way 28 5 Conclusions 30 A Dark matter model 32 B Dark matter halo density and temperature profile 33 C Dark forest 34 D Convergence of distribution functions 37 E Radiative transfer equation in an expanding Universe 37 F Limits on CMB spectral distortions from COBE/FIRAS 39 G Milky Way model for constraining the radiative coupling from CMB anisotropy maps 39 1 Introduction Dark matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' although making up a major fraction of the matter content in our Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' continues to remain a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Owing to its elusive nature, the different experiments search- ing for it [1, 2] have till now only succeeded in placing stringent constraints on its possible interactions beyond the usual gravitational force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The allowed parameter space for possible dark matter candidates is huge, with masses ranging from 10−22 eV ultralight bosons to 1043 GeV compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus a more clear picture of its true nature can only start emerging if we find new methods to look for dark matter in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The current searches for – 1 – dark matter broadly fall into three categories, direct detection: where detectors search for nuclear/electronic recoil due to dark matter, indirect detection: where experiments look for emission signals of different standard model particles produced from dark matter annihila- tion, decay, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=', or collider searches: where high energy accelerators try to produce dark matter by colliding standard model particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absence of a distinct signature of dark matter in any of these searches so far not only hints at its far more nuanced nature, but also calls for new detection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we propose a novel method to look for dark matter in the absorption lines of a background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The main advantage of absorption lines is their ability to probe very weak interactions between dark matter and photons, which is possible if the background source is sufficiently bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Absorption lines are a generic feature of a class of models where dark matter is a compos- ite particle with a discrete energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The presence of a small electromagnetic coupling can allow the transitions between different dark matter energy states via emission/absorption of a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As a specific example, we will consider dark matter to be a composite parti- cle made of two elementary particles of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A strong dark attractive potential between the constituents makes dark matter stable on cosmological scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As a whole dark matter stays electromagnetically neutral, while the constituents carry a millicharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This simple model allows us to describe dark matter as a bound state with weak electromagnetic transitions similar to a hydrogen atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The generic signatures of such models will include both absorption as well as emission lines in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' While a lot of work in the past has been on dark matter induced emission lines [3–6] and electromagnetic signals in colliders [7, 8], only a few touch upon the absorption signatures of dark matter [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In addition to electromagnetic/radiative transitions, the transitions between different energy states can happen via inelastic scattering between dark matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This has been studied in the context of small scale structure problems like the core-cusp problem and the missing satellite problem [11–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we focus on the less studied and more promising signature of composite dark matter: the absorption of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption line from dark matter inside a single galaxy cluster at gamma ray frequencies was studied in [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, there is no a priori reason to be confined to the gamma-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, the detection of an absorption line unidentifiable with the known transitions in baryonic atoms or molecules in any part of the electromagnetic spectrum is a tell-tale signature of such dark matter models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In addition to a single absorption line, we can even have a collection of absorption lines or dark forest similar to the Ly-α or 21 cm forest generated by the neutral hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark forest arises due to absorption of light by dark matter halos along the line of sight (LoS) to a quasar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We show that the dark forest opens a new window to the large scale structure as it traces the evolution of dark matter temperature and distribution inside the dark matter halos through the cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We make a detailed study of the evolution of dark forest from redshift of 7 to 0 for dark matter transitions at radiowave frequency of 156 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Interestingly, we find that the amount of absorption by a dark matter halo of a given mass is sensitive to the presence of dark matter self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Moreover, the density of absorption lines has a strong dependence on the smallest dark matter sub-structures present in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In general, the dark forest can appear in any part of the electromagnetic spectrum including radio, microwave, infrared, optical, X-ray, and gamma ray bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, the detection of absorption lines in the spectrum of a bright quasar at z ∼ 6 at frequencies < 200 MHz, below the 21 cm forest of neutral hydrogen, will be a smoking gun signature for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This may be possible with – 2 – the upgraded Giant Meterwave Telescope (uGMRT) [28, 29] and the Square Kilometer Array (SKA) [30] which have lowest frequency bands in 50-350 MHz and 125-250 MHz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' When the isotropic cosmic microwave background (CMB) acts as a background source, the absorption of CMB photons by inelastic composite dark matter gives rise to a global absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The origin of the global absorption signal from transitions in dark matter is similar to the global absorption feature caused by the hyper-fine transitions in neutral hydrogen during the Dark Ages [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' After dark matter decouples from the electron-baryon plasma, it cools as (1 + z)2, with the temperature soon becoming much lower than the CMB temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At very high redshifts, the strong inelastic collisions between dark matter particles bring the two dark matter energy levels in kinetic equilibrium with the dark matter temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark matter temperature being much lower than the CMB temperature implies that the dark matter particles in the ground state can absorb the CMB photons and generate an absorption signal in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As the Universe cools and the number density of dark matter particles gets diluted, the radiative transitions due to CMB photons take over the dark matter collisional transitions, bringing the level population in equilibrium with the CMB temperature and the signal vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' An important difference from 21 cm cosmology is the role of bremsstrahlung, which is important before recombination when the Universe has ample number of electrons and protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This process can erase spectral distortions in the low frequency tail of the CMB before recombination and establish an almost the perfect black body spectrum in the Rayleigh-Jeans tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus the high redshift (low frequency) edge of the absorption signal is entirely determined by bremsstrahlung, while the low redshift (high frequency) edge of the signal depends on the ratio of collisional to radiative coupling of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Tantalizingly, we find that the absorption of CMB photons by dark matter with a transi- tion frequency of 156 GHz at redshifts ∼ 3000−1000 can produce a global absorption feature that matches the amplitude, width and location in frequency measured by the Experiment to Detect the Global Epoch of reionization Signal (EDGES) [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The EDGES collaboration reported a strong absorption feature which is almost double in amplitude compared to the maximum absorption expected from 21cm absorption by hydrogen in the standard model of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However a recent experiment Shaped Antenna measurement of the background Radio Spectrum (SARAS 3) [37] disfavors the EDGES absorption profile being cosmological in origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A recent paper from the EDGES collaboration [38] does a Bayesian analysis jointly constraining the receiver calibration, foregrounds, and the measured signal reaffirming the presence of an absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Several other groups such as Large aperture Experiment to detect the Dark Ages (LEDA) [39–41], Probing Radio Intensity at high Z from Marion (PRIZM) [42], Radio Experiment for the Analysis of Cosmic Hydrogen (REACH) [43], Sonda Cosmol´ogica de las Islas para la Detecci´on de Hidr´ogeno Neutro (SCI-HI) [44], and Cosmic Twilight Polarimeter (CTP) [45] are working on validating these claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even if the exact profile measured by EDGES turns out to be due to systematics, an anomalously large ampli- tude of the absorption signal would still require beyond standard model physics, if confirmed by other experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We begin with a discussion of the theoretical framework of composite dark matter in the section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We make a detailed study of the unique absorption signatures of such dark matter models in section 3, where we discuss the physics of dark forest in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1, and the global absorption feature in the CMB spectrum in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We then proceed towards the implications of different astrophysical and cosmological experiments on the allowed parameter space for our dark matter model in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We conclude our results – 3 – in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We use the Planck 2018 [46] cosmological parameters (Hubble constant: H0 = 100 h = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='66 km s−1 Mpc−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3111, and Ωb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='049).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also use the publicly available codes Recfast++ [47, 48], Colossus [49], and FeynCalc [50–52] in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 2 A theoretical framework for the composite dark sector In this section, we set up a theoretical framework for the underlying physics of composite dark sector whose principal modes of observation are absorption features in the spectrum of a background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Specifically, we provide a proof of principle scenario that can accom- modate a dark sector which leaves its signature through photon absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Before we build such a model, we note the following set of considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even though each of these conditions need not be fulfilled strictly, stating them is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This not only allows us to stay general but also provides hints at the emerging model for the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (i) When the dark sector states make transitions among themselves, photons get absorbed/ emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This immediately suggests that the dark sector consists of multi-level states with mass gap(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Further, if there exists a symmetry that makes at least the lowest lying state cosmologically stable, the conservation of the associated quantum number dictates that all such transitioning states must have identical quantum numbers/charges under the same symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) In order to emit/absorb a photon when a transition takes place, there must be contact operators between the dark sector states and electromagnetic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such a coupling can arise if dark matter possesses an electromagnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, different observations such as CMB [53, 54], virialization in dark matter halos, elliptical shape of galaxies [55], bullet cluster [56], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' strongly constrain the electromagnetic charge dark matter can possess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' To evade such constraints, we consider the transitioning states to be neutral composits with electrically charged constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such states are bound states analogous to an atom of the visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Unfortunately, the electric charge of the constituents will lead to non-zero electromagnetic moment [10, 57] of the low lying dark state that can give strong signals in the CMB, direct, and indirect dark matter searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such bounds weaken when the constituents have very small electric charge or millicharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iii) The ionization of dark matter, if accessible, would provide much stronger signals in ex- periments compared to the absorption/emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Ionized dark matter in the early Universe will be subject to radiation pressure similar to the baryons which can modify the CMB acoustic peaks, imprint acoustic oscillation features in the dark matter power spec- trum, and erase structure on small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At late times, the Coulomb scattering between ionized dark matter particles inside a halo can give rise to cored central density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This suggests that scenarios where the transition energy and binding/ionization energy are similar (Etransition ≃ Ebinding), the signals from ionization are a far better probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work we concentrate on scenarios where the lines are the primary signatures of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Therefore the model we construct should have an added feature that Etransition ≪ Ebinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) (iv) One can use data from CMB experiments to put a strong constraint on the ionization of dark matter (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 for details), Ebinding ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) – 4 – All this information makes it quite clear that, The presence of a small electric charge of the constituents makes electromagnetism respon- sible only for a small fraction of the binding energy of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark sector cannot be a simple scaled version of hydrogen-like atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' While the first conclusion is obvious, the second demands a more careful explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For a hydrogen-like dark matter with mass Mbound state, satisfying the condition in (iii) implies that the transitions in dark matter cannot be Lyman-α like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even though the hyper-fine splitting (∆Ehf ≪ Ebinding) in these systems perfectly satisfies (iii), we find the simplicity of the set-up over-constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For example, a hydrogen-like system satisfies the following relation, Mbound state ∼ Ebinding �Ebinding ∆Ehf � =⇒ Mbound state ≫ Ebinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus, the desired hierarchy in ∆Ehf and Ebinding gets translated into a hierarchically large Mbound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If such a state represents dark matter, a large mass would result in small dark matter number density which reduces the signal of absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The task therefore is to build a framework which is calculable to a large extent and can accommodate the hierarchy in (iii) while keeping the bound state mass within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we propose a scenario where the leading order results get quantum cor- rected due to additional light degrees of freedom (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=') in the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This is reminis- cent of the proton-neutron mass difference occurring in nature, where the photons in the loop cancel the tree level contribution from the mass difference between the up and down quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Summarizing, we require the dark sector to consist of composite states that are massive and cosmologically stable, and couple strongly to the light d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As a starting point, we take the dark states to be bound states of mass mχ where the attractive potential between the constituents is dominantly due to dark-gluon exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' To be specific, we take the underlying physics to be a non-abelian SU(N) gauge theory (we will refer to it as dark color) with a suitably designed matter content and associated charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The strength of the gauge coupling constant roughly at the mass of the bound state i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' αN(mχ) determines most the features of the tower of dark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For instance, for αN(mχ) ∼ O(1), the bound state is relativistic and generically all energy splittings in the spectrum are of the same order as the binding energy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Etransition ∼ Ebinding, which does not satisfy condition (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Therefore, we need a separate mass scale, MQ in the theory which would ultimately yield mχ to be different from the scale (ΛN) at which gauge theory becomes strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A mild hierarchy between MQ and ΛN would allow the theory to be perturbative αN(mχ) ≪ 1, resulting in non-relativistic dark bound states where the transition energy is much smaller than the binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' An additional flavor symmetry can play a constructive role, partly in providing stability to the dark state and partly in giving rise to naturally light pseudo Nambu Goldstone bosons (pNGBs) which couple strongly to the dark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The details of the model we employ are provided in table 1, where we list the particle content in the ultraviolet (UV) and their quantum numbers under the gauge as well as global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark quarks making up the matter content follow the mass hierarchy: mq < ΛN < MQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The mass term for the dark quarks in Weyl representation is given by, L ⊃ MQ QDQc D + mqδab qDaqc Db + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) – 5 – SU(N) SU(2)D L SU(2)D R U(1)D U(1)em qD N 2 1 0 +ϵ qc D ¯N 1 ¯2 0 −ϵ QD N 1 1 +1 +ϵ Qc D ¯N 1 1 −1 −ϵ Table 1: The dark quarks in Weyl representation and their charges under gauge and global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' where a and b are the flavor indices which take values 1, 2 and the hermitian conjugate is abbreviated as h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='. The mass term for the light quarks in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) ensures that the condensate ⟨qDqc D⟩ is flavor diagonal signaling that the global SU(2)D L × SU(2)D R × U(1)D L × U(1)D R gets sponta- neously broken into a diagonal SU(2)D V × U(1)D, with an additional axial U(1)A broken by the dark color itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The axial SU(2)D A is explicitly broken by the light quark mass mq in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' After symmetry breaking, there are no massless NGB modes but there exist three light pNGBs or dark pions πD whose mass mπD is controlled by light dark quark mass mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We take these non relativistic heavy-light bound states (made of heavy quark QD and light quark qD) to form the dark tower of states protected by the darkness number (quantum number) of the global U(1)D symmetry listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Further, the dark states with light quarks as constituents also carry the flavor charges of the light quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This implies that the chiral effective Lagrangian generated by the dark color will automatically include the dark state - pNGB interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The model proposed here may appear complicated, but it should be noted that the physics of the dark sector described here mimics aspects of the physics of the visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In nature the strong interactions of Quantum chromodynamics (QCD) play a similar role in producing heavy-light bound states such as D-mesons or B-mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The spontaneous breaking of the approximate chiral symmetry associated with the light quarks of the visible sector gives rise to pions (pNGBs of the visible sector) with substantial interactions with the heavy light mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Following the formalism of the chiral Lagrangian for the heavy flavor [58–60], one can write down the form of interaction between the dark states and the dark pions, and estimate the hyper-fine splitting, transition rates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In appendix A, we chalk out some aspects of this formalism in the context of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the main body of this work we simply outline the main features of the model: The dark pions are pNGBs lying the coset SU(2)D L × SU(2)D R � SU(2)D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' One can use the usual exponential parametrization to express πD as, ΣD ≡ exp �2iπD fD � where πD ≡ πa D ˜ta, and ΣD SU(2)D L ×SU(2)D R −−−−−−−−−−−→ LDΣDR† D (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) The broken generators ˜ta belong to the coset, fD is the energy scale of the dark condensate ⟨qDqc D⟩, and LD and RD are transformation operators for SU(2)D L and SU(2)D R respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We take ¯QD qD bound state (darkness number -1) as the candidate for dark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Similar to the matter anti-matter asymmetry in the visible sector, the asymmetry in the number – 6 – of dark (darkness number -1) versus anti-dark states (darkness number +1) yields the observed dark matter abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We designate the lowest lying pseudoscalar spin 0 state as χ and the spin 1 state as χ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A convenient way to represent these four physical states collectively is using a matrix field Xv which is an eigenstate of the velocity v of the dark bound state, Xv ≡ P+ � χ∗ µγµ − χγ5� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) The projection operator P+ = 1 2 � 1 + /v � captures only the small fluctuations (≪ MQ) for this Heavy Quark Effective Field theory (HQET) [61–64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' χ-χ∗ exhibits a nearly degenerate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The mass difference between χ and χ∗, known in the literature as the hyper-fine splitting, arises from operators shown in of appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This splitting is suppressed by the heavy quark mass MQ, but contains a number of unknown parameters and in the limit MQ → ∞, χ and χ∗ become exactly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The spontaneous emission rate for χ∗ → χ γ process is computable within the paradigm of the chiral Lagrangian for the heavy-light systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The rate is also MQ suppressed like the hyper-fine splitting but depends on a different set of unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In order to derive the χ∗χ∗πD or χχ∗πD couplings, one needs to take into consideration the symmetry properties of the heavy-light bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' It is conventional to write the interactions between the bound states and dark pions by defining the vectorial and axial currents which contain the dark pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' These couplings also contribute non-trivially to the hyper-fine splitting as well as the transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even though the formalism of HQET seems to describe the different properties of the B and D meson states, such as, the pattern of their couplings and mass gaps extremely well, there is one crucial drawback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The physics of these states is described in terms of unknown constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' While in the context of states in the visible sector, the existence of data allows us to determine these constants (which in turn, makes it possible to predict several other observables rather precisely), such a procedure is not practical for our dark bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also cannot just scale up QCD to predict these, since the number of colors, flavors, and pattern of masses are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Of course, lattice QCD can make definitive statements about the size of splittings and photon transition rates, but such an exercise is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For the purpose of phenomenology, it is sufficient to consider a simplified model where we only keep a handful of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Instead of using unknown parameters in the Lagrangian of the theory, we parameterize the simplified model in terms of parameters which quantify the energy splittings and transition rates of various physical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The underlying EFT is depicted in terms of a simplified model in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The main features of this model relevant for the estimation of absorption lines arising due to hyper-fine transitions in the dark sector can be summarized as follows, Since the only relevant transitions happening in the dark states are the hyper-fine tran- sitions, the physics is rather simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We can disregard all the other higher energy states and treat this as a two-state system with states 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The state 0 corresponds to en- ergy E0 = mχc2 and degeneracy factor g0, and state 1 corresponds to energy E1 = mχ∗c2 – 7 – Figure 1: The left part shows the different energy scales of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The right part is a zoomed-in version of the hyper-fine splitting in the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' and degeneracy factor g1, where c is the speed of light in vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The energy difference between two states is given by, ∆Ehf = E1 − E0 = hν0 = kBT∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) where h is the Planck’s constant, kB is the Boltzmann constant, ν0 is the transition fre- quency, and T∗ is the transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We will further assume ∆Ehf < mπD, which implies that as far as χ∗ ↔ χ transitions are concerned, we can disregard the dark pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, for simplicity we take g1/g0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The population of dark matter particles in the two states is decided by the collisional and radiative transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The ratio of the number density of dark matter particles in the ground state (n0) with respect to the excited state (n1) is parameterized by the excitation temperature Tex, n0 n1 ≡ g0 g1 exp(T∗/Tex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) In our simplified model, the occupation number of 0 and 1 states gives the total dark matter number density, nχ = n0 + n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8) The transition between the two states can happen via emission/absorption of a photon which is parameterized in terms of Einstein A and B coefficients in the following way: The number of radiative transitions per unit time per unit volume from level 0 to level 1 is proportional to the Einstein coefficient B01, dn0→1 dt = n0B01 ¯J, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9) where ¯J is the mean intensity of incident light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The number of radiative transitions per unit time per unit volume from level 1 to level 0 is a sum total of spontaneous emission which proportional to the Einstein coefficient A10 and stimulated emission which is proportional to B10, dn1→0 dt = n1(A10 + B10 ¯J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10) – 8 – excited states m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' AEh x AN m TDThe Einstein coefficients A10, B01, and B10 are related to each other via the Einstein relations which follow from the principle of detailed balance, A10 = 2hν3 0 c2 B10, g0B01 = g1B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11) In this work, we parameterize the Einstein coefficient for hyper-fine transitions in the dark sector in terms of the Einstein coefficient for hyper-fine transitions in the hydrogen atom, A10 = αA AHI 10, where AHI 10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='85 × 10−15s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) We will set αA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35 in our numerical computations (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 for justification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The transition between the two states can also happen via inelastic collisions between dark matter particles parameterized in terms of the collisional excitation and de-excitation coefficients C01 and C10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The number of collisional transitions per unit volume from level i to level j is given by, dni→j dt = niCij, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13) where i ̸= j and i, j run from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For a thermal velocity (Maxwell Boltzmann) distribu- tion of dark matter particles at temperature Tχ, the two collisional coefficients are related as, Cij(Tχ) = gj gi exp(−T∗/Tχ) Cji(Tχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='14) In case of a quasar, we simply consider the implications for two extreme scenarios, one where inelastic collisions are completely absent (collisionless dark matter) and the other where the inelastic collisions are strong (collisional dark matter) (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In case of CMB as a background source, we use intermediate collisional cross-section parameters as a function of dark matter temperature (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10) of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3 Experimental signatures The composite dark matter particle can make an electromagnetic transition from the ground state to an excited state by absorbing a photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such transitions can give rise to unique experimental signatures in the form of absorption lines in the spectrum of a bright background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, the detection of a new absorption line, not identifiable with a known atomic or molecular transition in any part of the electromagnetic spectrum, would be a smoking gun signature for such dark matter models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The high dark matter density in structures like dark matter halos and dwarf galaxies makes these sites ideal targets that can generate such absorption signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, when one such object lies along the line of sight (LoS) to a compact source, the absorption of light by the composite dark matter particles inside these objects produces an absorption line in the source spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The shape of the line is characterized by the density and velocity dis- tribution profile of dark matter particles inside the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In reality, we will have multiple – 9 – Figure 2: Schematic diagram of the line of sight intersecting an absorber at an impact parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' such absorbers along the LoS to a distant quasar resulting in a series of absorption lines at different frequency locations in the observer’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The frequency location of the absorp- tion lines is decided by the transition frequency and the redshift of the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We will study the absorption lines in the spectrum of a compact source for a single absorber by taking an example of a dwarf galaxy and a general dark matter halo in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We then proceed towards the case of multiple absorbers along the LoS to a quasar in subsubsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' When the source is isotropic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' the CMB, the composite dark matter particles absorb the CMB radiation giving rise to a broad global absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We study such a dark global absorption feature in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 Absorption lines in the spectrum of a compact source When the LoS to the compact source passes through an absorber, such as a dark matter halo, the composite dark matter particles inside these structures can absorb the incident light, imprinting an absorption feature in the source spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Similar to absorption, we can also have emission lines from dark matter imposed on the average spectrum of a galaxy inside the dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the rest frame of a point-like absorber situated at redshift z0, the absorption/emission happens at the transition frequency ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Due to the expansion of the Universe, the absorp- tion/emission line is observed today at a frequency ν = ν0/(1 + z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, complication arises for absorbers of finite size and non-trivial density and velocity profiles in different possible astrophysical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Dark matter density profile: For an extended absorber intersected by the LoS to the source at an impact parameter p (as shown in figure 2), the cumulative net absorption (true absorption minus stimulated emission) gets contribution from all the particles present along the LoS, which is denoted by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Lets consider a line element ds (in figure 2) along the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The true absorption (stimulated emission) is proportional to the number density of dark matter particles in the ground (excited) state, which in turn is proportional to the total number density of dark matter particles nχ(r) = ρ(r)/mχ, where ρ(r) is the dark matter density at a distance r from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Excitation temperature: The population of dark matter particles in the two states at a radius r inside the dark matter halo is determined by the excitation temperature Tex – 10 – Line of sight s ds S=0 Absorberdefined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The excitation temperature is determined by two processes, namely, the radiative transitions due to the CMB photons which try to bring the two levels in kinetic equilibrium with the CMB temperature (Tγ(z0)), and the collisional transitions due to inelastic collisions between dark matter particles inside the halo, which try to bring the two levels in kinetic equilibrium with the temperature of the halo (Th(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work we will study two extreme scenarios for dark matter (DM) inelastic self-interactions, Tex(r) = � Tγ (z0) collisionless DM, Th (r) collisional DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) The general scenario would lie somewhere between these two extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We will indeed find that the absorption lines are sensitive to the collisional nature of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Doppler broadening: The non-trivial velocity profile of the dark matter particles along the LoS gives rise to the Doppler broadening of the absorption line around ν0 in the halo’s rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This broadening is characterized by the line profile, φ (νh, r, p) = 1 √π∆νD (r) exp � −(νh − ν0 (1 + vLoS(r, p)/c))2 ∆νD (r)2 � , where ∆νD (r) = ν0 (1 + vLoS(r, p)/c) c � 2kBTh (r) mχ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) νh being the absorption frequency in the halo’s rest frame, and vLoS being the peculiar velocity of the dark matter halo along the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The effect of vLoS is simply to shift the frequency location of the line in observer’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we will not be calculating the two-point correlations but only the one-point statistics of the dark matter forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' So we will ignore the halo peculiar velocity and set vLoS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the presence of absorption, the flux density measured by the observer falls exponentially with the column density along the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Conventionally, the observed absorption line is quan- tified by the optical depth τν which is defined as, τν = log �F 0 ν Fν � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) where Fν and F 0 ν are the flux densities of the source in the presence and absence of absorption respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For a halo intersected at an impact parameter p (as shown in figure 2), the optical depth profile in the halo’s rest frame is given by [65], τ(νh, p) = � do −ds g1 g0 αAAHI 10 c2 8πν2 0 ρ (r) mχ φ (νh, r, p) � 1 − e− T∗ Tex(r) 1 + (g1/g0) e− T∗ Tex(r) � ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) where r2 = p2 + s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We set the origin s = 0 to be the position where the impact parameter intersects the LoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We integrate the LoS from the source to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The distance between the source and the absorber is denoted by ds and the distance between the observer and the absorber is denoted by do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the frame of the observer on Earth, the optical depth profile is obtained by mapping νh → νh/(1 + z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 11 – In the rest of the analysis we choose the following dark matter model parameters: mχ = 1 MeV, ν0 = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 GHz, and αA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35 (see section 4 for justification) for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our main results are however quite general and we leave the full exploration of the parameter space for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 A dark line: absorption by a single dark matter halo The dark matter halos are gravitationally bound structures which form the building block of the non-linear matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We want to study how the different properties of dark matter halos influence the absorption profile generated by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A dark matter halo is characterized by its mass parameter (Mh), a length parameter (virial radius rvir), a temperature (Th), and a dark matter density profile ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Some of these parameters are related (see appendix B for exact expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We assume a NFW density profile [16] for the dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The Doppler line profile for the halo is decided by the halo temperature Th (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) in appendix B for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We proceed towards calculating the optical depth or the absorption profile generated by a given halo mass Mh using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The optical depth depends crucially on the dark matter density profile, the impact parameter (p expressed in rvir units), and the Doppler broadening due to random motion of the dark matter particles parameterized by the effective temperature (Th) of the dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We present our results in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We plot the optical depth profiles for a given halo mass (106M⊙/h) at different redshifts (z = 6, 4 and 2) intersected at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 rvir in the top two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the bottom two panels, we plot the optical depth profiles at a given redshift (z = 5) for different halo masses (in M⊙/h units) intersected at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 rvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We make the following observations: (i) There is stronger absorption in collisional DM compared to collisionless DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) As we go to higher redshifts, the total absorption by a halo, which is equal to the area under the optical depth profile, grows (top two panels of figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iii) The width of the absorption profile increases with halo mass and redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iv) In collisionless DM case, the peak amplitude of the absorption profile increases with the halo mass (third panel of figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (v) In collisional DM case, the peak amplitude of the absorption profile decreases with the halo mass (fourth panel of figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We explain these findings using figure 4 where we compare the halo temperature at the virial radius rvir for different halo masses with the CMB temperature in the first panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also compare the dark matter number density profile and the halo temperature profile at different redshifts in the last two panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Corresponding to the above observations, the explanations are as follows: (i) For halos of mass ≲ 108M⊙/h, Th < Tγ (first panel of figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Therefore the excitation temperature (defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1)) for collisional DM is less than that for collisionless DM, resulting in stronger absorption in the case of collisional DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) As we go to higher redshifts, the number density of dark matter particles increases which results in stronger absorption (second panel of figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 12 – −6 −4 −2 0 2 4 6 ∆ f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='20 Optical depth Collisionless DM z = 6 z = 4 z = 2 −6 −4 −2 0 2 4 6 ∆ f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Optical depth z = 2 z = 4 z = 6 Collisional DM −20 −10 0 10 20 ∆ f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='035 Optical depth Mh in M⊙/h Collisionless DM Mh = 108 Mh = 107 Mh = 106 −20 −10 0 10 20 ∆ f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 Optical depth Mh in M⊙/h Collisional DM Mh = 108 Mh = 107 Mh = 106 Figure 3: Top: The optical depth profiles for 106M⊙/h halo at impact parameters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 rvir in the halo’s rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Bottom: The optical depth profiles for different halo masses intersected at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 rvir at z = 5 in the halo’s rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The solid lines refer to collisionless DM and dashed lines refer to collisional DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iii) The width of the optical depth profile in the halo’s rest frame ∝ √Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The halo temperature, Th increases with both redshift and halo mass (third panel of figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iv) In collisionless case, Tex = Tγ is independent of the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The number density of dark matter particles increases with the halo mass resulting in stronger absorption profiles for collisionless DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (v) In the collisional case, Tex = Th increases with the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even though the total dark matter number density increases with the halo mass, a higher value of Tex implies less dark matter particles in the ground state, which combined with broadening of the profile results in smaller peak amplitude of the absorption profile for higher halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Note that in this analysis we have assumed the dark matter density and velocity distribution profiles to be the same in both collisionless and collisional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The presence of dark – 13 – 0 2 4 6 Redshift 100 101 102 Th(rvir) (K) Mh = 109 CMB Mh = 108 Mh = 107 Mh in M⊙/h 0 1 2 3 4 Radius (in rvir units) 100 102 104 DM number density(cm−3) z = 6 z = 4 z = 2 0 1 2 3 4 Radius (in rvir units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Th (K) z = 6 z = 4 z = 2 Figure 4: From left: The evolution of halo temperature at rvir as a function of redshift for halo masses 107, 108, and 109 M⊙/h, the dark matter number density profile and the halo temperature profile for 106 M⊙/h halo as a function of radius (in rvir units) at different redshifts z = 6, 4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='010 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='010 ∆ f (MHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Optical depth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 rs Figure 5: Optical depth profile for Leo T subhalo for impact parameters p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 rs, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs for collisionless DM represented by solid lines and collisional DM represented by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' matter self-interactions does not change the Maxwellian velocity profile as discussed in [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In principle, strong dark matter collisions may modify the density profile of dark matter halos which can modify the shape of the absorption profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As a detour, to showcase the possibility of hunting for dark matter absorption signatures in satellite galaxies of Milky Way, we take the example of absorption line generated by the dark matter subhalo that hosts the Leo-T dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Leo-T dwarf galaxy: Low mass dwarf galaxies are excellent venues to study dark matter since they have low star formation activity and weak electromagnetic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Some of the best current constraints on emission signatures of dark matter come from the dwarf satellite galaxies of Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [67–72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus we also expect that the absorption of light from a background source by composite dark matter particles in dwarf galaxies would provide strong tests for such dark matter models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We consider one such MW satellite galaxy, namely, – 14 – Leo-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We model its dark matter density profile using a Burkert profile from [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We assume the velocity distribution of dark matter to be Maxwellian with a velocity dispersion ( � ⟨v2⟩) equal to that of hydrogen ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9 km/s [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The temperature of the halo is defined by the relation, kBTh = 1 3mχ⟨v2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For mχ = 1 MeV, we find Th ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In figure 5, we show the absorption profiles of Leo-T intersected at impact parameters 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 rs, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 rs, where rs is the scale radius of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We note that since Th ∼ Tγ for Leo-T, the absorption profiles in collisional and collisionless cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However the small difference in Tex still gives a noticeably stronger absorption in case of collisional dark matter compared to collisionless dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Dark forest: absorption by multiple dark matter halos If the LoS to the source passes through multiple halos (located at different redshifts z0), each intersection gives rise to an absorption profile at ν = νh/(1 + z0) to an observer on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Collectively, a large number of absorption lines coming from the same transition in dark matter at different redshifts, and hence separated in frequency, are called forest in spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this section we describe the procedure to simulate a dark forest and discuss its qualitative and quantitative aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The simulation consists of discretized frequency bins in a given frequency range with the bin width adjusted such that each bin has an identical probability of net absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In a pseudo experiment, we then simulate the absorption line by generating a random number to select the bin where absorption occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We plot the observed dip in intensity in terms of the relative transmission e−τν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We summarize the algorithm for generating the dark forest spectra below [65, 75, 76]: We begin by selecting the frequency range of simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For an instrument sensitive in νmin to νmax range, the absorption lines correspond to halos in zmax = ν0/νmin − 1 to zmin = ν0/νmax − 1 redshift range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We find the equiprobable bin width ∆ν at a given ν by relating it to the probability of finding a halo in redshift bin ∆z centered at z = ν0/ν − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This probability is equal to the fraction of the area on the sky covered by halos of all masses in ∆z redshift bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus the probability of intersecting a halo in a frequency range ν to ν + ∆ν is given by, ∆Nh = ∆ν dNh dν = ∆z dNh dz = ∆z c (1 + z)2 H (z) � Mmax Mmin dMh dn dMh (z)A (Mh, z) , where A (Mh, z) = πrmax (Mh, z)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) The halo mass function dn/dMh in comoving units is taken from [77], Mmin and Mmax denote the minimum and maximum halo mass at a given redshift respectively, and rmax is the physical radius of the halo at which the dark matter number density is equal to the mean dark matter number density in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We choose the bin width ∆ν at each ν such that the probability of absorption ∆Nh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We generate a random number from a uniform distribution in [0, 1] in each frequency bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The bin is selected for absorption if the random number is ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption profile is characterized by the halo’s redshift z0, mass Mh, and impact parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For the selected bin, we choose Mh from the probability distribution function – 15 – of the area fraction occupied by halos of mass Mh at redshift z0, p (Mh, z0) ∝ dn dMh A (Mh, z0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) We choose the impact parameter from a uniform distribution over the cross-sectional area of the halo A(Mh, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We then generate the absorption profile in the halo’s rest frame using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) and map it to the observer’s frame by transforming νh → νh/(1+z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We simulate the synthetic dark forest spectra for 100 different LoS in the redshift range 7 to 0 (see appendix C for one such sample spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We quantify the information in the dark forest by studying the distribution functions of the peaks (τpeak) and widths (b2) of the absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The width is defined in terms of b2 given by [78], b2 = −2 � c ν0 �2 τ 2 ττ ′′ − τ ′2 = −2 � c ν0 �2 τpeak τ ′′ peak , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) where τ ′ = dτ/dν, τ ′′ = d2τ/dν2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' By combining the spectra for 100 different LoS, we calculate the mean count N and standard deviation in ∼ 30 bins to get the distribution functions for τpeak and b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Note that these distribution functions are unnormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The shape of the distribution functions at very low values is an artifact of the maximum impact parameter at which a halo is intersected (∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5rvir in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' So we place a cutoff on τpeak and b2 at the lower end and plot the distribution functions only above this cutoff, where we are not affected by the choice of the maximum impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We present our results in figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We observe: (i) The distribution functions for both τpeak and b2 are higher for Mmin = 104M⊙/h compared to Mmin = 106M⊙/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) The tails of both τpeak and b2 distribution functions in both collisionless and collisional case extend to larger values at lower redshifts (redshift range 2-1 compared to redshift range 6-5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iii) The τpeak distribution function for collisional DM rises above the collisionless DM at large values of τpeak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iv) In the collisionless DM case, the τpeak distribution function rises monotonically at lower values of τpeak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For collisional DM, the curve rises at lower values of τpeak, reaches a peak, falls, and again rises at τpeak ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (v) The b2 distribution function for collisionless and collisional DM almost coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (vi) The tail of b2 distribution function for Mmin = 106M⊙/h extends to larger values compared to Mmin = 104M⊙/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' To explain the findings above, we make two plots in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the first plot we compare the number of halos intersected per unit redshift for three different values of Mmin and in the second plot we compare the unnormalized probability of intersecting a halo of mass (Mh) at redshifts 1, 4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Corresponding to the above observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' the explanations are as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='– 16 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='τpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/dτpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 106M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 6 - 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='τpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/dτpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 104M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 6 - 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='τpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/dτpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 106M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 2 - 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='τpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/dτpeak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 104M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 2 - 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Figure 6: Distribution function for the optical depth peaks for minimum halo mass 106 M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='and 104 M⊙/h in redshift ranges 6-5 (top) and range 2-1 (bottom) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (i) The lower limit of the halo mass function Mmin decides the contribution of the low mass end of the halo mass function to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus the probability of intersecting a halo is higher for Mmin = 104M⊙/h compared to Mmin = 106M⊙/h (first panel of figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) As the matter overdensities grow, the collapse fraction increases and the higher mass halos start contributing to the mass function at lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In addition at lower redshifts, a given redshift interval dz corresponds to a larger comoving distance interval dη = dz/H(z) resulting in more number of halo intersections (second panel of figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This in turn gives rise to higher τpeak and b2 values at low redshifts as there is a greater chance of intersecting more massive halos as well as intersecting halos close to the center where dark matter density and halo temperature is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Moreover, the line width for a halo also increases at lower redshifts due to smaller Doppler shift (b2 ∝ 1/(1 + z)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iii) At a given redshift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' the probability of hitting low mass halos along the LoS is higher,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='since the halo mass function falls exponentially at larger masses (second panel of figure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='– 17 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='b2 (km2/s2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/d(b2[km2/s2]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 106M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 6 - 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='b2 (km2/s2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/d(b2[km2/s2]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 104M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 6 - 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='b2 (km2/s2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/d(b2[km2/s2]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 106M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 2 - 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='b2 (km2/s2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='dN/d(b2[km2/s2]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mmin = 104M⊙/h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='z : 2 - 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisionless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Collisional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Figure 7: Distribution function for line widths for minimum halo mass 106 M⊙/h and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 M⊙/h in redshift ranges 6-5 (top) and 2-1 (bottom) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As explained before, the absorption is stronger for collisional DM compared to collisionless DM in halos of masses ≲ 108M⊙/h (first panel of figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (iv) A new absorption peak is generated when the tails of two or more absorption profiles overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This can be seen in figure 9 where we compare the dark forest spectrum for collisional and collisionless DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Due to stronger absorption in collisional DM case, the new lines give rise to an extra feature in the τpeak distribution function for collisional case compared to collisionless case at the low τpeak end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (v) We consider the same velocity distribution profiles for DM inside the halo for both collisionless and collisional DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The small differences mostly arise when the absorption profiles of two or more halos overlap and give rise to new lines which can have different line widths in collisionless versus collisional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (vi) More massive halos have higher halo temperatures resulting in a larger line width (b2 ∝ Th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 18 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Redshift 0 2000 4000 6000 8000 10000 12000 14000 dN/dz Mmin = 104 Mmin = 105 Mmin = 106 Mmin in M⊙/h 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Log10(Mh) 10−25 10−21 10−17 10−13 10−9 10−5 10−1 (dn/dMh) × A(Mh) z=1 z=4 z=7 Figure 8: On the left we plot the number of halos intersected per unit redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' On the right we plot the unnormalized probability distribution for intersecting a halo of mass Mh along line of sight at redshifts 1, 4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='14 Redshift 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='400 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='425 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='450 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='475 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='500 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='525 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='550 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='575 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='600 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='000 Relative transmission 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='14 Redshift 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='400 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='425 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='450 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='475 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='500 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='525 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='550 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='575 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='600 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='998 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='000 Relative transmission Figure 9: Dark forest spectrum for collisionless DM (above) and collisional DM (below) for Mmin = 106M⊙/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption lines in the regions of overlap of absorption profiles of two halos are stronger in collisional case compared to collisionless case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Both the spectra are for identical halo intersections (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' same simulation), the only difference being the collisional property of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We check the convergence of the distribution functions by increasing the line of sight direc- tions from 100 to 1000 in appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 19 – 10−2 101 104 107 1010 Frequency (GHz) 10−5 10−4 10−3 10−2 Resolution (∆ν/ν) uGMRT LOFAR SKA1 JWST X-shooter BOSS XMM-Newton Chandra Athena 10−4 10−3 10−2 10−1 100 101 τpeak 10−2 10−1 100 101 102 103 dN/dτpeak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 αA mχ αA mχ 10 αA mχ Sensitivity Figure 10: On the left we plot the spectral resolution of different spectroscopic experiments over the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' On the right we show the scaling of the τpeak distribution function (for collisionless DM in redshift range 2 to 1 and Mmin = 104M⊙/h) with αA/mχ (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) for the definition of αA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dashed black line shows the minimum sensitivity (τpeak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01) of an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 Detectability of dark forest The dark forest is a collection of absorption lines, where each line is characterized by a frequency, width and a peak amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For our choice of ν0 = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 GHz, the absorption lines were generated at radiowave frequencies with a typical width ∆ν/ν ≈ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A different ν0 would give rise to the dark forest in a different part of the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The existence of a large number of spectroscopic experiments spanning different frequency ranges already make the detection of new dark absorption lines an exciting possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' To name a few,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' experiments like the Square Kilometer Array (SKA1) [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Low-Frequency Array (LOFAR) [79],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' and the upgraded Giant Meterwave Radio Telescope (uGMRT) [28,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 29] operate at radiowave frequencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' the James Webb Space Telescope (JWST) [80] covers the infra-red,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Baryon Oscillation Spectroscopic Survey (BOSS) [81] and X-shooter [82] look for quasars at redshifts ∼ 2 to 5 at the optical frequencies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' while Chandra [83] and X-ray Multi Mirror Mission (XMM-Newton) [84] operate at UV and X-ray frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The frequency coverage and spectral resolution for these experiments are shown in the first panel of figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We can see that existing experiments operating in radio and optical frequencies already have the required spectral resolution (∆ν/ν ≳ 10−3) to detect the new dark absorption lines in the spectrum of bright quasars/blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A high signal to noise ratio can be achieved by increasing the duration of observation or the integration time which would allow the detec- tion of the weak absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Note that the peak amplitude of an absorption line scales as αA/mχ as shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our choice of αA is at the boundary of being disallowed for dark matter of mass 1 MeV by the CMB constraints (shown in the third panel of figure 13 in section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, note that our constraints are very conservative and a more careful analysis would considerably weaken them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Also for a different value of ν0 outside the observ- able band of CMB, higher values of the radiative coupling αA would be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus, all our results and plots can be readily scaled for a different values of αA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 20 – We show the scaling of the τpeak distribution function with αA/mχ obtained by averaging over 100 different LoS directions for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01 and 10 times αA/mχ compared to αA/mχ chosen in this work in the second panel of figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We find that if an instrument can detect absorption lines with τpeak ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01, the full distribution function for 10 × αA/mχ can be probed with the spectra of ∼ 100 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The sensitivity decreases for weaker absorption lines and a sufficiently large quasar sample is required to probe the tail of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For instance, in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 × αA/mχ case, only ∼ 1 in 100 quasar spectra contributes to τpeak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01 in the tail of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Global absorption signal in the CMB spectrum The absorption of photons of a particular frequency also leave a tell-tale signature in the sky-averaged spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such features are called global signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The much studied 21 cm global signal [31–34] in the CMB spectrum due to neutral hydrogen, for example, carries within important information about the growth of structure and first stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Not surprisingly, we expect a similar global absorption in CMB in case of transition among the dark sector states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The underlying physics of the global absorption feature is more or less similar to the absorption along the LoS to a bright source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, there are some crucial differences: In case of absorption along the LoS to a compact source, the observed signal is equal to absorption minus stimulated emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The effect of spontaneous emission is negligible as it gets distributed along all directions in the 4π solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' However, spontaneous emission is important in case of CMB because it is an isotropic source and the observed signal along a given LoS gets contribution from spontaneous emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We assume that dark matter is in kinetic equilibrium with the baryonic plasma and CMB till zdec > 105 (see subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As long as dark matter is kinematically coupled to the CMB, its temperature is equal to the CMB temperature ∝ (1+z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Since the dark matter is non-relativistic at decoupling, it cools faster than the CMB with temperature evolving with redshift ∝ (1 + z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Tχ(z) = � � � Tγ(z) z ≥ zdec, Tγ(zdec) � 1+z 1+zdec �2 z < zdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8) First consider the redshift at which dark matter starts absorbing CMB photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Let z0 be the redshift at which dark matter absorbs a photon of frequency ν0 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The optical depth per unit redshift is given by, dτχ dz = − g1/g0 1 + (g1/g0) e− T∗ Tex(z0) � 1 − e− T∗ Tex(z0) � αAAHI 10 c4 nχ (z0) 8πν3 0H (z) �1 + z0 1 + z � δ (z − z0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9) The optical depth τχ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9) is obtained by integrating the line profile along the LoS in an expanding Universe using Sobolev approximation [31–34, 47, 48, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The Sobolev approximation is valid as long as the Doppler line width is negligible compared to the width of the global absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In case of a single source we analyzed two extreme limits: collisionless DM and highly colli- sional DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The effect of inelastic collisions is however essential to have a global absorption – 21 – signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The physics of dark matter inelastic collisions is described in detail in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13) of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The exact functional form of the collisional coefficients depends on the details of the dark matter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even for a simple system of a hydrogen atom, collision cross- sections have a complicated temperature dependence [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For simplicity, we will assume the dark matter collision cross-sections qualitatively similar to the inelastic cross-sections of hyper-fine transitions in hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We parameterize C10 as a power law in Tχ: C10 = nχ⟨σv⟩, where ⟨σv⟩ = � � � � � a1 � Tχ(z) Tχ(zrec) �β z < zsat, a1 � Tχ(zsat) Tχ(zrec) �β z ≥ zsat, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10) where a1 is the value of ⟨σv⟩ at zrec = 1100 (the redshift of hydrogen recombination), zsat is the saturation redshift parameter, and β is the power law index which is taken to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the presence of both radiative transitions as well as inelastic collisions, the change in the population of dark matter particles in the ground state (n0) can be calculated from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10), and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13), dn0 dz − 3n0 1 + z = − 1 H(z)(1 + z) � n1C10 − n0C01 + n1A10 + (n1B10 − n0B01) ¯J � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11) where ¯J is the CMB intensity at ν0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11), we find that the level population of dark matter particles which is pa- rameterized in terms of the excitation temperature Tex (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7)) is determined by the competition between the collisional rate and the radiative transition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' By differ- entiating eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) with respect to redshift and substituting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='11) into it, the evolution equation for excitation temperature Tex becomes, dTex dz = T 2 ex T∗ � 1 n1 dn1 dz − 1 n0 dn0 dz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) We note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) is quite general and applies to any two level system, not just the spin flip transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, we have not made any assumptions about the smallness of T∗ with respect to other temperatures in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' It is customary to express the specific intensity at frequency ν i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Iν in terms of the brightness temperature as, Tb = c2 2ν2kB Iν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='13) Prior to recombination, the collisions between the free electrons and ions create and destroy photons by the bremsstrahlung process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Bremsstrahlung plays a vital role in preserving the blackbody spectrum of CMB by erasing any distortion that may have originated in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Even when it becomes unimportant in maintaining the CMB blackbody spectrum over most of the frequency range at z ≤ 106, it is still important in the low energy Rayleigh- Jeans tail of the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The bremsstrahlung process tries to bring the brightness – 22 – temperature in equilibrium with the gas/baryon temperature Tg, which is equal to the CMB temperature (Tb → Tg = Tγ) until z ≈ 500 [87, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Consequently, the quantity x ≡ hν (z) kBTg (z) = hν0 kBTγ (z0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='14) remains invariant till z ≈ 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The optical depth due to bremsstrahlung (τbr) per unit redshift is given by [89, 90], dτbr dz (x) = − cσT αnenB (24π3)1/2 H (1 + z) gbr(x, z) �kBTg mec2 �−7/2 � h mec �3 �1 − e−x x3 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='15) where α is the fine structure constant, σT is the Thomson scattering cross-section, me is the mass of the electron, nB(z) and ne(z) are the number densities of baryons and electrons respectively, gbr(x) ≡ Z2 i ni⟨gff(x)⟩/nB, where Zi is the charge of the ith ion having number density ni, and ⟨gff(x)⟩ is the thermally averaged Gaunt factor which has been taken from [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At high redshifts, the number density of dark matter particles is high which results in stronger collisional transitions between the two dark matter states, compared to radiative transitions due to CMB photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus, initially Tex is in kinetic equilibrium with the dark matter temperature which is much lower than the CMB temperature (Tex → Tχ ≪ Tγ) at z < zdec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark matter particles absorb the CMB photons and a net flow of energy takes place from CMB to dark matter resulting in an absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption of CMB photons by dark matter at redshift z0 generates an absorption line at x (defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='14)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If this line lies in the low frequency Rayleigh Jeans tail (x ≪ 1) of the CMB spectrum, it gets erased by the bremsstrahlung emission at subsequent times (z < z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As the number density and the temperature of dark matter falls due to expansion of the Universe, the collision rate falls and the radiative transitions involving the CMB photons begin to dominate over the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This brings Tex in kinetic equilibrium with the CMB temperature (Tex → Tγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' When this happens there is no net emission or absorption of the CMB photons by dark matter and the absorption signal vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus we expect to see a broad absorption feature in the CMB spectrum, starting from the time dark matter decouples until a later time when the radiative transitions take over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The evolution of brightness temperature at ν = ν0/(1 + z0) incorporating the effect of absorption by dark matter at z = z0 (from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9)) and bremsstrahlung (from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='15)) is given by, dTb(ν) dz − Tb(ν) 1 + z = dτχ dz � −Tb(ν) + hν kB 1 (ehν/kBTex(z) − 1) � +dτbr(x) dz (−Tb(ν) + Tg) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='16) The differential brightness temperature δTb observed at frequency ν = ν0/(1+z0) is defined as the brightness temperature (obtained by solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='16)) minus the CMB temperature today i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' in the observer’s frame, δTb (ν, z = 0)observer’s frame ≡ Tb (ν, z = 0) − Tγ (z = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='17) – 23 – 6000 3000 2000 1500 1200 1000 Redshift 50 100 150 Frequency (MHz) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Brightness temperature (K) 99 % CI EDGES best fit Case 1 Case 2 Figure 11: Dark absorption features for two cases denoted by red and blue lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The choice of model parameters is given in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The grey band shows the 99% CI for the EDGES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' C10 ∝ T 4 χ in case 1 results in a narrower right edge compared to C10 ∝ T 2 χ in case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Parameter Case 1 Case 2 Mass (mχ (MeV)) 1 1 Binding energy (Ebinding (MeV)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='06 Transition frequency (ν0 (GHz)) 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Transition temperature (T∗ (K)) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 Radiative coupling (αA) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35 Collisional coupling � a1 � cm3s−1� , β, zsat � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='18 × 10−22, 2, 2700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='49 × 10−23, 4, 2000 Table 2: Two sets of parameters that can give the amplitude and width required by the EDGES experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 The EDGES anomaly The EDGES collaboration [35] reported a strong absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This feature is almost twice in amplitude compared to the maximum possible signal expected from the 21 cm transitions in neutral hydrogen during the cosmic dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we propose that this anomalous signal is caused by the absorption of CMB photons by composite dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, we show below that we can get an absorption feature which has the same central frequency, amplitude, and width as the EDGES signal by a suitable plausible choice of dark matter model parameters listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Although there are many free model parameters that decide the final shape of the signal, we would like to emphasize a few important points: The left (low frequency) edge of the signal is entirely decided by the bremsstrahlung pro- cess which erases the absorption by dark matter at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This also fixes the – 24 – location of the maximum absorption which happens around the redshift of recombina- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In particular, the rapid decrease in bremsstrahlung efficiency around recombination provides a sharp edge to the signal at the low frequency end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The right (high frequency) edge of the signal is decided by the strength of inelastic collision cross-section relative to the radiative coupling of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We find that it is relatively easy to get a narrow absorption feature, at least close to the peak, which has the same amplitude as EDGES as shown in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The shape of the high frequency right edge is a strong function of the temperature dependence of the collision cross-section which can be tuned by having the collision cross-section to depend weakly or strongly on the dark matter temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 General predictions for the shape of the dark absorption feature To understand the role of different model parameters in determining the shape of the global absorption signal, we vary each model parameter one by one keeping all the other parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We keep the fiducial parameters as given in column 1 of table 2, however the qualitative behavior and results are valid for any ν0 in the Rayleigh Jeans part of the CMB spectrum at recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The variation in the absorption feature of the CMB for different dark matter model parameters is shown in figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Mass: Given the abundance of dark matter, smaller mass of dark matter implies a higher dark matter number density which increases the strength of the absorption signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Binding energy: Dark matter with a higher binding energy decouples earlier (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) of section 4), resulting in a lower dark matter temperature Tχ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Since the dark matter inelastic collisional coupling ∝ T β χ , where β > 0 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10), smaller dark matter temperature implies weaker collisional coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A weaker collisional coupling compared to radiative coupling drives Tex → Tγ earlier, resulting in smaller amplitude of the signal for higher binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Transition frequency: When the transition frequency is varied, there is an overall shift in the position in frequency of the absorption signal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Moreover the bremsstrahlung rate eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='15) is a sensitive function of the transition frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As we go to lower frequencies, the bremsstrahlung is more efficient in erasing the absorption signal resulting in a smaller amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Inelastic collision cross-section: The high frequency or right edge of the signal is decided by the relative strength of collisional versus radiative coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If we change the power law index (β) of the collision cross-section while keeping the amplitude at zrec = 1100 fixed, a higher β would mean stronger collisional coupling at z > zrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus we see stronger absorption as we increase β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Radiative coupling: A higher spontaneous emission rate compared to collisional transi- tion rate couples Tex → Tγ earlier, shifting the absorption signal to higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The bremsstrahlung process is more efficient in erasing the signal at higher redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Hence we see a smaller amplitude of the absorption signal at higher values of αA (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) for the definition of αA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 25 – 5000 1200 700 500 400 330 Redshift 100 200 300 400 500 Frequency (MHz) −2 −1 0 Brightness temperature (K) mχ(MeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01 1 10 (a) Mass of dark matter 5000 2000 1200 900 700 Redshift 50 100 150 200 250 Frequency (MHz) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Brightness temperature (K) Ebinding(MeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='31 (b) Binding energy 5000 2000 1200 900 Redshift 50 100 150 200 Frequency (MHz) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Brightness temperature (K) ν0(GHz) 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 (c) Transition frequency 5000 2000 1200 900 Redshift 50 100 150 200 Frequency (MHz) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Brightness temperature (K) β 1 2 3 (d) Inelastic collision cross-section 5000 2000 1200 900 700 600 Redshift 100 200 300 Frequency (MHz) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Brightness temperature (K) αA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 (e) Radiative coupling Figure 12: Role of different model parameters in deciding the shape of the global absorption signal from dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We vary each parameter at a given time keeping all the other parameters fixed at the case 1 choice given in table 2 (solid blue line in figures 11 and 12 are identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 26 – 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 General predictions for the dark absorption feature in different parts of the CMB spectrum Irrespective of EDGES, composite dark matter predicts an absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our choice of transition frequency was motivated by the EDGES observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In general for a different transition frequency ν0 and absorption redshift z0, the absorption feature will be appear in a different part of the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The upcoming experiment called the Array of Precision Spectrometers for the Epoch of RecombinAtion (APSERA) [92], which aims to detect the recombination lines in the CMB spectrum will also be sensitive to the dark absorption feature in the 2-6 GHz frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Any dark absorption feature originating at z0 ≳ 2×106 in the CMB cannot be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This is because Compton scattering [93] along with the photon number changing processes like bremsstrahlung and double Compton scattering [94–97] are efficient in erasing any de- viations from the black body spectrum till z ∼ 2 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As the bremsstrahlung and double Compton scattering rates fall ∝ ν−2 with frequency, they decouple at z ∼ 2×106 for photons having x ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='01 [94, 98–103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' With only Compton scattering efficient in z ∼ 2 × 106 − 105 range, the equilibrium spectrum is the Bose-Einstein spectrum and the resulting deviations from blackbody are created in the form of µ-type spectral distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If the absorption hap- pens at z ≲ 105, we will have a broad absorption feature in the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The COsmic Background Explorer/Far-InfraRed Absolute Spectrophotometer (COBE/FIRAS) [104] ex- periment strongly limits the CMB spectral distortions in 60−600 GHz band (see appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus any absorption happening at z0 < 2 × 106 corresponding to 60 < ν0(GHz) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 × 109 range will be strongly constrained by COBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In addition, CMB photons having x ∼ 50 correspond to different energy states of hydro- gen and helium in 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 − 50 eV range at recombination (z ∼ 1100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If dark matter absorbs these photons, there would be fewer CMB photons that can excite and ionize hydrogen and helium speeding up recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' An early recombination would modify the position and amplitude of angular peaks and the Silk-damping tail of the CMB anisotropy power spec- trum which is strongly constrained by Planck [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We leave the detailed analysis of the constraining power of different CMB observations on our dark matter model to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 4 Observational constraints A crucial step towards establishing the viability of our dark matter model involves utilizing the current astrophysical and cosmological data to constrain the different model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We divide this section into two parts, first we look at the early Universe constraints coming from CMB observations and then we move towards the late Universe constraints from the Milky Way galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Note that we do a conservative analysis to find the absolutely allowed parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The goal here is not to derive the best constraints on our model but rather to show that a significant and interesting part of the parameter space is allowed and has unique signatures in experiments as explained before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1 Early Universe constraints At early times, the Universe is dominated by the cosmic microwave background radiation and the precise measurements of the CMB allow us to strongly constrain any possible electro- magnetic interaction dark matter had in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' when the temperature of the DM-SM plasma is higher than the binding energy of the composite dark matter particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' dark ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='matter exists in the form of electromagnetically charged free dark quarks which thermally ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='– 27 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mass of dark matter (keV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Binding energy (keV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='CMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Milky Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mass of dark matter (keV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='⟨σv⟩ (cm2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Bullet cluster ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Milky Way ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='106 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Mass of dark matter (keV) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='αA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='DM cooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='DM emission ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='Figure 13: We show the absolutely allowed parameter space for the different astrophysical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='parameters (left: binding energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' center: collision cross-section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' and right: radiative coupling) for our composite dark matter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The colored region in the plots is ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We choose the transition frequency ν0 = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 GHz which is motivated to solve the EDGES anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' couple the dark matter plasma to the baryon-photon fluid via Coulomb scattering and Comp- ton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such electromagnetic couplings of dark matter to baryon-photon plasma are strongly constrained by the CMB observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The Planck experiment [105] is sensitive to angular scales up to l ∼ 3000 or co-moving wave numbers less than k ≈ l/rLSS ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='22 Mpc−1 which enter the horizon at redshift z∗ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 × 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' These considerations demand that the rate of Coulomb and Compton scattering must be less than the Hubble expansion rate at redshift z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Using these conditions, we find that the electromagnetic charge ϵ of free dark quarks must obey an upper bound, ϵ ≲ 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We can avoid these constraints on ϵ if dark matter recombines into a stable and neutral composite state by z ≥ z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As the Universe expands and the CMB temperature falls, the peak of the CMB blackbody spectrum shifts towards lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' There are fewer high energy photons left in the Wein tail of the CMB spectrum that can ionize the composite dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Assuming the recombination history of hydrogen and dark matter to be similar, Binding Energy kBTγ(z = 1100) ���� hydrogen ≈ 50 ≈ Binding Energy kBTγ(z = z∗) ���� dark matter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) This imposes a lower bound on the binding energy of dark matter, Ebinding ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 keV, as shown in the first panel of figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this work, we assume that dark matter recombines before z∗ and kinematically decouples from the thermal plasma thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also note that the photo-ionization cross-section of dark matter ∝ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This together with the high binding energy implies that dark matter will not be reionized in most astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 Late Universe constraints from Milky Way Consistency with the precise observations of Milky Way (MW) gives additional constraints on the properties of dark matter at late times, in particular the dark matter self-interactions and the strength of electromagnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We derive these constraints for two extreme scenarios, one where the dark matter self-interaction cross-section is very weak such that it remains collisionless in the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In the other case, we consider the inelastic collisions of dark matter to be strong enough such that the states corresponding to electromagnetic transitions are in kinetic equilibrium with the virial temperature of the MW halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 28 – (i) Collisionless dark matter: If the timescale of dark matter collisions denoted by tcollision = (nχ⟨σv⟩)−1 in our local neighborhood (ρχ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 GeV cm−3) is longer than the age of the Milky Way (tMW ≈ 13 billion years [106]), the dark matter stays collisionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This puts an upper limit on the dark matter collision cross-section (elastic + inelastic), ⟨σv⟩ < 1 tMW �mχ ρχ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) We show this constraint in the second panel of figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also show constraints from Bullet cluster [56] (σ/mχ < 2 cm2g−1) assuming the relative velocity between the two clusters v ≈ 4700 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This indirectly puts constraints on dark matter inelastic collision cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We note that our back of the envelope constraints from MW are very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A more careful analysis taking into account the non-negligible dark matter self-interactions which maybe preferred by data, would relax these constraints [107, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (ii) Collisional dark matter: The observations in fact allow dark matter self-interactions as long as they do not significantly disturb the dark matter halo profile, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' by collapsing the halo into a disk [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We assume here the other extreme case, where dark matter self-interactions are strong enough that the excitation temperature of the two states re- sponsible for transitions are in kinetic equilibrium with the virial temperature of the MW halo i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Tex = Tvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark matter particles in the MW halo are in virial equilibrium at a temperature Tvir ∝ mχ (calculated using the virial theorem) and have an energy ∼ kBTvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For the dark matter to not ionise into charged dark quarks, the energy exchange in each inelastic collision (∼ kBTvir) must be less than the binding energy of dark matter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This imposes a lower bound on the binding energy as shown in the first panel of figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We note that this bound is weaker compared to the CMB bound derived in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Therefore we consider the CMB bound in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In a collisional excitation, the kinetic energy of dark matter particles is converted into its internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The excited dark matter particles can then de-excite by sponta- neously emitting a photon, converting the internal energy of dark matter into radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This would lead to a gravitationally unstable dark matter halo, which cools and starts collapsing into a disk similar to baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' From GAIA observations [110] such a scenario is ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The cooling timescale tcooling for this process is given by the ratio of the thermal energy density U in dark matter to the radiative cooling rate C, tcooling ≈ U C ≈ 3 2nχkBTvir C , where C = n1A10kBT∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) The MW dark matter halo remains gravitationally stable if the cooling timescale is longer than the age of MW [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This puts an upper limit on the radiative coupling of dark matter (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='12) for the definition of αA), αA < 3 2 Tvir T∗ 1 tMWAHI 10 � 1 + g0 g1 e T∗ Tvir � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) In this analysis, we have assumed that dark matter only cools via the dark hyper-fine transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In principle, if the levels with higher energies can be collisionally excited – 29 – and have a higher spontaneous emission rate, they will contribute to the cooling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' As discussed in section 2, in this model we assume that the other transitions correspond to very high energies which cannot be excited collisionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We plot this limit on αA in the third panel of figure 13 for transition frequency 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 GHz or T∗ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This choice of parameters was motivated by the EDGES observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The collisionally excited dark mater particles in the MW halo can de-excite by spon- taneously emitting a photon, creating a background radiation with a dipole anisotropy owing to our off-center position 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3 kpc [111] away from the MW halo center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The experiments like WMAP and Planck [105, 112] measure the fluctuations in the sky temperature in broad frequency bands in the frequency range of 10-500 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Considering j to be the integrated specific intensity and javg be the average integrated specific intensity due to dark matter emission (see appendix G for details), the temper- ature fluctuations along a given LoS in a particular frequency band (νmin to νmax with a central frequency νc) is given by, δTνc = � |j − javg| νmax − νmin � c2 2kBν2c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) We note that this dipole will be aligned with the galactic plane and thus obscured by Galactic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Also this dipole will appear in only one channel that includes ν0 and would be absent in other channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Without doing a detailed analysis, we can just require very conservatively that this dipole must be smaller than the cosmological dipole δTνc < 3 mK, imposing an upper bound on αA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The constraints on αA from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) are shown in the third panel of figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In our analysis, we have chosen ν0 = 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 GHz which falls in the frequency band of Planck centered at νc = 143 GHz with a bandwidth ∆ν/ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We note that CMB is the most precisely measured part of the electromagnetic spectrum over the whole sky and constraints from other parts of the spectrum (for other values of ν0) would be much weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We note that these constraints only apply if the collision cross-sections for these transitions are large enough such that the levels are in kinetic equilibrium at MW virial temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' These constraints would get relaxed significantly for weaker collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We also note that the collision cross-sections usually have a strong temperature dependence which suggests that dark matter collisions may be insignificant in MW but maybe important in other dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 5 Conclusions In this work, we explore the unique experimental signature of a class of composite dark matter models having electromagnetic transitions: the absorption lines in the spectrum of a background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such absorption signatures occur as distinct dark lines in the spectrum of a quasar, and as a global absorption feature or spectral distortion in the spectrum of the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption of light by multiple dark matter halos along the LoS to a quasar gives rise to a dark forest, similar to the Lyman-α forest from neutral hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In contrast to the Lyman-α or 21 cm forest, the dark forest is less prone to the uncertainties related to the non-linear baryonic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 30 – The dark forest is sensitive to the self-interactions of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark forest is also sensitive to the minimum halo mass and therefore the primordial power spectrum on small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Absorption of CMB photons by dark matter gives rise to µ-type spectral distortions at z ≳ 105 and a global absorption feature in the CMB spectrum at z ≲ 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The bremsstrahlung process plays an important role if the dark matter absorption fre- quency falls in the Rayleigh-Jeans tail of the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In this case, bremsstrahlung determines the high redshift edge of the global absorption feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The low redshift shape of the global signal is determined by the relative strength of inelastic collisional coupling with respect to the radiative coupling of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our dark matter model (with a suitable choice of parameters) can reproduce the absorp- tion feature that matches the frequency location, width, and amplitude but not the exact flattened shape of the anomalous signal measured by the EDGES collaboration [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' If the EDGES signal is indeed produced by dark matter absorption, then our model predicts that a dark forest must exist in a frequency band ten times higher than the 21 cm forest frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' A global absorption feature or spectral distortion due to dark matter is a general prediction of our model which can even occur in a different frequency band of the CMB spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Such signals could be detected in future experiments such as Primordial Inflation Explorer (PIXIE) [113] or APSERA [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The absorption of CMB photons in the UV band can affect the recombination history and this part of the parameter space can be probed by the CMB anisotropy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We have ignored the clustering of halos in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' One needs to perform N-body simu- lations to take into account the two point correlations of the dark forest which we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our results open up a new and unexpected interesting window into the nature of dark matter and a potential new probe of the cosmic web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Our work motivates the search for dark forest and global absorption features across the full electromagnetic spectrum, especially in the desert i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' the part of the electromagnetic spectrum where there are no strong lines expected from baryons in the standard model of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' With the current sensitivity of spectroscopic surveys in different frequency bands, it should already be possible to probe the dark forest at optical and radio-wave frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Acknowledgments We acknowledge the computational facilities offered by the Department of Theoretical Physics at TIFR, Mumbai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' This work is supported by Max Planck Partner Group for cosmology of Max Planck Institute for Astrophysics Garching at TIFR funded by Max-Planck-Gesellschaft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We acknowledge the support from Department of Atomic Energy (DAE) Government of India, under Project Identification No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' RTI 4002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' AG is thankful to Yidong Xu, Steven Furlanetto, Girish Kulkarni, Shadab Alam, Nissim Kanekar, Aritra Kumar Gon and Shikhar Mittal for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 31 – A Dark matter model By construction, the dark sector model is described by an asymptotically free non-abelian SU(N) gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At UV energies, the theory is described by weakly coupled dark quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The necessary d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' are given in table 1 in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At low energies, SU(N) is strong and spectra of the dark sector is given in terms dark color neutral bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The formulation we discuss here closely follows the chiral Lagrangian techniques developed in particle physics [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' One essential difference though is that there is no analogous weak force present in the dark sector which allows the heavy-light bound dark matter states to decay in comparison to the heavy mesons of the visible sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We begin this section with the spontaneous breaking of the chiral symmetry of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' At low energies, the strong dark QCD generates the flavor diagonal condensate ⟨qDa qc Db⟩ = δab Λ3 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The flavor symmetry gets spontaneously broken in the following fashion, SU(2)D L × SU(2)D R → SU(2)D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) This symmetry breaking results in three pNGBs/dark pions πD parameterized in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In order to discuss the physics of heavy-light bound states, it is convenient to introduce a field ξD defined via, ΣD(x) = ξ2 D(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' with ξD(x) SU(2)D L ×SU(2)D R −−−−−−−−−−−→ LD ξD(x) V † D(x) = VD(x) ξD(x) R† D, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) where VD(x) is the transformation operator for the unbroken vectorial symmetry SU(2)D V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We choose our dark matter candidate to be a heavy-light bound state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' To do this, we first redefine the heavy and light dark quark Dirac spinors in the following way, QD ≡ �QDw Qc † Dw � , qD ≡ � ξ† DqDw ξD qc † Dw � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) where QDw and qDw are the Weyl spinors for the heavy and light dark quarks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The dark matter candidate is defined by the matrix field X as, Xa ≡ ¯QD qDa SU(2)D L ×SU(2)D R −−−−−−−−−−−→ VDab(x)Xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) In order to construct the operators responsible for interactions between dark bound state and dark pions, we define the vector current VDµ and the axial current ADµ which incorporate the dark pions πD, VDµ = 1 2(ξ† D∂µξD + ξD∂µξ† D), ADµ = 1 2(ξ† D∂µξD − ξD∂µξ† D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) Since the dark state X is a non-relativistic heavy-light bound state, its velocity v is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Using the framework of HQET, we can project the small scale fluctuations [61, 62] and write the Lagrangian for Xv invariant under the full flavor symmetry SU(2)D L × SU(2)D R, LXv = i tr � ¯ X a v vµDab µ X b v � + ig1 tr � ¯ X a v γµγ5 Aab µ X b v � + 1 4f2 D ∂µΣab∂µΣ† ba +g2 f2 DΛNmqδab Σab + g3 mqδab tr � ¯ X a v X b v � ξD ξD + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) – 32 – where the covariant derivative is defined as Dµ = ∂µ + VDµ and the trace is taken over the gamma matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The Lagrangian in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) does not distinguish the spin 1 state χ∗ from the spin 0 state χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The terms that violate the spin symmetry occur as powers in 1/MQ and contribute to the mass gap between χ and χ∗ resulting in the hyper-fine splitting of the dark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The tree level operators that generate this mass gap are given by, λ MQ tr � ¯ X a v σµν X a v σµν � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' λ′ mq MQ tr � ¯ X a v σµν X a v σµν � ξDξD .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) We can decompose the matrix field Xv into a pseudoscalar field and a vector field using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) and compute the tree level hyper-fine splitting from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Further, the axial current operator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) gives rise to χ∗χ∗πD and χχ∗πD couplings which also contribute to the hyper-fine splitting at the loop level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' B Dark matter halo density and temperature profile The halo mass density is modeled as an NFW profile [16] with a concentration model taken from [114, 115] to describe the halo up to 10 times the virial radius [116] of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The halo mass Mh is defined as the total matter enclosed within a sphere around the halo center that encloses a fixed density equal to 200 times the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Thus r200c is defined as, M200c(r < r200c) 4πr3 200c/3 = 200ρc(z) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) where halo mass Mh = M200c = � r200c 0 d3r ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' For simplicity, we assume the dark matter to follow a Maxwell-Boltzmann velocity distribution with a radially dependent rms width σ described by the power law profile in [117], ρ σ3 (r) = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='46 �ρc(z) v3 vir �� r rvir �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) where r is the radius of the halo in physical units, ρc is the critical density of the Universe, ρ(r) is the NFW density profile of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' vvir and rvir are the virial velocity and virial radius of the dark matter halo respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' One can associate a temperature to the width of the velocity distribution using the relation, σ2(r) = 3kBTh(r)/mχ, which gives, Th(r) = mχ 3k �101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='46 ρ(r) �ρc(z) v3 vir �� r rvir �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9�−2/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) The random motion of dark matter particles along the LoS gives rise to a Doppler line profile whose line width in units of speed is given by, b(r) = c∆νD ν0 = (1 + vLoS(r, p)/c) � 2kBTh(r) mχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) – 33 – C Dark forest We show the sample dark forest spectra in the redshift range 7-0 for collisionless and collisional DM for three different choices of Mmin = 106, 105 and 104M⊙/h respectively keeping other dark matter model parameters fixed at: mχ = 1 MeV, T∗ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 K, and αA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 21 22 23 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' transmission (e−τ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 24 25 26 27 Frequency (GHz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 28 30 32 Frequency (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' transmission (e−τ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 45 50 55 Frequency (GHz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 60 70 80 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Redshift 100 120 140 160 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Relative transmission Figure 14: Dark forest spectrum for collisionless DM in for Mmin = 106M⊙/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 34 – 6.' metadata={'source': 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25 26 27 Frequency (GHz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 28 30 32 Frequency (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 3.' metadata={'source': 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5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 24 25 26 27 Frequency (GHz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' transmission (e−τ) 2.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' transmission (e−τ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 24 25 26 27 Frequency (GHz) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 28 30 32 Frequency (GHz) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' transmission (e−τ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 45 50 55 Frequency (GHz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='00 Redshift 60 70 80 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Redshift 100 120 140 160 Frequency (GHz) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='0 Relative transmission (e−τ) Figure 17: Dark forest spectrum for collisional DM for Mmin = 104M⊙/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' D Convergence of distribution functions We compare the τpeak and b2 distribution functions for collisionless DM in the redshift range 6-5 for 100 and 1000 LoS directions in figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The mean count N in a given τpeak or b2 bin follows a Poisson distribution with a standard deviation ∝ √ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' When the average is taken over spectra from 1000 different LoS directions, we find that the tail becomes smoother and the error bars on the distribution functions contract as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' E Radiative transfer equation in an expanding Universe Let uν(Ω) be the radiation energy density per unit frequency per unit solid angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' In terms of the photon distribution function f(ν), uν(Ω)dν dΩ = hνf(ν)(h/c)3ν2 dν dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) In an expanding Universe, frequency redshifts as 1/a, where a as the scale factor, while f(ν) remains conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' in the absence of a source or sink,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' uν(Ω)/ν3 or uν(Ω)a3 – 37 – 10−2 10−1 100 τpeak 10−3 10−2 10−1 100 101 102 dN/dτpeak Mmin = 106M⊙/h z : 6 - 5 nseed = 100 nseed = 1000 100 101 102 103 b2 (km2/s2) 10−3 10−2 10−1 100 101 102 dN/d(b2[km2/s2]) Mmin = 106M⊙/h z : 6 - 5 nseed = 100 nseed = 1000 Figure 18: The distribution functions for peak optical depth (left) and line width (right) for collisionless DM for Mmin = 106M⊙/h in the redshift range 6-5 for 100 and 1000 LoS directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' remains constant with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' d(uνa3) dt = 0, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) a3 ∂uν ∂t + a3 �dν dt �∂uν ∂ν + 3a2 ˙auν = 0, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) a3 ∂uν ∂t − νa2 ˙a∂uν ∂ν + 3a2 ˙auν = 0, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) ∂uν ∂t − ν ˙a a ∂uν ∂ν + 3 ˙a auν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='5) In the presence of a source having emission coefficient jν and absorption coefficient αν, the radiative transfer equation becomes [118], ∂uν ∂t − ν ˙a a ∂uν ∂ν + 3 ˙a auν = −cανuν + jν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) The specific intensity Iν, defined as the energy per unit area per unit time per unit frequency per unit solid angle is related to the energy density per unit frequency per unit solid angle as uν = Iν/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The specific intensity in temperature units is given by the following relation, cuν = Iν = 2ν2 c2 kBTb, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) where Tb is the brightness temperature at a fixed frequency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' We can use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) to cast eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='6) in terms of the brightness temperature [89, 119], dTb dt + ˙a aTb = cαν � − Tb + c2 2ν2kB jν αν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='8) If the level population defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='7) is in kinetic equilibrium at temperature Tex, by using the Einstein relations we can find the ratio between the emission and absorption coefficients – 38 – 100 200 300 400 500 600 Frequency (GHz) 10−5 10−4 10−3 10−2 10−1 ∆Iν/Iν Figure 19: 1-σ limits on the fractional deviation from the CMB blackbody spectrum (∆Iν/Iν) as a function of frequency from COBE/FIRAS data [104, 120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' which is given by [89], jν αν = 2hν3 c2 1 (ehν/kBTex − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='9) This relation following from the principle of detailed balance depends only on the excitation temperature Tex of the two levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' F Limits on CMB spectral distortions from COBE/FIRAS We show the limits on spectral distortions of CMB from a perfect blackbody from COBE/FIRAS [104, 120] in the 60-600 GHz band in figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' G Milky Way model for constraining the radiative coupling from CMB anisotropy maps We assume that dark matter in MW is distributed according to the following NFW [16] density profile, ρχ(r) = ρ0 1 r r0 � 1 + ( r r0 ) �2 , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='1) with ρ0 (106 M⊙ kpc−3) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='486 and r0 (kpc) = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='77 [79] and has a virial temperature Tvir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' The emissivity due to dark matter emission from MW along the LoS is given by [89], jν = hν 4π φ(ν)n1(r)αAAHI 10, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='2) where ν is the frequency of the emitted photon and φ(ν) is the Doppler line profile centered at ν0 with a line width, ∆νD = ν0 c � 2kBTvir mχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='3) – 39 – For a band covering a frequency range from νmin to νmax, the integrated specific intensity along a particular LoS is given by, j = � νmax νmin dν jν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='4) – 40 – References [1] Kimberly K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Boddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Astrophysical and Cosmological Probes of Dark Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' arXiv e-prints, page arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='06380, March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='06380, [ADS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Workman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' and Particle Data Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Review of Particle Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Progress of Theoretical and Experimental Physics, 2022(8):083C01, August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [DOI], [ADS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [3] David E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Kaplan, Gordan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Krnjaic, Keith R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Rehermann, and Christopher M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Atomic dark matter.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' Astrophysics for Physicists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [ADS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' [120] CMB Monopole Spectrum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' URL: https://lambda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='gov/product/cobe/firas_monopole_get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} +page_content=' – 48 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf'} diff --git a/X9E3T4oBgHgl3EQf1guW/content/2301.04747v1.pdf b/X9E3T4oBgHgl3EQf1guW/content/2301.04747v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7a07fcfbe060e8cb2eb01897e53a698dcfae9510 --- /dev/null +++ b/X9E3T4oBgHgl3EQf1guW/content/2301.04747v1.pdf @@ -0,0 +1,3 @@ +version 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Franka,1, G.V. Kulina,2, M.A. Zakharova, +a Joint Institute for Nuclear Research, Dubna, Russia +The paper considers the concept of an ultracold neutron source (UCN) based +on the deceleration of very cold neutrons (VCN) by a local decelerating device. As +the latter, it is proposed to use a gradient spin flipper. It is shown that in this +case, the flux of VCNs, which after deceleration are converted into the UCN, has a +pulse structure. In this case, the duration of neutron bunches can be significantly +less than their repetition period. Accordingly, the density of the neutron flux in +the bunch will significantly exceed the average value. This opens up the possibility +of pulse filling of the UCN trap, without preliminary time focusing. +PACS: 29.25.Dz Neutron sources; 03.75.Be Atom and neutron optics +INTRODUCTION +It is known that ultracold neutrons (UCN) were first observed by Shapiro’s +group in an experiment performed at a reactor with an average power of 6 +kW [1,2]. Probably, it was then that there was an understanding of the im- +portance of the fact that the pulse density of the UCN generated by a periodic +source can significantly exceed the average value. The question arose how +to take advantage of this circumstance. A possible solution to this problem +was soon proposed in [3]. It consists in filling the UCN trap only during the +pulse and effectively isolating it the rest of the time. Ideally, when there are +no losses, the density of the UCN in the trap will correspond to the pulsed +neutron density, which may be several orders of magnitude higher than the +average in time. +Unfortunately, this idea has not yet been implemented, although the prob- +lem of using pulsed rather than medium UCN density has become even more +urgent due to the creation of new pulsed neutron sources [4–6]. The IBR +2 – IBR 2M pulse reactor [7, 8] with an average power of 1.5–2 MW and a +pulse flux of about 1016 n/cm2 has been successfully operating in Dubna for +many decades. The design of a new "Neptune" reactor [9,10] with a signifi- +cantly large pulse flux is underway. The construction of the European pulsed +neutron source (ESS) is also close to completion [11]. +The implementation of the idea of pulse filling of the trap is hindered by +the fact that in practice it is remote from the moderator due to the presence +of biological protection. In this case, it is necessitates the appearance of a +transport neutron guide several meters long, feeding the trap. The placement +1E-mail: frank@jinr.ru +2E-mail: kulin@jinr.ru + +2 +of an insulating valve near the moderator – the source of the UCN, causes +the neutron guide to become part of this trap. Due to the small transverse +size of the neutron guide, the frequency of neutron collisions against its walls +is large enough, which greatly reduces the storage time of the UCN in the +trap–neutron guide system and significantly reduces the density of neutrons +accumulated in the trap. Placing the valve at the entrance to the trap, several +meters away from the source, is useful only in the case of sources with a low +repetition rate [4–6]. For sources with a repetition rate of several hertz, the +spread of the UCN transit times from the source to the trap will exceed the +intervals between pulses, and the presence of a valve at the entrance to the +trap does not make sense. +To solve the problem of pulsed filling of a remote trap, it was proposed +to use a special device - a time lens that dose-changes the energy of neutrons +as they come to the lens [12,13]. Such a device makes it possible to restore +the pulsed structure of the neutron beam immediately before entering the +trap. An important question is the method of changing the neutron energy +according to a given time law. In this regard, in +[12, 13] it was proposed +to turn to quantum nonstationary phenomena. Among the latter, the phase +modulation of a neutron wave, across the direction of propagation of which +a phase diffraction grating moves, and the resonant the neutron spin flip in +a magnetic field were considered. +Later, nonstationary diffraction of the UCN by a moving grating was +observed in the experiment [14] and some time later, in experiments with a +moving grating, the effect of focusing in time was also demonstrated [15,16]. +The possibility of time focusing based on the resonant the neutron spin flip +has also found its experimental confirmation [17,18]. +The concept of a UCN source on a periodic pulsed reactor, based on the +use of the time lens with pulse filling of the UCN trap, was considered in a +recent paper [19]. A similar approach was proposed in [20], in which it was +proposed to focus very cold neutrons (VCN) with velocities of about 50 m/s, +followed by their deceleration in a escaping trap. Such a deceleration method +was proposed in [21], but has not yet been applied in practice. +The extraction of neutrons with higher speeds than those of the UCN from +the moderator-converter provides better conditions for the transportation of +neutrons and allows to use more efficient converter. +In the UCN source +of the Institut Laue-Langevin [22], neutrons are slowed down rising to a +height of several meters, followed by Doppler "cooling" when reflected from +an escaping mirror. The deceleration of neutrons in the Earth gravity field +during their transportation in a vertical neutron guide was also successfully +used in the UCN sources at the WWR-M reactor of PNPI [23]. However, in +the case of pulsed neutron generation, the deceleration of VCN may lead to +some new and important consequences. The present work is devoted to their +discussion. + +3 +DECELERATION OF NEUTRONS GENERATED BY A PULSED +SOURSE USING A LOCAL DEVICE +We will consider a UCN source in which, for a relatively short time, pulse +generation of very cold neutrons occurs and their subsequent transportation +through a mirror neutron guide. +Suppose that a device slowing down neutrons is used, the purpose of +which is to produce ultracold neutrons, whose energy after deceleration is +small enough so that they can be stored in a material trap. Let’s call this +device a decelerator to avoid the term "moderator" widely used in neutron +physics. In contrast to the case considered in [20], we will assume that the +deceleration of neutrons by the decelerator occurs in a relatively short section +of their transport in close proximity to the trap. +To store in the trap the neutrons obtained due to deceleration, their +full velocity V should not exceed the boundary velocity of the trap matter +V < Vb = +� +2U/m, where U is the effective potential of the trap walls. +Since before getting into the trap, neutrons must pass a considerable +path in the neutron guide, which we assume to be a mirror, the transverse +velocity of neutrons normal to the surface of the walls of the latter is limited +by the boundary energy value of the neutron guide walls Egd, so that v⊥ < +� +2Egd/m. Having obtained a limit for the total and transverse velocity of +neutrons capable of being stored in a trap, we thereby obtained a limit for +the longitudinal velocity of such neutrons directed along the Z axis of the +neutron guide. +vzfin = +� +V 2 +b − v2 +⊥. +(1) +The decelerator, which forms the neutron flux immediately before neu- +trons enter the trap, changes the neutron energy by a certain amount ED. If +it is constructed correctly, then this change in kinetic energy is mainly due +to a change in the longitudinal velocity of neutrons. +Being interested in the future only in the distribution of the longitudinal +velocity of neutrons and the kinetic energy associated with it, we will skip +below the z index of the quantities we are interested in. Then the energy +of neutrons entering the trap and being able to be stored in it lies in the +range from zero to Efin = mv2 +fin/2. Before deceleration , the energy of these +neutrons should be in the range ED < E < Efin + ED. At a sufficiently large +energy value ED, the range of neutron energies that can be trapped after +deceleration can be much smaller than the energy itself δE ≈ Efin ≪ ED. +But this means that the spread of the flight times δt from the pulsed source +to the decelerator and, accordingly, to the trap, of the "useful" neutrons of +interest to us are not only small, but may be much shorter than the time of +flight itself t = L/V , where L is the length of the transport neutron guide +δt +t = δV +V +≃ δE +2E ≃ Efin +2ED +≪ 1. +(2) + +4 +Under good conditions, δt can also be significantly less than the pulse +repetition period of the reactor T. In this case, the flux of "useful" neutrons, +which after deceleration will be converted to the UCN, will have a pulsed +structure, although during transport through the neutron guide, the pulse +duration will inevitably increase due to velocity dispersion δV . +Let us now consider the question of the fluxes of UCN coming into the +trap. Obviously, the process of deceleration does not affect the number of +neutrons in any way and, consequently, the decelerator itself does not change +the flux of neutrons with the corresponding spin projection. Therefore, for- +getting about polarization for now, and assuming that the transmission of +neutron guides for neutrons with the above velocity distribution is ideal, +let’s compare the flux of UCN Φ1z that would enter the trap directly from +the source with the flux of "useful" neutrons Φ2z coming from the source to +the decelerator and only then into the trap. Assuming that the distribution +over the "normal" velocities v⊥is the same in both cases, we will continue to +omit the z index, being interested only in the components of the velocities +directed along the beam and the corresponding fluxes. +The flux Φ1 is carried by neutrons with velocities ranging from zero to vfin, +and the flux Φ2 is carried by neutrons with velocities from V1 = +� +2ED/m +to V2 = +� +2 (ED + Efin) /m. Since the UCN energy is very small, then in the +Maxwell distribution for the velocity distribution of the flux in the source +the linear approximation is valid dΦ(V ) = nV dV , where n is the neutron +density. Then for both fluxes we have +Φ1 = n +� √ +2Efin/m +0 +V dV, +Φ2 = n +� √ +2(ED+Efin)/m +√ +2ED/m +V dV. +(3) +It is easy to see that Φ1 = Φ2. +Thus, neutrons entering the trap directly from the source and neutrons +obtained by converting from VCN to UCN carry the same flux, but have a +significantly different temporal and spatial structure. In the first case, the +spread of the flight times δt is much larger than the pulse repetition period +T. In this case, the pulse structure practically disappears and an essentially +uniform flux enters the trap, which corresponds to the average value of the +neutron density. +In the second case, when the duration of the bunch is +significantly less than the pulse repetition period δt/T ≪ 1, the length of the +bunches is less than the distance between them. Accordingly, the neutron +density in the bunch exceeds the average by a value of G = T/δt. +ADIABATIC SPIN-FLIPPER FOR NEUTRON DECELERATION. +QUANTITATIVE ESTIMATES +Let us make some estimates now. +For certainty, as a decelerator, we +consider a flipper in which a spin flip occurs under the action of an alternating +high-frequency field directed perpendicular to a large permanent field. As +such, the so-called adiabatic or gradient flipper can be used [24–26]. Passing + +5 +the flipper, the neutron energy changes by an amount ED = 2µB, where +µ is the magnetic moment of the neutron and B is the magnitude of the +permanent magnetic field, and under good conditions this energy change is +mainly due to a change in the longitudinal velocity of the neutrons. +The energy change at the neutron spin reversal in the such flipper was +demonstrated in [27], and the possibility of creating a flipper with the per- +manent field of the order of 1 T was demonstrated in [17, 18]. There is no +reason to doubt that it is possible to create a flipper with a field of the order +of 15-20 T, which is achievable in modern superconducting systems. +For the final velocity vfin, we assume a value of 3 m/s, so that the total +velocity in a circular neutron guide with a boundary energy of 5.7 m/s does +not exceed 6.5 m/s. This velocity corresponds to an energy of the order of +50 neV. For the magnitude of the magnetic field B, in which the spin flip +occurs, we take the value of 20 T. +When the spin is reversed in such a field, the energy changes by the +value ED = 2.4 × 103 neV. Therefore, the spread of neutron energies, whose +longitudinal velocity after deceleration will not exceed 3 m/s, should be about +50 neV, while the energy itself will be somewhat greater than ED. This energy +corresponds to the neutron velocity of 21 m/s. +From formula (2) follows the estimate +δt +t ≈ 0.01. +(4) +Assuming for a rough estimate the length of the neutron guide is L ≈ +10 m, we obtain for the time of flight and its dispersion the values t = 0.48 s +and δt ≈ 5 ms. +The last of these values determines the duration of the bunch of "useful" +neutrons at the entrance to the flipper-decelerator. This duration δt may +be significantly less than the pulse repetition period of the source T. +In +particular, for the IBR-2 reactor, T = 200 ms. +At the same time, at the direct transport of neutrons from the source +to the trap, the spread of the flight times of the order of the flight time +itself is δt ≈ L/vfin ≈ 3 s, which is an order of magnitude greater than the +repetition period T and the density of neutrons reaching the trap in this case +corresponds to the average value. Remind that if the fluxes are equal, that +is following from (3)l, the compression of a bunch of neutrons by their local +deceleration means an increase in the neutron density proportional to the +value of G. +Above, we assumed that the deceleration time in the flipper is the same +for all neutrons. +This is obviously not the case. +The process of neutron +deceleration in the flipper is caused by the slowing down of neutrons in the +increasing magnetic field when entering it, and then, after the spin flip, the +same slowing down in the decreasing field when exiting the flipper. These +two stages are complemented by some time of flight in the weakly inhomo- +geneous field of the flipper itself. The deceleration time and its dispersion +are determined by the design of the flipper and the range of initial and fi- + +6 +nal neutron velocities. This should be taken into account at its designing. +Rough estimates based on the assumption of the constant neutron decelera- +tion in a region l of an inhomogeneous magnetic field lead to the magnitude +of the dispersion of the deceleration times is δt ≈ 2lV2/V 2 +1 ≈ 10 ÷ 15 ms, +where V1 and V2 are the neutron velocities before and after deceleration. This +value can be reduced if neutrons with the lowest velocities are excluded from +consideration, which will lead to a very insignificant loss in neutron flux. +CONCLUSION +Thus, it is shown that decelerating the neutrons generated by a pulsed +source, with the help of a local device, it is possible to obtain a noticeable gain +in the pulse density of the UCN and without time focusing. The question of +the combination of these two approaches, partially raised in [20], probably +requires additional analysis. +The authors are grateful to E. V. Lychagin, O. V. Karamyshev, S. V. +Mironov, A. Yu. Muzychka and M. S. Novikov for useful discussions. +REFERENCES +1. Luschikov V.I., Pokotilovsky Yu.N., Strelkov A.V., Shapiro F.L. Obser- +vation of ultracold neutrons // Journal of Experimental and Theoretical +Physics Letters. — 1969. — V. 9, no. 1. — P. 23–26. +2. Strelkov A.V. The history of the discovery of ultracold neutrons // +Shapiro F. L. Collection of works. Neutron Studies. — Moscow: Nauka, +2015. — P. 362–364. — in Russian. +3. Shapiro F.L. Remarks on the measurement of phases of structural am- +plitudes in neutron diffraction and on the accumulation of neutrons // +Physics of Elementary Particles and Atomic Nuclei. — 1971. — V. 2, +no. 4. — P. 975–979. — in Russian. +4. Anghel A., Atchison F., Blau B., . et al. The PSI ultra-cold neutron +source // Nuclear Instruments & Methods in Physics Research Section +A-accelerators Spectrometers Detectors and Associated Equipment. — +2009. — V. 611. — P. 272–275. +5. Saunders A., Makela M., Bagdasarova Y. et al. 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Production, stor- +age, and polarization of ultracold neutrons // Soviet Journal of Nuclear +Physics. — 1974. — V. 19, no. 2. — P. 147–152. +25. Luschikov V.I., Taran Y.V. On the calculation of the neutron adiabatic +spin-flipper // Nuclear Instruments and Methods in Physics Research. — +1984. — V. 228. — P. 159–160. +26. Grigoriev S.V., Okorokov A., Runov V. Peculiarities of the construction +and application of a broadband adiabatic flipper of cold neutrons // +Nuclear Instruments and Methods in Physics Research A. — 1997. — V. +384. — P. 451–456. +27. Weinfurter +H., +Badurek +G., +Rauch +H., +Schwahn +D. +Inelas- +tic action of a gradient radio-frequency neutron spin flipper // +Zeitschrift f¨ur Physik B Condensed Matter. — 1988. — V. 72, no. 2. — +P. 195–201. — URL: https://doi.org/10.1007/BF01312135. + diff --git a/Y9AyT4oBgHgl3EQf9frR/content/tmp_files/load_file.txt b/Y9AyT4oBgHgl3EQf9frR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e3561a3307ecf92de0c6a004b39c55cd13fc02e4 --- /dev/null +++ b/Y9AyT4oBgHgl3EQf9frR/content/tmp_files/load_file.txt @@ -0,0 +1,474 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf,len=473 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='00877v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='ins-det] 2 Jan 2023 On the new possibility of pulse accumulation of UCN in a trap A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Franka,1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Kulina,2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Zakharova, a Joint Institute for Nuclear Research, Dubna, Russia The paper considers the concept of an ultracold neutron source (UCN) based on the deceleration of very cold neutrons (VCN) by a local decelerating device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' As the latter, it is proposed to use a gradient spin flipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' It is shown that in this case, the flux of VCNs, which after deceleration are converted into the UCN, has a pulse structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In this case, the duration of neutron bunches can be significantly less than their repetition period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Accordingly, the density of the neutron flux in the bunch will significantly exceed the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This opens up the possibility of pulse filling of the UCN trap, without preliminary time focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' PACS: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='Dz Neutron sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='Be Atom and neutron optics INTRODUCTION It is known that ultracold neutrons (UCN) were first observed by Shapiro’s group in an experiment performed at a reactor with an average power of 6 kW [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Probably, it was then that there was an understanding of the im- portance of the fact that the pulse density of the UCN generated by a periodic source can significantly exceed the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The question arose how to take advantage of this circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' A possible solution to this problem was soon proposed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' It consists in filling the UCN trap only during the pulse and effectively isolating it the rest of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Ideally, when there are no losses, the density of the UCN in the trap will correspond to the pulsed neutron density, which may be several orders of magnitude higher than the average in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Unfortunately, this idea has not yet been implemented, although the prob- lem of using pulsed rather than medium UCN density has become even more urgent due to the creation of new pulsed neutron sources [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The IBR 2 – IBR 2M pulse reactor [7, 8] with an average power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='5–2 MW and a pulse flux of about 1016 n/cm2 has been successfully operating in Dubna for many decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The design of a new "Neptune" reactor [9,10] with a signifi- cantly large pulse flux is underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The construction of the European pulsed neutron source (ESS) is also close to completion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The implementation of the idea of pulse filling of the trap is hindered by the fact that in practice it is remote from the moderator due to the presence of biological protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In this case, it is necessitates the appearance of a transport neutron guide several meters long, feeding the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The placement 1E-mail: frank@jinr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='ru 2E-mail: kulin@jinr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='ru 2 of an insulating valve near the moderator – the source of the UCN, causes the neutron guide to become part of this trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Due to the small transverse size of the neutron guide, the frequency of neutron collisions against its walls is large enough, which greatly reduces the storage time of the UCN in the trap–neutron guide system and significantly reduces the density of neutrons accumulated in the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Placing the valve at the entrance to the trap, several meters away from the source, is useful only in the case of sources with a low repetition rate [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' For sources with a repetition rate of several hertz, the spread of the UCN transit times from the source to the trap will exceed the intervals between pulses, and the presence of a valve at the entrance to the trap does not make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' To solve the problem of pulsed filling of a remote trap, it was proposed to use a special device - a time lens that dose-changes the energy of neutrons as they come to the lens [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Such a device makes it possible to restore the pulsed structure of the neutron beam immediately before entering the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' An important question is the method of changing the neutron energy according to a given time law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In this regard, in [12, 13] it was proposed to turn to quantum nonstationary phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Among the latter, the phase modulation of a neutron wave, across the direction of propagation of which a phase diffraction grating moves, and the resonant the neutron spin flip in a magnetic field were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Later, nonstationary diffraction of the UCN by a moving grating was observed in the experiment [14] and some time later, in experiments with a moving grating, the effect of focusing in time was also demonstrated [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The possibility of time focusing based on the resonant the neutron spin flip has also found its experimental confirmation [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The concept of a UCN source on a periodic pulsed reactor, based on the use of the time lens with pulse filling of the UCN trap, was considered in a recent paper [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' A similar approach was proposed in [20], in which it was proposed to focus very cold neutrons (VCN) with velocities of about 50 m/s, followed by their deceleration in a escaping trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Such a deceleration method was proposed in [21], but has not yet been applied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The extraction of neutrons with higher speeds than those of the UCN from the moderator-converter provides better conditions for the transportation of neutrons and allows to use more efficient converter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In the UCN source of the Institut Laue-Langevin [22], neutrons are slowed down rising to a height of several meters, followed by Doppler "cooling" when reflected from an escaping mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The deceleration of neutrons in the Earth gravity field during their transportation in a vertical neutron guide was also successfully used in the UCN sources at the WWR-M reactor of PNPI [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' However, in the case of pulsed neutron generation, the deceleration of VCN may lead to some new and important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The present work is devoted to their discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' 3 DECELERATION OF NEUTRONS GENERATED BY A PULSED SOURSE USING A LOCAL DEVICE We will consider a UCN source in which, for a relatively short time, pulse generation of very cold neutrons occurs and their subsequent transportation through a mirror neutron guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Suppose that a device slowing down neutrons is used, the purpose of which is to produce ultracold neutrons, whose energy after deceleration is small enough so that they can be stored in a material trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Let’s call this device a decelerator to avoid the term "moderator" widely used in neutron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In contrast to the case considered in [20], we will assume that the deceleration of neutrons by the decelerator occurs in a relatively short section of their transport in close proximity to the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' To store in the trap the neutrons obtained due to deceleration, their full velocity V should not exceed the boundary velocity of the trap matter V < Vb = � 2U/m, where U is the effective potential of the trap walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Since before getting into the trap, neutrons must pass a considerable path in the neutron guide, which we assume to be a mirror, the transverse velocity of neutrons normal to the surface of the walls of the latter is limited by the boundary energy value of the neutron guide walls Egd, so that v⊥ < � 2Egd/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Having obtained a limit for the total and transverse velocity of neutrons capable of being stored in a trap, we thereby obtained a limit for the longitudinal velocity of such neutrons directed along the Z axis of the neutron guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' vzfin = � V 2 b − v2 ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' (1) The decelerator, which forms the neutron flux immediately before neu- trons enter the trap, changes the neutron energy by a certain amount ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' If it is constructed correctly, then this change in kinetic energy is mainly due to a change in the longitudinal velocity of neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Being interested in the future only in the distribution of the longitudinal velocity of neutrons and the kinetic energy associated with it, we will skip below the z index of the quantities we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Then the energy of neutrons entering the trap and being able to be stored in it lies in the range from zero to Efin = mv2 fin/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Before deceleration , the energy of these neutrons should be in the range ED < E < Efin + ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' At a sufficiently large energy value ED, the range of neutron energies that can be trapped after deceleration can be much smaller than the energy itself δE ≈ Efin ≪ ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' But this means that the spread of the flight times δt from the pulsed source to the decelerator and, accordingly, to the trap, of the "useful" neutrons of interest to us are not only small, but may be much shorter than the time of flight itself t = L/V , where L is the length of the transport neutron guide δt t = δV V ≃ δE 2E ≃ Efin 2ED ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' (2) 4 Under good conditions, δt can also be significantly less than the pulse repetition period of the reactor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In this case, the flux of "useful" neutrons, which after deceleration will be converted to the UCN, will have a pulsed structure, although during transport through the neutron guide, the pulse duration will inevitably increase due to velocity dispersion δV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Let us now consider the question of the fluxes of UCN coming into the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Obviously, the process of deceleration does not affect the number of neutrons in any way and, consequently, the decelerator itself does not change the flux of neutrons with the corresponding spin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Therefore, for- getting about polarization for now, and assuming that the transmission of neutron guides for neutrons with the above velocity distribution is ideal, let’s compare the flux of UCN Φ1z that would enter the trap directly from the source with the flux of "useful" neutrons Φ2z coming from the source to the decelerator and only then into the trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Assuming that the distribution over the "normal" velocities v⊥is the same in both cases, we will continue to omit the z index, being interested only in the components of the velocities directed along the beam and the corresponding fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The flux Φ1 is carried by neutrons with velocities ranging from zero to vfin, and the flux Φ2 is carried by neutrons with velocities from V1 = � 2ED/m to V2 = � 2 (ED + Efin) /m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Since the UCN energy is very small, then in the Maxwell distribution for the velocity distribution of the flux in the source the linear approximation is valid dΦ(V ) = nV dV , where n is the neutron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Then for both fluxes we have Φ1 = n � √ 2Efin/m 0 V dV, Φ2 = n � √ 2(ED+Efin)/m √ 2ED/m V dV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' (3) It is easy to see that Φ1 = Φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Thus, neutrons entering the trap directly from the source and neutrons obtained by converting from VCN to UCN carry the same flux, but have a significantly different temporal and spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In the first case, the spread of the flight times δt is much larger than the pulse repetition period T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In this case, the pulse structure practically disappears and an essentially uniform flux enters the trap, which corresponds to the average value of the neutron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In the second case, when the duration of the bunch is significantly less than the pulse repetition period δt/T ≪ 1, the length of the bunches is less than the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Accordingly, the neutron density in the bunch exceeds the average by a value of G = T/δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' ADIABATIC SPIN-FLIPPER FOR NEUTRON DECELERATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' QUANTITATIVE ESTIMATES Let us make some estimates now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' For certainty, as a decelerator, we consider a flipper in which a spin flip occurs under the action of an alternating high-frequency field directed perpendicular to a large permanent field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' As such, the so-called adiabatic or gradient flipper can be used [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Passing 5 the flipper, the neutron energy changes by an amount ED = 2µB, where µ is the magnetic moment of the neutron and B is the magnitude of the permanent magnetic field, and under good conditions this energy change is mainly due to a change in the longitudinal velocity of the neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The energy change at the neutron spin reversal in the such flipper was demonstrated in [27], and the possibility of creating a flipper with the per- manent field of the order of 1 T was demonstrated in [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' There is no reason to doubt that it is possible to create a flipper with a field of the order of 15-20 T, which is achievable in modern superconducting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' For the final velocity vfin, we assume a value of 3 m/s, so that the total velocity in a circular neutron guide with a boundary energy of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='7 m/s does not exceed 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='5 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This velocity corresponds to an energy of the order of 50 neV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' For the magnitude of the magnetic field B, in which the spin flip occurs, we take the value of 20 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' When the spin is reversed in such a field, the energy changes by the value ED = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='4 × 103 neV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Therefore, the spread of neutron energies, whose longitudinal velocity after deceleration will not exceed 3 m/s, should be about 50 neV, while the energy itself will be somewhat greater than ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This energy corresponds to the neutron velocity of 21 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' From formula (2) follows the estimate δt t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' (4) Assuming for a rough estimate the length of the neutron guide is L ≈ 10 m, we obtain for the time of flight and its dispersion the values t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content='48 s and δt ≈ 5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The last of these values determines the duration of the bunch of "useful" neutrons at the entrance to the flipper-decelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This duration δt may be significantly less than the pulse repetition period of the source T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' In particular, for the IBR-2 reactor, T = 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' At the same time, at the direct transport of neutrons from the source to the trap, the spread of the flight times of the order of the flight time itself is δt ≈ L/vfin ≈ 3 s, which is an order of magnitude greater than the repetition period T and the density of neutrons reaching the trap in this case corresponds to the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Remind that if the fluxes are equal, that is following from (3)l, the compression of a bunch of neutrons by their local deceleration means an increase in the neutron density proportional to the value of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Above, we assumed that the deceleration time in the flipper is the same for all neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This is obviously not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The process of neutron deceleration in the flipper is caused by the slowing down of neutrons in the increasing magnetic field when entering it, and then, after the spin flip, the same slowing down in the decreasing field when exiting the flipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' These two stages are complemented by some time of flight in the weakly inhomo- geneous field of the flipper itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The deceleration time and its dispersion are determined by the design of the flipper and the range of initial and fi- 6 nal neutron velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This should be taken into account at its designing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Rough estimates based on the assumption of the constant neutron decelera- tion in a region l of an inhomogeneous magnetic field lead to the magnitude of the dispersion of the deceleration times is δt ≈ 2lV2/V 2 1 ≈ 10 ÷ 15 ms, where V1 and V2 are the neutron velocities before and after deceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' This value can be reduced if neutrons with the lowest velocities are excluded from consideration, which will lead to a very insignificant loss in neutron flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' CONCLUSION Thus, it is shown that decelerating the neutrons generated by a pulsed source, with the help of a local device, it is possible to obtain a noticeable gain in the pulse density of the UCN and without time focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The question of the combination of these two approaches, partially raised in [20], probably requires additional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' The authors are grateful to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Lychagin, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Karamyshev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Muzychka and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Y9AyT4oBgHgl3EQf9frR/content/2301.00877v1.pdf'} +page_content=' Novikov for useful 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a/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/2301.00638v1.pdf.txt b/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/2301.00638v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..367f39daa0c6f200d3dbd8c44e59a75f23262f0e --- /dev/null +++ b/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/2301.00638v1.pdf.txt @@ -0,0 +1,1881 @@ +arXiv:2301.00638v1 [hep-th] 2 Jan 2023 +Holographic Anisotropic Background in 5D +Einstien-Gauss-Bonnet Gravity +S. N. Sajadi∗ +School of Physics, Institute for Research in Fundamental Sciences (IPM), +P. O. Box 19395-5531, Tehran, Iran +Abstract +In this paper, we extend the work on the AdS/QCD model to quadratic gravity to gain +insight into the influence of gravity. +We obtain an anisotropic black brane solution to a +5D Einstein-Gauss-Bonnet-two Maxwell-dilaton system. The background is specified by an +arbitrary exponent, a dilaton field, a time component of the first Maxwell field, and a magnetic +component of the second Maxwell field. The system in three cases has been investigated and in +each case the effect of the parameter of theory, the anisotropic parameter has been considered. +The blackening function supports the thermodynamical phase transition between small/large +and AdS/large black brane for a suitable chemical potential and other parameters. +1 +Introduction +Quantum chromodynamics (QCD) is a non-abelian gauge theory that describes the strong in- +teraction between quarks and gluons. QCD at low temperatures exhibits confinement whereas +at high temperature undergoes a phase transition to a chiral symmetry. The investigation and +understanding of the phase diagram of QCD and the search for new phases of matter are of +attracting attention in the theoretical and experimental communities. The gauge/gravity duality +provided another way to further understand the dynamics of the strong-couple system, where +standard methods do not work [1], [2]. The quark-gluon plasma (QGP) is one such system created +in a short time in heavy ion collisions, it is believed to be anisotropic during this time [3], [4]. +Therefore, various properties of QCD have been investigated in an anisotropic background [5]. In +[6], [7] the confinement-deconfinement phase transition in the framework of the Einstein-dilaton- +Maxwell theory for the isotropic case has been studied. In [8], the confinement-deconfinement +phase transition in the framework of 5D Einstein-dilaton two-Maxwell theory with an anisotropic +background has been studied. In [9], the authors extended the work of [8], by introducing a back- +ground magnetic field to gain insight into the influence of such field on QCD observables. +Higher-order gravitational models have recently received attention [10]-[14], in part because string +theory predicts that at low energies Einstein’s equations are subject to first-order corrections [15]. +In AdS/CFT context, higher-order gravities have been used as tools to characterize numerous +properties of strongly coupled conformal field theories [16]-[18]. From quantum gravity viewpoint, +in order to unify quantum mechanics and gravitational interactions, going beyond the Einstein +gravity is necessary [19]. The first correction of Lovelock gravity to the Einstein-Hilbert action +appears in five and higher dimensions and is given by a precise combination of quadratic curva- +ture terms yields the second-order field equations known as the Gauss-Bonnet density [20], [21], +∗Electronic address: naseh.sajadi@gmail.com +1 + +[22]. Cosmological models, including in the inflation, and in the framework of Brane cosmology +have been well studied in this theory [23]. Black hole solutions of the theory have been studied +in [24]-[27]. The thermodynamics of black holes has also been studied in the framework of this +theory [28]. The Gauss-Bonnet term in 4D gives a non-zero contribution to the field equations in +the presence of the dilatonic scalar field φ [29], [30], [31]. In this paper, we extend the work of [8] +to the Einstein Quadratic Gravity, which is general relativity extended by quadratic curvature +invariants in the action to find the effect of higher derivative terms on QCD. +The paper is organized as follows. In section 2 we construct the anisotropic 5-dimensional solu- +tion with an arbitrary dynamical exponent, an exponential quadratic warp function, a non-zero +time component of the first Maxwell field and a non-zero magnetic component of the second +Maxwell field in the framework of EGB gravity. In section 2.1 first we consider zero warp func- +tion and obtain the exact solution for blackening function and other unknown quantities. We +have shown the behavior of the quantities with plots and we discuss the thermodynamics of the +constructed background. In section 2.2, we consider exponential quadratic warp function and +zero chemical potential and solved the differential equations approximatly and show that for neg- +ative exponential warp function the dilaton field is real. Then, we discuss the thermodynamics +of the constructed background and find out the small/large phase transition. In section 2.3 we +consider the non-zero warp function and non-zero chemical potential and obtained the approx- +imatly solution for the unknown functions. In this case we study the thermodynamics of the +black brane and find out the small/large and AdS/large phase transitions. We finish the paper +with some concluding remarks in section 3. +2 +Basic Formalism +We consider a 5-dimensional Einstein-quadratic-dilaton-two-Maxwell system. The action of the +system in the Einstein frame is specified as [8] +S = +1 +16πG5 +� +d5x√−gL, +(1) +where the Lagrangian is +L = R + γRabcdRabcd + βRabRab + αR2 − 1 +4f1(φ)F 2 +(1) − 1 +4f2(φ)F 2 +(2) − 1 +2∂µφ∂µφ − V (φ), +(2) +and F 2 +(i) = FµνF µν, φ is the dilaton field, f1(φ) and f2(φ) are the gauge functions representing +the coupling between the two U(1) gauge fields on one hand and the dilaton on the other hand. +V (φ) is the potential of the dilaton field, and G5 is the Newton constant in five dimensions. +(α, β, γ) are coupling constants of theory. We use the metric ansatz gµν, dilaton field φ and field +strength tensor F µν +(i) in the following form: +ds2 = l2b(z) +z2 +� +−g(z)dt2 + dz2 +g(z) + dx2 + P(z)(dy2 +1 + dy2 +2) +� +, +(3) +with +A(1) +µ += At(z)δ0 +µ, +F(2) = qdy1 ∧ dy2, +φ = φ(z), +(4) +where b(z) is the warp function, g(z) is the metric function and l is the AdS length scale. z = 0 +corresponds to the boundary of the 5d spacetime. The first gauge field (F (1)) is the electric part +of the Maxwell tensor which causes the black hole to become electrically charged. In relation +2 + +(11), we relate the charge of the black hole to the chemical potential of the dual quantum field +system. The second gauge field (F (2)) is the magnetic part of the Maxwell tensor on a plane y1y2 +and causes the anisotropy of the metric spatial part. The variation of the action (1) over metric +gµν, the scalar field φ and At gives the field equations as follows +Eµν = Gµν + α +� +2R(Rµν − 1 +4gµνR) + 2(gµν□ − ∇µ∇ν)R +� ++ β +� +(gµν□ − ∇µ∇ν)R + □Gµν + 2Rλρ(Rµλνρ +−1 +4gµνRλρ) +� ++ γ +� +−1 +2gµνRαβγηRαβγη + 2RµλρσRν +λρσ + 4RµλνρRλρ − 4RµσRσ +ν + 4□Rµν − 2∇µ∇νR +� += 1 +2f(i) +� +F (i) +µρ F (i)ρ +ν +− 1 +4gµνF 2(i) +� ++ 1 +2 +� +∂µφ∂νφ − 1 +2gµν(∂φ)2 − gµνV +� +, +∇2φ = ∂V +∂φ + 1 +4 +∂f(i) +∂φ (F 2(i)), +∇µ +� +f(i)F µν(i)� += 0, +(i = 1, 2) +(5) +where Gµν is the Einstien tensor. Using the ansatz of the metric, the Maxwell fields and the +dilaton field (5), it is easy to obtain the equations of motion for the background fields. The +explicit components of the field equation are large and bulky and we have not included them +here. The field equations for φ and At are given by: +− P 2z4f ′ +1h′2 + q2z4 df2 +dφ + 2b2l4P 2 dV +dφ − 3l2P 2gz2φ′b′ − 2l2bP 2z2gφ′′ + 6l2zbP 2gφ′ +− 2l2z2bP 2φ′g′ = 0, +(6) +f1A′ +tb′zP − 2f1A′ +tbP + 2f1bPA′′ +t z + 2bPzf ′ +1A′ +t = 0, +(7) +where prime is differential with respect to z. One can check that the equation of motion for +the second Maxwell field will not give any additional equation. To find the solution for the field +equations, we assume [8] +b(z) = e− cz2 +2 , +f1 = z−2+ 2 +ν , +P(z) = z2− 2 +ν , +(8) +where ν is a parameter that specified the anisotropic backgrounds. To solve the background, we +also impose the boundary conditions in the form +b(0) = 1, +g(0) = 1, +g(zh) = 0, +At(0) = µ, +At(zh) = 0 +(9) +where zh is the horizon and µ is the chemical potential of the boundary theory. The boundary +conditions are used to fix the integration constants. Now, we are going to solve the field equations. +First, by solving the differential equation (7), one can get +At(z) = c1 + c2e +cz2 +4 , +(10) +where +c1 = +µe +cz2 +h +4 +−1 + e +cz2 +h +4 +, +c2 = +µ +−1 + e +cz2 +h +4 +. +(11) +By inserting solution (10) into the equation Ett, one can obtain V (φ). Then, by inserting V (φ) +into the field equation Exx, one can obtain f2(φ). By inserting f2(φ) and V (φ) into Ezz one can +3 + +obtain φ′. Finally, from equation Ey1y1, the differential equation for g(z) obtains as follows +4ν4z4l(4α + β)e− cz2 +4 g(z)g′′′′ − 4ν3lz3e− cz2 +4 [(4α + β)(νcz2 − 2ν + 4)g − 2zν(α − γ)g′]g′′′ +− 6νzle− cz2 +4 [−4l2zν3e− cz2 +2 − 4ν2z2g′(20γ − 4α + 4β + (10α + 6β + 14γ)ν + cνz2(5β + 3α + 17γ)) ++ νg(c2ν2z4(5β + 24γ) + 4cνz2(16γ + 2β − 4α + ν(4α + β)) + ν2(20β + 32α + 48γ) + 16ν(β + 4γ) ++ 48γ − 32α)]g′′ − νzle− cz2 +4 [l2ν2e− cz2 +2 (4ν + 6νcz2 + 8) − g((3β + 12γ + 2α)c3ν3z6 + c2ν2z4(80γ+ +16α + 20β − (19β + 16α + 72γ) + 4cνz2(3(β + 2α + 2γ)ν2 − ν(12α + 4β + 8γ) + 12α + 10β + 38γ) ++ 4(ν + 2)(ν2(5β + 8α + 12γ) + ν(16γ − 8α + 2β) + 4β + 12γ + 8α)))]g′ − 4z4ν3l(2γ + β + 2α) +e− cz2 +4 g′′2 − νz2le− cz2 +4 g′2(112α + 32β + 32γ + (128α + 32β + 8cz2(18α + 5β + 4γ))ν + (48α + 20β ++ 32γ − 16cz2(3α + β + γ) + c2z4(40α + 8γ + 11β))ν2) − c2c2 +2ν3le +cz2 +4 z +4ν+2 +ν += 0. +(12) +For generic coupling constant α, β and γ this fourth order differential equation analytically cannot +be solved, therefore we consider the case where γ = α, β = −4α. In this case, the theory reduced +to Einstien-Gauss-Bonnet gravity (EGB), and the field equation (12), reduced to second order +differential equation for metric function g(z) as follows +− 4zl((2 + νcz2)2zνg(z)αe− cz2 +4 − zν3l2e− 3cz2 +4 )g′′ − 4νz2lα(2 + νcz2)2e− cz2 +4 g′2− +2zl((2 + νcz2)(z4c2ν2 − 6cz2ν2 + 6νcz2 + 4ν + 8)g(z)αe− cz2 +4 + 2l2ν2(4 + 2ν + 3νcz2)e− 3cz2 +4 )g′ +− c2ν3lc2 +2z +4ν+2 +ν e +cz2 +4 = 0. +(13) +In the following, we solve the above differential equations in special cases: +2.1 +The case c = 0 +In this case the warp function b(z) = 1 and the field equation for g(z) becomes: +zν(l2ν2 − 4αg(z))g′′ − g′(4ναzg′ + (2 + ν)(l2ν2 − 4αg)) = 0, +(14) +one can exactly solve it and obtain analytic solution for g(z) as +g(z) = +1 +4α(1 + ν)[l2ν2 + l2ν3− +� +l4ν4 + 2l4ν5 + l4ν6 − 4ν2αc1z +2(1+ν) +ν +− 16ναc2 − 8ν2αc2 − 4αc1νz +2(ν+1) +ν +− 8αc2], +(15) +and by taking into account the boundary conditions (9), we get +c1 = −2(ν3l2 + ν2l2 − 2αν − 2α) +νz +2(ν+1) +ν +h +, +c2 = ν2l2 − 2α. +(16) +For α ≪ 1, the metric function is given as +g(z) ≈ 1 − +� z +zh +� 2(ν+1) +ν +− 2α +ν2l2 +� z +zh +� 2(ν+1) +ν + +1 − +� z +zh +� 2(ν+1) +ν + + + O(α2). +(17) +4 + +The second term is the correction from the Gauss-Bonnet gravity and in the case of α → 0 the +metric function goes to [8] for Einstein gravity. +(a) ν = 4.5, l = 1, zh = 2 +(b) α = 0.1, l = 1, zh = 2 +(c) α = 0.1, l = 1, ν = 4.5 +Figure 1: Plot of g(z) in terms of z for α = 0, 0.5, 1, 1.5, 2 (left), ν = 1, 2, 3, 4, 5, 6 (middle) and +zh = 1, 2, 3, 4, 5, 6 (right). +The behavior of the metric function is depicted in Fig.(1). The main feature is that the metric +function values decrease faster for larger α (Fig.1a). In the isotropic case (ν = 1) the metric +function values are larger than in the anisotropic ones (ν ̸= 1) (Fig.1b). In this panel by increasing +ν the metric function values decrease faster. Changing the values of α and ν does not influence +the horizon position. In the following we look at the behavior of Ricci and Kretschmann scalar +K = RabcdRabcd of the black brane. The Ricci scalar is given as follows +R = +1 +2αν2l2 +� +(ν2l2 − 4α)2z +2ν+2 +ν +h ++ 8α(ν2l2 − 2α)z +2ν+2 +ν +� 3 +2 +[−l2ν2(4ν + 3(ν2 + 1)) +� +(ν2l2 − 4α)2z +2ν+2 +ν +h ++ 8α(ν2l2 − 2α)z +2ν+2 +ν +� 3 +2 +(3(ν2 + 1) + 4ν)(ν2l2 − 4α)4z +3ν+3 +ν +h ++ 4(ν2l2 − 2α)((ν2l2 − 4α)2(9ν2 + 11ν + 10)(zhz2) +ν+1 +ν ++ 8α(ν2l2 − 2α)(2ν2 ++ ν + 3)z +−ν−1 +ν +h +z +4ν+4 +ν )]. +(18) +The scalars are smooth inside the black hole and start to diverge for z > zh. In larger α it +happens earlier for R, while for K it happens earlier for smaller α. +5 + +(a) ν = 4.5, l = 1, zh = 2 +(b) ν = 4.5, l = 1, zh = 2 +Figure 2: Plot of Ricci scalar and Kretschmann scalar in terms of z for α = 0.1, 0.2, 0.3, 0.4, 0.5. +By inserting (15) in Ett, one can obtain φ as follows: +φ(z) = +−2 +� +−2K(ν − 1) +�� +H +√ +D arctan +�� +G +F +� +− L +√ +F arctan +�� +G +D +� +− 2 +� +DFG(l4ν4 − 8αc2) +� +(ν + 1)l +√ +GFEν +√ +l4ν4 − 8αc2 +� +l2ν3 + l2ν2 − +√ +E ++ c3, +(19) +by imposing the condition φ(zh) = 0, one can obtain c3 = 0. The constants K, H, ... are provided +in (59). In the case of α ≪ 1, one can get +φ(z) = 2√ν − 1 +ν +ln +� z +zh +� +− 4α√ν − 1 +l2ν2(ν + 1) +� +(ν + 1) ln +� z +zh +� ++ ν +2 +� +1 − +� z +zh +� 2ν+2 +ν +�� ++ O(α2). +(20) +The first term is the contribution of the Einstien term and the second term is from the Gauss- +Bonnet term. In figure (3), the real and imaginary parts of the scalar field in terms of z for +different values of parameters have been shown. As can be seen the imaginary part of scalar +field inside and outside the black brane has a non-zero value and is unstable. By increasing ν in +0 < z < zh, the real part and imaginary part of the scalar field increase and decrease respectively +and for z > zh vice versa (Fig.(3)a). In panel b, by increasing the coupling of theory in 0 < z < zh +the real and imaginary parts decrease and increase respectively, and for z > zh vice versa. +6 + +(a) α = 0.1, l = 1, zh = 2 +(b) α = 0.1, l = 1, ν = 4.5 +(c) zh = 2, α = 0.1, l = 1 +Figure 3: Plots of imagenary (solid lines) and real part (dashed lines) of φ in terms of z for +ν = 2, 3, 4, 5, 6 (left), for α = 0.1, 0.2, 0.3, 0.4, 0.5 (middle), for zh = 2, 3, 4, 5, 6 (right). +By inserting (20) into Exx, one can obtain f2 as follows: +f2 = − +2 +ν3q2α(ν + 1) ¯ +A +5 +2 (ν2l2(ν + 1) − +√ ¯ +A) +[−6ν(ν − 1)(ν + 1)5αc1 ¯C(ν4l4 − 8αc2)(νl2(ν − 1 +2) +� +¯ +A − ¯K)z +2ν−2 +ν +− 8α3c3 +1 ¯Cν2(2ν2 − 5ν − 2 + ν3)(ν + 1)3z +2+6ν +ν ++ 2z4ν(ν + 1)4α2c2 +1 ¯C(ν2l2� +¯ +A +(2ν2 + 5 − 5ν) − ¯F) + z− 4 +ν [1 +2(ν2l2(ν + 1)2 ¯E ¯ +A +3 +2 ) + 1 +2(ν2l2(ν2 − 1) ¯B ¯ +A +5 +2 ) + νl2(ν − 1)(ν + 1)6 +(ν − 1 +2) ¯C +� +¯ +A(ν4l4 − 8αc2)2 − ν15l8 ¯C ¯G − 5ν14l8 ¯C ¯G − 9ν13l8 ¯C ¯G − 5ν12l8 ¯C ¯G + 5ν11l4 ¯C ¯G(l4+ +16 +5 αc2) + 9ν10l4 ¯C ¯G(l4 + 80 +9 αc2) + 5ν9l4 ¯C ¯G(l4 + 144 +5 αc2) + l4ν8 ¯C ¯G(l4 + 80αc2) − 80αν7c2 ¯C ¯G +(l4 + 4 +5αc2) − 144αc2ν6 ¯C ¯G(l4 + 20 +9 αc2) − 80αν5c2 ¯C ¯G(l4 + 36 +5 αc2) − 16αν4c2 ¯C ¯G(l4 + 20αc2) ++ 320α2ν3c2 +2 ¯C ¯G + ν2 ¯L + ν ¯ +J + 1 +2 +¯B ¯ +A3 + 64α2c2 +2 ¯C ¯G]], +(21) +where the constants ¯ +A, ¯B... are provided in (60). In the case of α ≪ 1, one can get +f2 = 4l2(ν2 − 1)z− 4 +ν +ν2q2 +− +8(ν2 − 1)z4α +q2ν4 +� +z +2ν+2 +ν +− z +2ν+2 +ν +h +�2 × +� +−4ν +�zh +z +� 2ν+2 +ν ++ 3 +� z +zh +� 4ν+4 +ν +− 6 +� z +zh +� 2ν+2 +ν ++ ν +�zh +z +� 4ν+4 +ν ++ (2ν + 3) +� ++ O(α2). +(22) +The first terms is the contribution of Einstien gravity and the second term is related to the +Gauss-Bonnet gravity. In figure (4), the behavior of f2 in terms of z are shown. As can be seen, +by increasing α and q, f2 decreases and by increasing ν, f2 increases. +7 + +(a) ν = 4.5, l = 1, zh = 2, q = 0.5 +(b) ν = 4.5, l = 1, zh = 2, α = 0.1 +(c) l = 1, α = 0.1, q = 0.5, zh = 2 +Figure 4: Plots of f2 in terms of z for α = 0.1, 0.2, 0.3, 0.4, 0.5 (left), q = 0.1, 0.2, 0.3, 0.4, 0.5 +(middle), ν = 1, 2, 3, 4, 5 (right). +Finally from Ett, one can get V (z) (We did not bring it here due to its bulk). In the case of +α ≪ 1, one can get +V (z) = −2(ν + 1)(2ν + 1) +ν2l2 ++ +α +� +z +2ν+2 +ν +− z +2ν+2 +ν +h +�3 +ν4l4 +[−4z +4ν+4 +ν z +2ν+2 +ν +h +(−1 − 7ν + 6ν3 + 20ν2) ++ 8ν(3ν2 + 7ν − 1)z +2ν+2 +ν z +4ν+4 +ν +h ++ (8ν3 + 64ν2 − 36ν − 12)z +6ν+6 +ν ++ 4(ν − 1)(2ν + 1)z +10ν+10 +ν +z +−4ν−4 +ν +h +− 4(ν − 1)(8ν + 3)z +8ν+8 +ν z +−2ν−2 +ν +h +− 8ν2(ν + 2)z +6ν+6 +ν +h +] + O(α2). +(23) +In Fig. (5), the behavior of scalar potential in terms of z for anisotropic case has been shown. In +the left panel, between 0 < z < zh by increasing α, the potential decreases. In the right panel, +between 0 < z < zh by increasing ν the scalar potential increases. +(a) ν = 4.5, zh = 2, l = 1 +(b) ν = 4.5, l = 1, zh = 2 +Figure 5: Plots of V in terms of z for α = 0.1, 0.2, 0.3, 0.4, 0.5 (left), ν = 1, 2, 3, 4, 5 (right). +8 + +2.1.1 +Thermodynamics of the background +In this subsection, we explore the thermodynamics of the black brane solution (15). In order to +investigate the thermodynamic properties of the black brane we need to obtain some relevant +thermodynamic quantities. The temperature of the black brane is obtained as follows: +T = +���� +g′ +4π +���� = ν + 1 +2πzhν +� +1 − 2α +ν2l2 +� +. +(24) +It is noticed that, the temperature monotonically decreases with the increase of the horizon. By +increasing α, the temperature is decreased, and for α = 0, one can get the result of [8]. The +entropy is given as follows [32],[33] +S = −1 +8 +� +Σ +dn−2x√η +δL +δRµαβν +ǫµαǫβν, +(25) +where +δL +δRµαβν += +�1 +2 + αR +� � +gµβgαν − gµνgαβ� ++ 1 +2β +� +Rµβgαν − Rαβgµν − Rµνgαβ + Rανgµβ� ++ 2γRµαβν, +(26) +and ǫµν = −2 +√ +−hδt +[µδz +ν]. For c = 0 we have +s = S +V = l3P(zh)b(zh) +3 +2 +4z3 +h += +l3 +4z +ν+2 +ν +h +, +(27) +which is independent of parameter of the EGB gravity. In terms of the temperature, the entropy +is given as +s = l3 +4 +� +2πTν3l2 +(ν + 1)(ν2l2 − 2α) +� ν+2 +ν +. +(28) +For isotropic case s ≈ T 3 and for anisotropic case s ≈ T +ν+1 +ν . The free energy density F(T) can +be calculated from the entropy density s(T) by integrating as follows +F = +� +sdT = +ν +2(ν + 1)Ts, +(29) +which is related to temperature as F ≈ T +2ν+1 +ν . The sound velocity c2 +s which can directly measure +the conformality of the system, can be obtained from the temperature and entropy: +c2 +s = d log T +d log s = +ν +ν + 2. +(30) +For isotropic case (ν = 1), c2 +s = 1/3, the system is conformal, for anisotropic (ν ̸= 1), c2 +s ̸= 1/3 +the system is non-conformal. The heat capacity is given as +CV = T ds +dT = (ν + 2) +ν +s = s +c2s +. +(31) +In terms of temperature, +CV +T 3 = l3(ν + 2) +4ν +� +2πν3l2 +(ν + 1)(ν2l2 − 2α) +� ν+2 +ν +T +2−2ν +ν . +(32) +9 + +For isotropic case, the right hand side of (32) has a constant value and for ν > 1 depends to +the temperature and at high temperature goes to zero. Since entropy is positive therefore CV is +positive and the black hole is stable. In figure 6, the behavior of s, F and CV in terms of T for +isotropic (dashed lines) and anisotropic (solid lines) and different values of α have been shown. +As can be seen, by increasing α, the thermodynamical quantities s, F, and CV increase. For +T < T i +cross, the entropy, free energy, and heat capacity of anisotropic case is larger than isotropic, +and for T > T i +cross vice versa. Where i = s, F, CV and T i +cross are given as follows +(a) l = 1 +(b) l = 1 +(c) l = 1 +Figure 6: Plots of s, F and CV in terms of T for α = 0.1, 0.2, 0.3, 0.4, 0.5 and ν = 4.5 (solid line) +and ν = 1 (dashed lines) . +T s +cross = +� +l4((l2 − 2α)3ν +3(ν+2) +ν +π +2(1−ν) +ν +2 +ν+2 +ν )ν +l4ν((ν + 1)(l2ν2 − 2α))2(ν + 1)(l2ν2 − 2α)ν +� +1 +2ν−2 +, +(33) +T F +cross = +� +(l2 − 2α)24 +ν+1 +ν π +2−2ν +ν +ν +−3ν−6 +ν ++ ν +−4ν−6 +ν +� +ν +2(ν−1) +[(ν + 1)(l2ν2 − 2α)ν+2l4(ν−1)] +1 +2ν−2 +, +(34) +T CV +cross = +� +l4(ν+1)((l2ν2 − 2α)3(ν + 2)ν +2(ν+1) +ν +π +2(1−ν) +ν +2 +ν+2 +ν )ν +3ν(ν + 1)(l2ν2 − 2α)ν+2 +� +1 +2ν−2 +. +(35) +2.2 +The case c ̸= 0, µ = 0 +In this case At = 0 and the differential equation (12), becomes +4νlz2e− cz2 +4 (l2ν2e− cz2 +2 − α(2 + νcz2)2g(z))g′′ − 2zle− cz2 +4 (ν2l2(4 + (2 + 3z2)ν)− +α(2 + νcz2)(8 + 4ν + 6νcz2 − 6cν2z2 + c2ν2z4)g)g′ − 4ανz2le− cz2 +4 (2 + νcz2)2g′2 = 0. +(36) +In order to solve equation (36), we assume g(z) as follows +g(z) = 1 + ǫg1(z) + O(ǫ2), +(37) +by inserting it into the (36), one can achieve a homogeneous differential equation for g1(z) as +g′′ +1 + [−l2ν3(2 + 3cz2) − 4ν2(l2 + 4αcz2) + 4αν(2 + cz2) + 16α]g′ +1 +νz(l2ν2 − 4α) += 0. +(38) +10 + +Solving (38) give g1(z) as +g1(z) = c1 + c2erf + +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +z + + , +(39) +where c1 and c2 are constants of integration. Using (39), the metric (37) becomes +g(z) = 1 + c1 + c2erf + +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +z + + . +(40) +The conditions (9) give us c1 and c2 as +c1 = 0, +c2 = − +1 +erf +� +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +zh +�. +(41) +Finally, using (41) the metric function becomes +g(z) = 1 − +erf +� +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +z +� +erf +� +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +zh +�. +(42) +In the case of α ≪ 1 and c < 0, the blakening function become +g(z) ≈ 1− +erf +�√−6cz +2 +� +erf +�√−6czh +2 +� − +4(2ν + 1)√−6cα +� +ze +3cz2 +2 erf +�√−6czh +2 +� +− zherf +�√−6cz +2 +� +e +3cz2 +h +2 +� +3√πl2ν2erf +�√−6czh +2 +�2 ++O +� +α2� +, +(43) +and for α ≪ 1 and c > 0, we have +g(z) ≈ 1 − z +zh +e +3c(z2−z2 +h) +2 +− 2αc(2ν + 1) +ν2l2 +z(z2 − z2 +h) +zh +e +3c(z2−z2 +h) +2 ++ O(α2). +(44) +In figure (7), the behavior of g(z) for positive and negative values of warp function is depicted. +As c increases, the metric slope becomes more decreasing. Also, changing the value of c has no +effect on the value of the horizon. By substituting the obtained metric (42), we arrive at the +differential equation for the scalar field as: +11 + +(a) ν = 4.5, α = 0.1, l = 1, zh = 2 +(b) α = 0.1, l = 1, zh = 2 +Figure 7: +Plots of g(z) in terms of z for c += +0.5, 0.4, 0.3, 0.2, 0.1 (dashed lines) and +−0.5, −0.4, −0.3, −0.2, −0.1 (solid lines) (left), ν = 1, 2, 3, 4, 5, 6, c = 0.5 and c = −0.5 (right). +φ′2 = +e +cz2 +2 +z2l2ν32π(l2ν2 − 4α)erf +� √−2cAzh +2 +� � +erf +� √−2cAzh +2 +� +− erf +� √−2cAz +2 +��[2α +√ +−2cπAz(2 + νcz2) +� +−erf +�√ +−2cAzh +2 +� ++ erf +�√ +−2cAz +2 +�� +e +Acz2 +2 (ν2c2z4(5l2ν2 + 32αν − 4α) − 4(ν + 2)(l2ν2 − 4α) ++ 2νcz2(3ν3l2 + 20αν + 3l2ν2 + 4α)) − 8cανz2(2 + νcz2)2(3l2ν2 + 16αν − 4α)eAcz2 ++ 2l2ν2z +√ +−2cπAerf +�√ +−2cAzh +2 +� +e +cz2ν(νl2+8α) +ν2l2−4α +(νcz2(3l2ν2 + 32αν + 4α) − 2(ν + 2)(l2ν2 − 4α)) +− π(l2ν2 − 4α) +� +erf( +√ +−2cAz +2 +) − erf( +√ +−2cAzh +2 +) +� +(l2νe− cz2 +2 erf( +√−2cAzh +2 +)(−8 + 8ν + 3ν2c2z4 ++ 18ν2cz2) + α(2 + νcz2)(erf( +√ +−2cAz +2 +) − erf( +√ +−2cAzh +2 +))(−16 + 16ν + 3ν2z6c3 + 2νc2z4(1+ +11ν) + 4cz2(6ν2 − 2 + 5ν)))] +(45) +where +A = 3l2ν2 + 16να − 4α +l2ν2 − 4α +. +(46) +In figure 8, the behavior of imaginary and real part of φ(z) in terms of z for different values of +parameters has been shown. As can be seen, the imaginary part of scalar field inside the black +hole is zero and outside the black hole the scalar field is unstable, and by increasing α instability +increase. +12 + +(a) ν = 4.5, α = 0.1, l = 1, zh = 2 +(b) ν = 4.5, l = 1, zh = 2, c = −0.5 +Figure 8: Plots of real (dashed lines) and imagenary (solid lines) part of φ in terms of z for +c = −0.1, −0.2, −0.3, −0.4, −0.5 (left), α = 0.1, 0.2, 0.3, 0.4,0.5 (right). +In equation (47) the exact coupling function f2 and approximatly to first order in α in equation +(48) has been obtained. +In figure (9), f2 for positive/negative c and for different values of +parameters has been plotted. The important feature of the figures is that for the negative c, f2 +goes to the negative values by increasing z, while it does not become negative anywhere for the +positive c. +f2 = +(1 − ν)z− 4 +ν +π +3 +2 ν3q2(l2ν2 − 4α)erf +� √−2cAzh +2 +�2 [πzα +√ +Ae +cAz2 +2 +� +erf +�√ +−2cAz +2 +� +− erf +�√ +−2cAzh +2 +�� +(cz2l2ν4(cz2 − 6) ++ ν3(32c2z4α − 2l2(cz2 + 10)) + ν2(−8l2 + α(12c2z4 + 88cz2)) + 40αν(cz2 + 2) + 32α) − 4√παcz2ν(2+ +νcz2)(3l2ν2 + 16αν − 4α)ecAz2 + (l2ν2 − 4α)(−2l2ν2πz +√ +−2cAe +cz2ν(l2ν+8α) +l2ν2−4α +erf +�√ +−2cAzh +2 +� ++ π +3 +2 +� +erf +�√ +−2cAz +2 +� +− erf +�√ +−2cAzh +2 +�� +(l2νerf +�√ +−2cAzh +2 +� +e− cz2 +2 (3νcz2 + 4ν + 4) + [erf +�√ +−2cAz +2 +� +− +erf +�√ +−2cAzh +2 +� +]α(16 + 8cz2 + ν2(z6c3 − 2c2z4) + ν(16 + 6c2z4 + 12cz2))], +(47) +13 + +In the case of α ≪ 1 and c < 0 one can get +f2 ≈ +l2(ν − 1)e− cz2 +2 +� +2√−6czνe +3cz2 +2 +− π +1 +2 � +4 + 4ν + 3νcz2� � +erf +� √−6cz +2 +� +− erf +� √−6czh +2 +��� +ν2q2z +4 +ν q2π +1 +2 erf +�√−6czh +2 +� +− +(ν − 1)α +ν4q2π3z +4 +ν erf +� √−6czh +2 +�2 [−12z2π2cν2(2 + νcz2)e3cz2 − 4(2ν + 1)zecz2(4π2cνzhe +3czh +2 ++ π +5 +2 +√ +−6cerf +�√−6czh +2 +� � +νcz2 − 2 +3ν − 4 +3 +� ++ π +5 +2 √ +−6cνz)e +3cz2 +2 (c2z4ν2 − 6cν2z2 − 2νcz2 − 20ν − 8) +� +erf +�√−6cz +2 +� +− erf +�√−6czh +2 +�� +− 4(2ν + 1) +√ +−6cπ +5 +2 zh(4 +3 + 4 +3ν + νcz2)erf +�√−6cz +2 +� +e− cz2 +2 e +3cz2 +2 ++ νπ3 +� +erf +�√−6cz +2 +� +− erf +�√−6czh +2 +��2 +(16 + 8cz2 + ν(16 + 16c2z4 + 12cz2) + ν2(z6c3 − 2c2z4))] ++ O(α2). +(48) +(a) ν = 4.5, α = 0.1, l = 1, zh = +2, q = 0.5 +(b) α = 0.1, l = 1, zh = 2, q = 0.5 +(c) l = 1, zh = 2, q = 0.5, ν = 4.5 +Figure 9: +Plots of f2(z) in terms of z for c = 0.1, 0.2, 0.3, 0.4, 0.5 (solid lines) and c = +−0.1, −0.2, −0.3, −0.4, −0.5 (dashed lines) (left), ν = 2, 3, 4, 5, c = 0.5 (solid lines) and c = −0.5 +(dahsed lines) (middle), α = 0.1, 0.2, 0.3, 0.4, 0.5, c = 0.5 (solid lines) and c = −0.5 (dashed lines) +(right). +14 + +From equation Ett we get the expression for the scalar potential V as a function of z as follows: +V = +ecz2 +4ν3l4π +3 +2 (l2ν2 − 4α)erf +� √−2cAzh +2 +�2 [−2απze +cAz2 +2 √ +−2cA +� +erf +�√ +−2cAz +2 +� +− erf +�√ +−2cAzh +2 +�� +[l2z2cν5(5cz2 + 3c2z4 + 6) + ν4(−32c2z4α + l2(20cz2 + 20 + c2z4)) + ν3(4l2 + 10l2cz2 − α(88cz2 ++ 4c2z4 + 12c3z6)) − ν2(8l2 + α(80 + 36c2z4 + 48cz2)) − να(8cz2 + 48) + 32α] + ν[8cαz2√π(1 +− ν)(2 + νcz2)(3l2ν2 + 16αν − 4α)ecAz2 + (l2ν2 − 4α)[−2πνzl2√ +−2cA(3νcz2 + 4ν + 2)erf +�√ +−2cAzh +2 +� +e +cνz2(l2ν+8α) +l2ν2−4α ++ +� +erf +�√ +−2cAz +2 +� +− erf +�√ +−2cAzh +2 +�� +[l2erf +�√ +−2cAzh +2 +� +e− cz2 +2 (8 + ν2(9c2z4 + 12cz2 + 16) ++ ν(6cz2 + 8)) + (2 + cz2)α +� +erf +�√ +−2cAz +2 +� +− erf +�√ +−2cAzh +2 +�� +(ν2(3z6c3 − 2c2z4) + ν(20cz2 + 16+ +14c2z4 + 32 + 16cz2))]]]]. +(49) +(a) ν = 4.5, α = 0.1, l = 1, zh = 2 +(b) α = 0.1, l = 1, zh = 2 +Figure 10: Plots of V (z) in terms of z for c = 0.1, 0.2, 0.3, 0.4, 0.5 (dashed lines) and c = +−0.1, −0.2, −0.3, −0.4, −0.5 (solid lines) (left), α = 0.1, 0.2, 0.3, 0.4, 0.5, c = 0.5 (dashed lines) +and c = −0.5 (solid lines) (right). +Regardless of the sign of c, V (z) is negative under the horizon. In panel a, with the increase of +|c|, |V | decreases. In panel b, as α increases, |V | increases. +2.2.1 +Thermodynamics of background +The Hawking temperature can be obtained by using the surface gravity interpretation +T = κ +2π = +���� +g′ +4π +���� = +e +cAz2 +h +2 √ +−2cA +4π +3 +2 erf +�zh +2 +√ +−2cA +�, +(50) +15 + +where A provided in (46). In the case of small α and c < 0, one can get +T ≈ +√−6ce +3cz2 +h +2 +4π +3 +2 erf +� √−6czh +2 +� + +(2ν + 1)e +3cz2 +h +2 α[π√−6c(cz2 +h + 1 +3)erf +� √−6czh +2 +� +− 2√πczhe +3cz2 +h +2 ] +π +5 +2 l2ν2erf +� √−6czh +2 +�2 ++ O +� +α2� +. +(51) +The behavior of temperature is shown in Figs. (11). In Fig. (11)a, the variation of temperature +with respect to the horizon radius zh for different values of the coupling of theory α is shown. As +can be seen there exists a minimum temperature Tmin below which no black hole solution exist +(thermal gas). However, for T > Tmin, there are two black hole solutions, a large and a small +one (deconfined quark gluon plasma phase). The small black hole phase for which T increases +with zh whereas the large black hole phase for which T decreases with zh. In Fig. (11)b, to study +the stability of the solutions, we have shown the behavior of heat capacity CV and temperature. +As can be seen the large black hole has positive heat capacity and therefore is stable and small +black hole is unstable and thus not physical. +(a) ν = 4.5, l = 1, c = −0.5 +(b) ν = 4.5, l = 1, c = −0.5 +Figure 11: Plots of T in terms of zh for α = 0.1, 0.2, 0.3, 0.4, 0.5 and ν = 4.5 (left). Plots of CV +and T in terms of zh for α = 0.1, 0.2, 0.3, 0.4, 0.5 (right) +Following the standard Bekenstein-Hawking formula (27), one can easily read the black hole +entropy density s, which is defined as +s = l3e− +3cz2 +h +4 +4z +ν+1 +ν +h +. +(52) +The scaled entropy density s/T 3 as a function of scaled temperature T/Tmin is shown in Fig.12a. +The red lines correspond to the large stable solution and blue lines are for the small unstable +solution. The numerical result of the square of the sound velocity is shown in 12b. At Tmin, +the sound velocity square is around 0 which is in agreement with lattice data 0.05. +At high +temperature, the sound velocity square goes to 0.45 for ν = 1, which means that the system is +approximatly asymptotically conformal, while for ν = 4.5, the sound velocity square goes to 0.8. +The numerical result of the specific heat is shown in Fig. 12c. It can be clearly seen that the +16 + +specific heat CV diverges at Tmin. At T → ∞, the scaled specific heat CV /T 3 approaches to the +zero for this approximate solution [34]. +(a) l = 1, c = −0.5, α = 0.1 +(b) l = 1, c = −0.5, α = 0.1 +(c) l = 1, c = −0.5, α = 0.1 +Figure 12: Plots of scaled entropy density s/T 3 (left), c2 +s (middle) and CV /T 3 (right) in terms +of scaled temperature T/Tmin for ν = 1, Tmin = 0.19457 (dashed lines), ν = 4.5, Tmin = +0.1357 (solid lines). +In each panel red lines correspond to the large stable solution and blue +lines correspond to unstable small solution. +In figure 13, the behavior of c2 +s and CV have been shown. As can be seen from the figure and +obtained the result in the previous section, it can be concluded that the heat capacity has an +inverse relationship with the speed of sound. +(a) l = 1, c = −0.5, α = 0.1 +Figure +13: +Plots +of +c2 +s +(red +lines) +and +CV +(blue +lines) +in +terms +of +zh +for +ν += +1 (dashed lines), 4.5 (solid lines). +17 + +2.3 +The case c ̸= 0, µ ̸= 0 +In this case the differential equation (12), becomes +4νlz2e− cz2 +4 (l2ν2e− cz2 +2 − α(2 + νcz2)2g(z))g′′ − 2zle− cz2 +4 (ν2l2(4 + (2 + 3z2)ν)− +α(2 + νcz2)(8 + 4ν + 6νcz2 − 6cν2z2 + c2ν2z4)g)g′ − 4ανz2le− cz2 +4 (2 + νcz2)2g′2 +− c2ν3lc2 +2z +4ν+2 +ν e +cz2 +4 = 0 +(53) +in order to solve the above differential equation, we assume +g(z) = 1 + ǫg1(z) + O(ǫ2), +(54) +by inserting it, one can achieve a non-homogeneous differential equation as +g′′ +1 + [−l2ν3(2 + 3cz2) − 4ν2(l2 + 4αcz2) + 4αν(2 + cz2) + 16α]g′ +1 +νz(l2ν2 − 4α) ++ cνc2 +2z +2 +ν e +cz2 +2 +4α(4 + cνz2) = 0. +(55) +In order to solve the non-homogeneous differential equation (55) in the case of c < 0, c2 ̸= 0, we +consider the following particular solution: +g2(z) = e +cz2 +2 z− 2(ν−1) +ν +� +i +hizi, +(56) +where hi are coefficients of expansions. +By inserting the above solution into the differential +equation (55), and solving order by order for coefficients, one can get +h0 = h1 = h3 = h5 = 0, +h2 = +c2 +2ν2 +8α(ν − 2), +h4 = − +cν3c2 +2(−20αν + l2ν3 + l2ν2 − 12α) +16α(ν − 2)(ν4l2 + 3l2ν3 − 4αν2 + 2l2ν2 − 12αν − 8α), +h6 = +ν4c2c2 +2 +32α(2 + 3ν)(l2ν2 − 4α)2(1 + 2ν)(1 + ν)(ν2 − 4)(3l4ν6 + 6l5ν5 − 40ν4l2α + 3ν4l4 + 256α2ν3 +− 56ν3l2α + 624ν2α2 − 24ν2l2α + 448α2ν + 112α2), +(57) +where c2 used from (11). Therefore, the metric function by considering a particular solution +becomes +g(z) = 1 − +erf +� +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +z +� +erf +� +1 +2 +� +c(−6l2ν2 − 32να + 8α) +(l2ν2 − 4α) +zh +� + e +cz2 +2 z− 2(ν−1) +ν +� +i +hizi, +(58) +with hi are provided in (57). In figure 14, the effect of µ on the behavior of f2 and V have been +shown. As can be seen in left panel by increasing chemical potential, f2 decreases and in right +panel V increases. +18 + +(a) ν = 4.5, l = 1, α = 0.1, c = ±0.5 +(b) ν = 4.5, l = 1, α = 0.1, c = ±0.5 +Figure 14: Plots of f2 and V in terms of z for µ = 0, 0.05, 0.1, 0.15, 0.2, 0.25. +(a) ν = 4.5, l = 1, c = −0.5, α = +0.1 +(b) ν = 4.5, l = 1, c = −0.5, α = +0.1 +(c) ν = 4.5, l = 1, c = −0.5, α = +0.1 +Figure 15: Plots of s, T and CV in terms of zh for µ = 0.1, 0.15, 0.2, 0.22728, 0.3 (left) and +µ = 0.2, 0.22728, 0.3 (middle and right). +In fig. 15a, the behavior of entropy is shown in terms of temperature. The figure shows the +minimum and maximum temperature. As µ increases, the minimum temperature decreases. This +plot also shows that for 0 < µ < µc, there are minimal Tmin and maximal Tmax temperatures, +between which the entropy is a function of T with three branches. +When we decrease the +temperature, the entropy decreases along the first branch (Tmin < T < ∞). Then the entropy +decreases along the second branch with an increase of temperature from Tmin to Tmax, i.e. here +the black holes are unstable. Finally the entropy increases along the third branch with an increase +of temperature for 0 < T < Tmax. Such a behavior in terms of event horizon, one can see in the +15b and c. In each panel, the critical point has been shown in red color curve. Upon varying +the Hawking temperature, a phase transition from the large black hole phase to the thermal +AdS phase takes place at a critical temperature Tc = 0.1242. This is the famous black hole- +thermal AdS Hawking-Page phase transition which occurs in the presence of chemical potential +µc = 0.22728. In fig. 16 by using the approximate analysis, the phase diagram of the holographic +QCD model for anisotropic background and for Einstien gravity (long dashed line), Einstien- +19 + +Gauss-Bonnet gravity (solid line) has been shown. As one can see at µ = 0 the system undergoes +a black hole to thermal gas first order phase transition so that T (EGB)(µ = 0) > T (E)(µ = 0). +For 0 < µ < µc (in the transition lines), the system undergoes a large black hole to a small black +hole first-order phase transition. For 0 < µ < µI, the temperature of the black hole to black hole +transition of Einstien gravity (T (E)(µ)) is less than Einstien-Gauss-Bonnet gravity (T (EGB)(µ)) +and for µI < µ < µc vice versa. The first order phase transition stops at the critical point (µc, Tc), +where the phase transition becomes second order, herewith T (EGB) +c +< T (E) +c +, µ(EGB) +c +< µ(E) +c +. For +µ > µc, the system has a sharp but smooth crossover. These thermal AdS and black hole phases +in the usual language of gauge-gravity duality are dual to the confinement and deconfinement +phases in the dual boundary theory. +(a) ν = 4.5, l = 1, c = −0.5 +Figure 16: The phase diagram in T and µ plane for anisotropic background. At small µ, the +system undergoes a first order phase transition at finite T. The first order phase transition stops +at the critical point (µc, Tc) ∼ (0.22728, 0.1242), where the phase transition becomes second +order. The solid line is for α = 0.1 and long dashed line is for α = 0. +3 +Conclusion +In this work, we extended the AdS/QCD model to quadratic gravity to gain insight into the +influence of gravity on QCD. To do so, we considered an anisotropic black hole metric as a solution +to a system of 5D Einstein-quadratic-two Maxwell-dilaton fields. The anisotropic background +is specified by an arbitrary exponent, a non-zero dilaton field, a non-zero time component of +the first Maxwell field, and a non-zero longitudinal magnetic component of the second Maxwell +field. The field equations for the considered theory are coupled and bulky differential equations +for six unknown functions. Therefore, obtaining the solution to such field equations is too hard, +this is why we considered the special case of field equation, i.e. +γ = α, β = −4α (Einstien- +Gauss-Bonnet gravity). The differential equation for the metric function in EGB gravity is a +nonlinear second-order equation that has been solved in special cases. At the first, we obtained +20 + +the exact solutions for the differential equations with zero warp functions. In this case, it doesn’t +occur any thermodynamical phase transitions to the black brane. The second case that we have +considered is the case with zero chemical potential. The blackening function in this case supports +the Van der Waals-like phase transition between small and large black holes for suitable values +of parameters. The third case that has been considered is nonzero warp function and chemical +potential. In this case, in addition to the small/large phase transition, the blackening function +supported the phase transition from the large black hole phase to the thermal AdS phase at +a critical temperature. Holographically, this phase transition corresponds to the confinement- +deconfinement phase transition in QCD. In each case, we investigated the anisotropy influence +and the effect of parameters of theory on the thermodynamic properties of our background, in +particular, on the small/large black holes phase transition diagram. In fig. (16), the effect of the +Gauss-Bonnet term on the phase transition has been shown. This figure shows that before/after +the intersection point for constant chemical potential, the temperature of a black hole in EGB +gravity is more/less than that of Einstien gravity. Clearly, before the intersection point, α has a +dominant impact on the temperature (compared to the effect of µ on the temperature). In the +isotropic case corresponding to ν = 1 (zero magnetic fields) and , α → 0 reproduces previously +known results [8]. +For future work, one can consider the Weyl-squared term by using the combination γ = 6α +and β = −4/3γ for the parameters of the theory. But since in this case, the field equation for +the metric has a 4th-order derivative, the differential equations should be solved numerically. +Also, following the paper [9], one can study the effect of the magnetic field on the system in the +framework of EGB. +Acknowledgements +I would like to thank the referee for her/his fruitful comments. I also would like to thank the +School of Physics of the Institute for Research in Fundamental Sciences (IPM). +Constants +The constants related to equation(19): +E = (ν + 1)((ν + 1)(ν4l4 − 8αc2) − 4ανc1z +2ν+2 +ν +), +D = νl2 − ν2l2 − +� +ν4l4 − 8αc2, +F = νl2 − ν2l2 + +� +ν4l4 − 8αc2, +G = ν3l2 − νl2 − +√ +E +ν + 1 +, +H = −l2ν(ν − 1) +� +ν4l4 − 8αc2 + ν4l4 − 8αc2, +K = −l2ν +� +ν − 1 +2 +� √ +E − 2αc1νz +2ν+2 +ν ++ (ν + 1) +� +ν4l4 − 4αc2 − l4ν3 +2 +� +, +L = l2ν(ν − 1) +� +ν4l4 − 8αc2 + ν4l4 − 8αc2. +(59) +21 + +The constants related to equation(21): +¯ +A = (ν + 1)(−4αc1νz +2ν+2 +ν ++ (ν + 1)(ν4l4 − 8αc2)), +¯B = −4αν2c1z +2ν+2 +ν ++ (ν + 1)(ν + 3 +2)(ν4l4 − 8αc2), +¯C = −4ανc1z +2ν+2 +ν ++ (ν + 1)(ν4l4 − 8αc2), +¯E = −4c2 +1α2ν2(3ν3 − 4ν2 + 3ν + 2)z +2(2ν+2) +ν ++ +2c1να(ν2 − 1)(ν4l4 + 2l4ν3 − 16ανc2 − 8αc2)z +2ν+2 +ν ++ +(ν − 1)(ν + 1)2(ν4l4 − 8αc2)(4αc2 + ν5l5 − l4ν3 − ν4l4 +2 ), +¯F = ν7l4 − 7 +2ν6l4 + 2ν5l4 + 5 +2ν4l4 + 4ν3αc2 + 24αν2c2 − 36ανc2 − 8αc2, +¯H = 4c2 +1α2(5ν + 2 − 2ν2 + ν3)z +2(2ν+2) +ν +− 4c1α(ν2 − 1)(l4ν3 − 8ανc2 + 4αc2)z +2ν+2 +ν +− (ν + 1)2(ν − 1)ν2l4(ν2 − 1 +2ν − 1)(ν4l4 − 8αc2), +¯G = ν5l4 + 2αc2 + 1 +2ν4l4 − 1 +2ν3l4, +¯K = ν5l4 + 1 +3ν4l4 − 1 +2l4ν3 + 4αc2, +¯L = 576α2c2 +2 ¯G ¯C + 1 +2 +¯ +A2 ¯H, +¯ +J = 320α2c2 +2 ¯G ¯C − 1 +2( ¯ +A2( ¯ +A ¯B − ¯H)). +(60) +References +[1] J. 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Yan, JHEP 09, 041 (2011) doi:10.1007/JHEP09(2011)041 +[arXiv:1103.5389 [hep-th]]. +24 + diff --git a/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/load_file.txt b/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0b89f09e744a92d55290808528c2ec94b09255b --- /dev/null +++ b/ZtAyT4oBgHgl3EQfvvnv/content/tmp_files/load_file.txt @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf,len=1156 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='00638v1 [hep-th] 2 Jan 2023 Holographic Anisotropic Background in 5D Einstien-Gauss-Bonnet Gravity S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Sajadi∗ School of Physics, Institute for Research in Fundamental Sciences (IPM), P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Box 19395-5531, Tehran, Iran Abstract In this paper, we extend the work on the AdS/QCD model to quadratic gravity to gain insight into the influence of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' We obtain an anisotropic black brane solution to a 5D Einstein-Gauss-Bonnet-two Maxwell-dilaton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The background is specified by an arbitrary exponent, a dilaton field, a time component of the first Maxwell field, and a magnetic component of the second Maxwell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The system in three cases has been investigated and in each case the effect of the parameter of theory, the anisotropic parameter has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The blackening function supports the thermodynamical phase transition between small/large and AdS/large black brane for a suitable chemical potential and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 1 Introduction Quantum chromodynamics (QCD) is a non-abelian gauge theory that describes the strong in- teraction between quarks and gluons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' QCD at low temperatures exhibits confinement whereas at high temperature undergoes a phase transition to a chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The investigation and understanding of the phase diagram of QCD and the search for new phases of matter are of attracting attention in the theoretical and experimental communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The gauge/gravity duality provided another way to further understand the dynamics of the strong-couple system, where standard methods do not work [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The quark-gluon plasma (QGP) is one such system created in a short time in heavy ion collisions, it is believed to be anisotropic during this time [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Therefore, various properties of QCD have been investigated in an anisotropic background [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In [6], [7] the confinement-deconfinement phase transition in the framework of the Einstein-dilaton- Maxwell theory for the isotropic case has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In [8], the confinement-deconfinement phase transition in the framework of 5D Einstein-dilaton two-Maxwell theory with an anisotropic background has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In [9], the authors extended the work of [8], by introducing a back- ground magnetic field to gain insight into the influence of such field on QCD observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Higher-order gravitational models have recently received attention [10]-[14], in part because string theory predicts that at low energies Einstein’s equations are subject to first-order corrections [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In AdS/CFT context, higher-order gravities have been used as tools to characterize numerous properties of strongly coupled conformal field theories [16]-[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' From quantum gravity viewpoint, in order to unify quantum mechanics and gravitational interactions, going beyond the Einstein gravity is necessary [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The first correction of Lovelock gravity to the Einstein-Hilbert action appears in five and higher dimensions and is given by a precise combination of quadratic curva- ture terms yields the second-order field equations known as the Gauss-Bonnet density [20], [21], ∗Electronic address: naseh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='sajadi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='com 1 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Cosmological models, including in the inflation, and in the framework of Brane cosmology have been well studied in this theory [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Black hole solutions of the theory have been studied in [24]-[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The thermodynamics of black holes has also been studied in the framework of this theory [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The Gauss-Bonnet term in 4D gives a non-zero contribution to the field equations in the presence of the dilatonic scalar field φ [29], [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this paper, we extend the work of [8] to the Einstein Quadratic Gravity, which is general relativity extended by quadratic curvature invariants in the action to find the effect of higher derivative terms on QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In section 2 we construct the anisotropic 5-dimensional solu- tion with an arbitrary dynamical exponent, an exponential quadratic warp function, a non-zero time component of the first Maxwell field and a non-zero magnetic component of the second Maxwell field in the framework of EGB gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 first we consider zero warp func- tion and obtain the exact solution for blackening function and other unknown quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' We have shown the behavior of the quantities with plots and we discuss the thermodynamics of the constructed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, we consider exponential quadratic warp function and zero chemical potential and solved the differential equations approximatly and show that for neg- ative exponential warp function the dilaton field is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Then, we discuss the thermodynamics of the constructed background and find out the small/large phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 we consider the non-zero warp function and non-zero chemical potential and obtained the approx- imatly solution for the unknown functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this case we study the thermodynamics of the black brane and find out the small/large and AdS/large phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' We finish the paper with some concluding remarks in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 2 Basic Formalism We consider a 5-dimensional Einstein-quadratic-dilaton-two-Maxwell system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The action of the system in the Einstein frame is specified as [8] S = 1 16πG5 � d5x√−gL, (1) where the Lagrangian is L = R + γRabcdRabcd + βRabRab + αR2 − 1 4f1(φ)F 2 (1) − 1 4f2(φ)F 2 (2) − 1 2∂µφ∂µφ − V (φ), (2) and F 2 (i) = FµνF µν, φ is the dilaton field, f1(φ) and f2(φ) are the gauge functions representing the coupling between the two U(1) gauge fields on one hand and the dilaton on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' V (φ) is the potential of the dilaton field, and G5 is the Newton constant in five dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (α, β, γ) are coupling constants of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' We use the metric ansatz gµν, dilaton field φ and field strength tensor F µν (i) in the following form: ds2 = l2b(z) z2 � −g(z)dt2 + dz2 g(z) + dx2 + P(z)(dy2 1 + dy2 2) � , (3) with A(1) µ = At(z)δ0 µ, F(2) = qdy1 ∧ dy2, φ = φ(z), (4) where b(z) is the warp function, g(z) is the metric function and l is the AdS length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' z = 0 corresponds to the boundary of the 5d spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The first gauge field (F (1)) is the electric part of the Maxwell tensor which causes the black hole to become electrically charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In relation 2 (11), we relate the charge of the black hole to the chemical potential of the dual quantum field system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The second gauge field (F (2)) is the magnetic part of the Maxwell tensor on a plane y1y2 and causes the anisotropy of the metric spatial part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The variation of the action (1) over metric gµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' the scalar field φ and At gives the field equations as follows Eµν = Gµν + α � 2R(Rµν − 1 4gµνR) + 2(gµν□ − ∇µ∇ν)R � + β � (gµν□ − ∇µ∇ν)R + □Gµν + 2Rλρ(Rµλνρ −1 4gµνRλρ) � + γ � −1 2gµνRαβγηRαβγη + 2RµλρσRν λρσ + 4RµλνρRλρ − 4RµσRσ ν + 4□Rµν − 2∇µ∇νR � = 1 2f(i) � F (i) µρ F (i)ρ ν − 1 4gµνF 2(i) � + 1 2 � ∂µφ∂νφ − 1 2gµν(∂φ)2 − gµνV � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ∇2φ = ∂V ∂φ + 1 4 ∂f(i) ∂φ (F 2(i)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ∇µ � f(i)F µν(i)� = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 2) (5) where Gµν is the Einstien tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Using the ansatz of the metric, the Maxwell fields and the dilaton field (5), it is easy to obtain the equations of motion for the background fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The explicit components of the field equation are large and bulky and we have not included them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The field equations for φ and At are given by: − P 2z4f ′ 1h′2 + q2z4 df2 dφ + 2b2l4P 2 dV dφ − 3l2P 2gz2φ′b′ − 2l2bP 2z2gφ′′ + 6l2zbP 2gφ′ − 2l2z2bP 2φ′g′ = 0, (6) f1A′ tb′zP − 2f1A′ tbP + 2f1bPA′′ t z + 2bPzf ′ 1A′ t = 0, (7) where prime is differential with respect to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' One can check that the equation of motion for the second Maxwell field will not give any additional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' To find the solution for the field equations, we assume [8] b(z) = e− cz2 2 , f1 = z−2+ 2 ν , P(z) = z2− 2 ν , (8) where ν is a parameter that specified the anisotropic backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' To solve the background, we also impose the boundary conditions in the form b(0) = 1, g(0) = 1, g(zh) = 0, At(0) = µ, At(zh) = 0 (9) where zh is the horizon and µ is the chemical potential of the boundary theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The boundary conditions are used to fix the integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Now, we are going to solve the field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' First, by solving the differential equation (7), one can get At(z) = c1 + c2e cz2 4 , (10) where c1 = µe cz2 h 4 −1 + e cz2 h 4 , c2 = µ −1 + e cz2 h 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (11) By inserting solution (10) into the equation Ett, one can obtain V (φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Then, by inserting V (φ) into the field equation Exx, one can obtain f2(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By inserting f2(φ) and V (φ) into Ezz one can 3 obtain φ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' from equation Ey1y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' the differential equation for g(z) obtains as follows ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4ν4z4l(4α + β)e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 g(z)g′′′′ − 4ν3lz3e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 [(4α + β)(νcz2 − 2ν + 4)g − 2zν(α − γ)g′]g′′′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− 6νzle− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 [−4l2zν3e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 − 4ν2z2g′(20γ − 4α + 4β + (10α + 6β + 14γ)ν + cνz2(5β + 3α + 17γ)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ νg(c2ν2z4(5β + 24γ) + 4cνz2(16γ + 2β − 4α + ν(4α + β)) + ν2(20β + 32α + 48γ) + 16ν(β + 4γ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 48γ − 32α)]g′′ − νzle− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 [l2ν2e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (4ν + 6νcz2 + 8) − g((3β + 12γ + 2α)c3ν3z6 + c2ν2z4(80γ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='16α + 20β − (19β + 16α + 72γ) + 4cνz2(3(β + 2α + 2γ)ν2 − ν(12α + 4β + 8γ) + 12α + 10β + 38γ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 4(ν + 2)(ν2(5β + 8α + 12γ) + ν(16γ − 8α + 2β) + 4β + 12γ + 8α)))]g′ − 4z4ν3l(2γ + β + 2α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 g′′2 − νz2le− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 g′2(112α + 32β + 32γ + (128α + 32β + 8cz2(18α + 5β + 4γ))ν + (48α + 20β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 32γ − 16cz2(3α + β + γ) + c2z4(40α + 8γ + 11β))ν2) − c2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2ν3le ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4ν+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (12) For generic coupling constant α, β and γ this fourth order differential equation analytically cannot be solved, therefore we consider the case where γ = α, β = −4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this case, the theory reduced to Einstien-Gauss-Bonnet gravity (EGB), and the field equation (12), reduced to second order differential equation for metric function g(z) as follows − 4zl((2 + νcz2)2zνg(z)αe− cz2 4 − zν3l2e− 3cz2 4 )g′′ − 4νz2lα(2 + νcz2)2e− cz2 4 g′2− 2zl((2 + νcz2)(z4c2ν2 − 6cz2ν2 + 6νcz2 + 4ν + 8)g(z)αe− cz2 4 + 2l2ν2(4 + 2ν + 3νcz2)e− 3cz2 4 )g′ − c2ν3lc2 2z 4ν+2 ν e cz2 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (13) In the following, we solve the above differential equations in special cases: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 The case c = 0 In this case the warp function b(z) = 1 and the field equation for g(z) becomes: zν(l2ν2 − 4αg(z))g′′ − g′(4ναzg′ + (2 + ν)(l2ν2 − 4αg)) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (14) one can exactly solve it and obtain analytic solution for g(z) as g(z) = 1 4α(1 + ν)[l2ν2 + l2ν3− � l4ν4 + 2l4ν5 + l4ν6 − 4ν2αc1z 2(1+ν) ν − 16ναc2 − 8ν2αc2 − 4αc1νz 2(ν+1) ν − 8αc2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (15) and by taking into account the boundary conditions (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' we get c1 = −2(ν3l2 + ν2l2 − 2αν − 2α) νz 2(ν+1) ν h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' c2 = ν2l2 − 2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (16) For α ≪ 1, the metric function is given as g(z) ≈ 1 − � z zh � 2(ν+1) ν − 2α ν2l2 � z zh � 2(ν+1) ν \uf8eb \uf8ed1 − � z zh � 2(ν+1) ν \uf8f6 \uf8f8 + O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (17) 4 The second term is the correction from the Gauss-Bonnet gravity and in the case of α → 0 the metric function goes to [8] for Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 (c) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 1: Plot of g(z) in terms of z for α = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, 2 (left), ν = 1, 2, 3, 4, 5, 6 (middle) and zh = 1, 2, 3, 4, 5, 6 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The behavior of the metric function is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The main feature is that the metric function values decrease faster for larger α (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the isotropic case (ν = 1) the metric function values are larger than in the anisotropic ones (ν ̸= 1) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this panel by increasing ν the metric function values decrease faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Changing the values of α and ν does not influence the horizon position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the following we look at the behavior of Ricci and Kretschmann scalar K = RabcdRabcd of the black brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The Ricci scalar is given as follows R = 1 2αν2l2 � (ν2l2 − 4α)2z 2ν+2 ν h + 8α(ν2l2 − 2α)z 2ν+2 ν � 3 2 [−l2ν2(4ν + 3(ν2 + 1)) � (ν2l2 − 4α)2z 2ν+2 ν h + 8α(ν2l2 − 2α)z 2ν+2 ν � 3 2 (3(ν2 + 1) + 4ν)(ν2l2 − 4α)4z 3ν+3 ν h + 4(ν2l2 − 2α)((ν2l2 − 4α)2(9ν2 + 11ν + 10)(zhz2) ν+1 ν + 8α(ν2l2 − 2α)(2ν2 + ν + 3)z −ν−1 ν h z 4ν+4 ν )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (18) The scalars are smooth inside the black hole and start to diverge for z > zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In larger α it happens earlier for R, while for K it happens earlier for smaller α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 5 (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2 Figure 2: Plot of Ricci scalar and Kretschmann scalar in terms of z for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By inserting (15) in Ett, one can obtain φ as follows: φ(z) = −2 � −2K(ν − 1) �� H √ D arctan �� G F � − L √ F arctan �� G D � − 2 � DFG(l4ν4 − 8αc2) � (ν + 1)l √ GFEν √ l4ν4 − 8αc2 � l2ν3 + l2ν2 − √ E + c3, (19) by imposing the condition φ(zh) = 0, one can obtain c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The constants K, H, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' are provided in (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the case of α ≪ 1, one can get φ(z) = 2√ν − 1 ν ln � z zh � − 4α√ν − 1 l2ν2(ν + 1) � (ν + 1) ln � z zh � + ν 2 � 1 − � z zh � 2ν+2 ν �� + O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (20) The first term is the contribution of the Einstien term and the second term is from the Gauss- Bonnet term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure (3), the real and imaginary parts of the scalar field in terms of z for different values of parameters have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen the imaginary part of scalar field inside and outside the black brane has a non-zero value and is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By increasing ν in 0 < z < zh, the real part and imaginary part of the scalar field increase and decrease respectively and for z > zh vice versa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(3)a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In panel b, by increasing the coupling of theory in 0 < z < zh the real and imaginary parts decrease and increase respectively, and for z > zh vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 6 (a) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (c) zh = 2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1 Figure 3: Plots of imagenary (solid lines) and real part (dashed lines) of φ in terms of z for ν = 2, 3, 4, 5, 6 (left), for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (middle), for zh = 2, 3, 4, 5, 6 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By inserting (20) into Exx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' one can obtain f2 as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='f2 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν3q2α(ν + 1) ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (ν2l2(ν + 1) − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='[−6ν(ν − 1)(ν + 1)5αc1 ¯C(ν4l4 − 8αc2)(νl2(ν − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A − ¯K)z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2ν−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− 8α3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 ¯Cν2(2ν2 − 5ν − 2 + ν3)(ν + 1)3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2+6ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 2z4ν(ν + 1)4α2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 ¯C(ν2l2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(2ν2 + 5 − 5ν) − ¯F) + z− 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν [1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2(ν2l2(ν + 1)2 ¯E ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2(ν2l2(ν2 − 1) ¯B ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ) + νl2(ν − 1)(ν + 1)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(ν − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2) ¯C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A(ν4l4 − 8αc2)2 − ν15l8 ¯C ¯G − 5ν14l8 ¯C ¯G − 9ν13l8 ¯C ¯G − 5ν12l8 ¯C ¯G + 5ν11l4 ¯C ¯G(l4+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 αc2) + 9ν10l4 ¯C ¯G(l4 + 80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='9 αc2) + 5ν9l4 ¯C ¯G(l4 + 144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 αc2) + l4ν8 ¯C ¯G(l4 + 80αc2) − 80αν7c2 ¯C ¯G ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(l4 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5αc2) − 144αc2ν6 ¯C ¯G(l4 + 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='9 αc2) − 80αν5c2 ¯C ¯G(l4 + 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 αc2) − 16αν4c2 ¯C ¯G(l4 + 20αc2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 320α2ν3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ¯C ¯G + ν2 ¯L + ν ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='J + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='¯B ¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A3 + 64α2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ¯C ¯G]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (21) where the constants ¯ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' are provided in (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the case of α ≪ 1, one can get f2 = 4l2(ν2 − 1)z− 4 ν ν2q2 − 8(ν2 − 1)z4α q2ν4 � z 2ν+2 ν − z 2ν+2 ν h �2 × � −4ν �zh z � 2ν+2 ν + 3 � z zh � 4ν+4 ν − 6 � z zh � 2ν+2 ν + ν �zh z � 4ν+4 ν + (2ν + 3) � + O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (22) The first terms is the contribution of Einstien gravity and the second term is related to the Gauss-Bonnet gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure (4), the behavior of f2 in terms of z are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen, by increasing α and q, f2 decreases and by increasing ν, f2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 7 (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (c) l = 1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, zh = 2 Figure 4: Plots of f2 in terms of z for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (left), q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (middle), ν = 1, 2, 3, 4, 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Finally from Ett, one can get V (z) (We did not bring it here due to its bulk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the case of α ≪ 1, one can get V (z) = −2(ν + 1)(2ν + 1) ν2l2 + α � z 2ν+2 ν − z 2ν+2 ν h �3 ν4l4 [−4z 4ν+4 ν z 2ν+2 ν h (−1 − 7ν + 6ν3 + 20ν2) + 8ν(3ν2 + 7ν − 1)z 2ν+2 ν z 4ν+4 ν h + (8ν3 + 64ν2 − 36ν − 12)z 6ν+6 ν + 4(ν − 1)(2ν + 1)z 10ν+10 ν z −4ν−4 ν h − 4(ν − 1)(8ν + 3)z 8ν+8 ν z −2ν−2 ν h − 8ν2(ν + 2)z 6ν+6 ν h ] + O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (23) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (5), the behavior of scalar potential in terms of z for anisotropic case has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the left panel, between 0 < z < zh by increasing α, the potential decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the right panel, between 0 < z < zh by increasing ν the scalar potential increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, zh = 2, l = 1 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2 Figure 5: Plots of V in terms of z for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (left), ν = 1, 2, 3, 4, 5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 Thermodynamics of the background In this subsection, we explore the thermodynamics of the black brane solution (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In order to investigate the thermodynamic properties of the black brane we need to obtain some relevant thermodynamic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The temperature of the black brane is obtained as follows: T = ���� g′ 4π ���� = ν + 1 2πzhν � 1 − 2α ν2l2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (24) It is noticed that, the temperature monotonically decreases with the increase of the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By increasing α, the temperature is decreased, and for α = 0, one can get the result of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The entropy is given as follows [32],[33] S = −1 8 � Σ dn−2x√η δL δRµαβν ǫµαǫβν, (25) where δL δRµαβν = �1 2 + αR � � gµβgαν − gµνgαβ� + 1 2β � Rµβgαν − Rαβgµν − Rµνgαβ + Rανgµβ� + 2γRµαβν, (26) and ǫµν = −2 √ −hδt [µδz ν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For c = 0 we have s = S V = l3P(zh)b(zh) 3 2 4z3 h = l3 4z ν+2 ν h , (27) which is independent of parameter of the EGB gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In terms of the temperature, the entropy is given as s = l3 4 � 2πTν3l2 (ν + 1)(ν2l2 − 2α) � ν+2 ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (28) For isotropic case s ≈ T 3 and for anisotropic case s ≈ T ν+1 ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The free energy density F(T) can be calculated from the entropy density s(T) by integrating as follows F = � sdT = ν 2(ν + 1)Ts, (29) which is related to temperature as F ≈ T 2ν+1 ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The sound velocity c2 s which can directly measure the conformality of the system, can be obtained from the temperature and entropy: c2 s = d log T d log s = ν ν + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (30) For isotropic case (ν = 1), c2 s = 1/3, the system is conformal, for anisotropic (ν ̸= 1), c2 s ̸= 1/3 the system is non-conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The heat capacity is given as CV = T ds dT = (ν + 2) ν s = s c2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (31) In terms of temperature, CV T 3 = l3(ν + 2) 4ν � 2πν3l2 (ν + 1)(ν2l2 − 2α) � ν+2 ν T 2−2ν ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (32) 9 For isotropic case, the right hand side of (32) has a constant value and for ν > 1 depends to the temperature and at high temperature goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Since entropy is positive therefore CV is positive and the black hole is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure 6, the behavior of s, F and CV in terms of T for isotropic (dashed lines) and anisotropic (solid lines) and different values of α have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen, by increasing α, the thermodynamical quantities s, F, and CV increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For T < T i cross, the entropy, free energy, and heat capacity of anisotropic case is larger than isotropic, and for T > T i cross vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Where i = s, F, CV and T i cross are given as follows (a) l = 1 (b) l = 1 (c) l = 1 Figure 6: Plots of s, F and CV in terms of T for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 and ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid line) and ν = 1 (dashed lines) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' T s cross = � l4((l2 − 2α)3ν 3(ν+2) ν π 2(1−ν) ν 2 ν+2 ν )ν l4ν((ν + 1)(l2ν2 − 2α))2(ν + 1)(l2ν2 − 2α)ν � 1 2ν−2 , (33) T F cross = � (l2 − 2α)24 ν+1 ν π 2−2ν ν ν −3ν−6 ν + ν −4ν−6 ν � ν 2(ν−1) [(ν + 1)(l2ν2 − 2α)ν+2l4(ν−1)] 1 2ν−2 , (34) T CV cross = � l4(ν+1)((l2ν2 − 2α)3(ν + 2)ν 2(ν+1) ν π 2(1−ν) ν 2 ν+2 ν )ν 3ν(ν + 1)(l2ν2 − 2α)ν+2 � 1 2ν−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (35) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 The case c ̸= 0, µ = 0 In this case At = 0 and the differential equation (12), becomes 4νlz2e− cz2 4 (l2ν2e− cz2 2 − α(2 + νcz2)2g(z))g′′ − 2zle− cz2 4 (ν2l2(4 + (2 + 3z2)ν)− α(2 + νcz2)(8 + 4ν + 6νcz2 − 6cν2z2 + c2ν2z4)g)g′ − 4ανz2le− cz2 4 (2 + νcz2)2g′2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (36) In order to solve equation (36), we assume g(z) as follows g(z) = 1 + ǫg1(z) + O(ǫ2), (37) by inserting it into the (36), one can achieve a homogeneous differential equation for g1(z) as g′′ 1 + [−l2ν3(2 + 3cz2) − 4ν2(l2 + 4αcz2) + 4αν(2 + cz2) + 16α]g′ 1 νz(l2ν2 − 4α) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (38) 10 Solving (38) give g1(z) as g1(z) = c1 + c2erf \uf8eb \uf8ed1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) z \uf8f6 \uf8f8 , (39) where c1 and c2 are constants of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Using (39), the metric (37) becomes g(z) = 1 + c1 + c2erf \uf8eb \uf8ed1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) z \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (40) The conditions (9) give us c1 and c2 as c1 = 0, c2 = − 1 erf � 1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) zh �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (41) Finally, using (41) the metric function becomes g(z) = 1 − erf � 1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) z � erf � 1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) zh �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (42) In the case of α ≪ 1 and c < 0, the blakening function become g(z) ≈ 1− erf �√−6cz 2 � erf �√−6czh 2 � − 4(2ν + 1)√−6cα � ze 3cz2 2 erf �√−6czh 2 � − zherf �√−6cz 2 � e 3cz2 h 2 � 3√πl2ν2erf �√−6czh 2 �2 +O � α2� , (43) and for α ≪ 1 and c > 0, we have g(z) ≈ 1 − z zh e 3c(z2−z2 h) 2 − 2αc(2ν + 1) ν2l2 z(z2 − z2 h) zh e 3c(z2−z2 h) 2 + O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (44) In figure (7), the behavior of g(z) for positive and negative values of warp function is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As c increases, the metric slope becomes more decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Also, changing the value of c has no effect on the value of the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By substituting the obtained metric (42), we arrive at the differential equation for the scalar field as: 11 (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 Figure 7: Plots of g(z) in terms of z for c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (dashed lines) and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (solid lines) (left), ν = 1, 2, 3, 4, 5, 6, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='φ′2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='z2l2ν32π(l2ν2 − 4α)erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='��[2α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cπAz(2 + νcz2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='Acz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (ν2c2z4(5l2ν2 + 32αν − 4α) − 4(ν + 2)(l2ν2 − 4α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 2νcz2(3ν3l2 + 20αν + 3l2ν2 + 4α)) − 8cανz2(2 + νcz2)2(3l2ν2 + 16αν − 4α)eAcz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 2l2ν2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cπAerf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cz2ν(νl2+8α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν2l2−4α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(νcz2(3l2ν2 + 32αν + 4α) − 2(ν + 2)(l2ν2 − 4α)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− π(l2ν2 − 4α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=') − erf( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(l2νe− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 erf( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=')(−8 + 8ν + 3ν2c2z4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 18ν2cz2) + α(2 + νcz2)(erf( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=') − erf( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='))(−16 + 16ν + 3ν2z6c3 + 2νc2z4(1+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='11ν) + 4cz2(6ν2 − 2 + 5ν)))] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='where ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='A = 3l2ν2 + 16να − 4α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='l2ν2 − 4α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (46) In figure 8, the behavior of imaginary and real part of φ(z) in terms of z for different values of parameters has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen, the imaginary part of scalar field inside the black hole is zero and outside the black hole the scalar field is unstable, and by increasing α instability increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 12 (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, zh = 2, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 8: Plots of real (dashed lines) and imagenary (solid lines) part of φ in terms of z for c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (left), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In equation (47) the exact coupling function f2 and approximatly to first order in α in equation (48) has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure (9), f2 for positive/negative c and for different values of parameters has been plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The important feature of the figures is that for the negative c, f2 goes to the negative values by increasing z, while it does not become negative anywhere for the positive c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='f2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(1 − ν)z− 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ν3q2(l2ν2 − 4α)erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�2 [πzα ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='Ae ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cAz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(cz2l2ν4(cz2 − 6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ ν3(32c2z4α − 2l2(cz2 + 10)) + ν2(−8l2 + α(12c2z4 + 88cz2)) + 40αν(cz2 + 2) + 32α) − 4√παcz2ν(2+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='νcz2)(3l2ν2 + 16αν − 4α)ecAz2 + (l2ν2 − 4α)(−2l2ν2πz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cz2ν(l2ν+8α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='l2ν2−4α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(l2νerf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (3νcz2 + 4ν + 4) + [erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=']α(16 + 8cz2 + ν2(z6c3 − 2c2z4) + ν(16 + 6c2z4 + 12cz2))],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(47) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='In the case of α ≪ 1 and c < 0 one can get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='f2 ≈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='l2(ν − 1)e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2√−6czνe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 + 4ν + 3νcz2� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−6cz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν2q2z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν q2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(ν − 1)α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν4q2π3z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ν erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�2 [−12z2π2cν2(2 + νcz2)e3cz2 − 4(2ν + 1)zecz2(4π2cνzhe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−6cerf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='νcz2 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3ν − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−6cνz)e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (c2z4ν2 − 6cν2z2 − 2νcz2 − 20ν − 8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6cz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− 4(2ν + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−6cπ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 zh(4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3ν + νcz2)erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6cz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ νπ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6cz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√−6czh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='��2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(16 + 8cz2 + ν(16 + 16c2z4 + 12cz2) + ν2(z6c3 − 2c2z4))] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ O(α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (48) (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (c) l = 1, zh = 2, q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 9: Plots of f2(z) in terms of z for c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines) and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (dashed lines) (left), ν = 2, 3, 4, 5, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines) and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (dahsed lines) (middle), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines) and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (dashed lines) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='From equation Ett we get the expression for the scalar potential V as a function of z as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='V = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='ecz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4ν3l4π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (l2ν2 − 4α)erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� √−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�2 [−2απze ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cAz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 √ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='[l2z2cν5(5cz2 + 3c2z4 + 6) + ν4(−32c2z4α + l2(20cz2 + 20 + c2z4)) + ν3(4l2 + 10l2cz2 − α(88cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ 4c2z4 + 12c3z6)) − ν2(8l2 + α(80 + 36c2z4 + 48cz2)) − να(8cz2 + 48) + 32α] + ν[8cαz2√π(1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− ν)(2 + νcz2)(3l2ν2 + 16αν − 4α)ecAz2 + (l2ν2 − 4α)[−2πνzl2√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cA(3νcz2 + 4ν + 2)erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='cνz2(l2ν+8α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='l2ν2−4α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='[l2erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e− cz2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 (8 + ν2(9c2z4 + 12cz2 + 16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='+ ν(6cz2 + 8)) + (2 + cz2)α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='− erf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='−2cAzh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='(ν2(3z6c3 − 2c2z4) + ν(20cz2 + 16+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='14c2z4 + 32 + 16cz2))]]]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (49) (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 (b) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, l = 1, zh = 2 Figure 10: Plots of V (z) in terms of z for c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (dashed lines) and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines) (left), α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (dashed lines) and c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Regardless of the sign of c, V (z) is negative under the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In panel a, with the increase of |c|, |V | decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In panel b, as α increases, |V | increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 Thermodynamics of background The Hawking temperature can be obtained by using the surface gravity interpretation T = κ 2π = ���� g′ 4π ���� = e cAz2 h 2 √ −2cA 4π 3 2 erf �zh 2 √ −2cA �, (50) 15 where A provided in (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the case of small α and c < 0, one can get T ≈ √−6ce 3cz2 h 2 4π 3 2 erf � √−6czh 2 � + (2ν + 1)e 3cz2 h 2 α[π√−6c(cz2 h + 1 3)erf � √−6czh 2 � − 2√πczhe 3cz2 h 2 ] π 5 2 l2ν2erf � √−6czh 2 �2 + O � α2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (51) The behavior of temperature is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (11)a, the variation of temperature with respect to the horizon radius zh for different values of the coupling of theory α is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen there exists a minimum temperature Tmin below which no black hole solution exist (thermal gas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' However, for T > Tmin, there are two black hole solutions, a large and a small one (deconfined quark gluon plasma phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The small black hole phase for which T increases with zh whereas the large black hole phase for which T decreases with zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (11)b, to study the stability of the solutions, we have shown the behavior of heat capacity CV and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen the large black hole has positive heat capacity and therefore is stable and small black hole is unstable and thus not physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 11: Plots of T in terms of zh for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 and ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Plots of CV and T in terms of zh for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (right) Following the standard Bekenstein-Hawking formula (27), one can easily read the black hole entropy density s, which is defined as s = l3e− 3cz2 h 4 4z ν+1 ν h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (52) The scaled entropy density s/T 3 as a function of scaled temperature T/Tmin is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The red lines correspond to the large stable solution and blue lines are for the small unstable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The numerical result of the square of the sound velocity is shown in 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' At Tmin, the sound velocity square is around 0 which is in agreement with lattice data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' At high temperature, the sound velocity square goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='45 for ν = 1, which means that the system is approximatly asymptotically conformal, while for ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, the sound velocity square goes to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The numerical result of the specific heat is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 12c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' It can be clearly seen that the 16 specific heat CV diverges at Tmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' At T → ∞, the scaled specific heat CV /T 3 approaches to the zero for this approximate solution [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (b) l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (c) l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 Figure 12: Plots of scaled entropy density s/T 3 (left), c2 s (middle) and CV /T 3 (right) in terms of scaled temperature T/Tmin for ν = 1, Tmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='19457 (dashed lines), ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, Tmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1357 (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In each panel red lines correspond to the large stable solution and blue lines correspond to unstable small solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure 13, the behavior of c2 s and CV have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen from the figure and obtained the result in the previous section, it can be concluded that the heat capacity has an inverse relationship with the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 Figure 13: Plots of c2 s (red lines) and CV (blue lines) in terms of zh for ν = 1 (dashed lines), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 The case c ̸= 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' µ ̸= 0 In this case the differential equation (12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' becomes 4νlz2e− cz2 4 (l2ν2e− cz2 2 − α(2 + νcz2)2g(z))g′′ − 2zle− cz2 4 (ν2l2(4 + (2 + 3z2)ν)− α(2 + νcz2)(8 + 4ν + 6νcz2 − 6cν2z2 + c2ν2z4)g)g′ − 4ανz2le− cz2 4 (2 + νcz2)2g′2 − c2ν3lc2 2z 4ν+2 ν e cz2 4 = 0 (53) in order to solve the above differential equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' we assume g(z) = 1 + ǫg1(z) + O(ǫ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (54) by inserting it,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' one can achieve a non-homogeneous differential equation as g′′ 1 + [−l2ν3(2 + 3cz2) − 4ν2(l2 + 4αcz2) + 4αν(2 + cz2) + 16α]g′ 1 νz(l2ν2 − 4α) + cνc2 2z 2 ν e cz2 2 4α(4 + cνz2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (55) In order to solve the non-homogeneous differential equation (55) in the case of c < 0, c2 ̸= 0, we consider the following particular solution: g2(z) = e cz2 2 z− 2(ν−1) ν � i hizi, (56) where hi are coefficients of expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' By inserting the above solution into the differential equation (55), and solving order by order for coefficients, one can get h0 = h1 = h3 = h5 = 0, h2 = c2 2ν2 8α(ν − 2), h4 = − cν3c2 2(−20αν + l2ν3 + l2ν2 − 12α) 16α(ν − 2)(ν4l2 + 3l2ν3 − 4αν2 + 2l2ν2 − 12αν − 8α), h6 = ν4c2c2 2 32α(2 + 3ν)(l2ν2 − 4α)2(1 + 2ν)(1 + ν)(ν2 − 4)(3l4ν6 + 6l5ν5 − 40ν4l2α + 3ν4l4 + 256α2ν3 − 56ν3l2α + 624ν2α2 − 24ν2l2α + 448α2ν + 112α2), (57) where c2 used from (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Therefore, the metric function by considering a particular solution becomes g(z) = 1 − erf � 1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) z � erf � 1 2 � c(−6l2ν2 − 32να + 8α) (l2ν2 − 4α) zh � + e cz2 2 z− 2(ν−1) ν � i hizi, (58) with hi are provided in (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In figure 14, the effect of µ on the behavior of f2 and V have been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As can be seen in left panel by increasing chemical potential, f2 decreases and in right panel V increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 18 (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, c = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, c = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 14: Plots of f2 and V in terms of z for µ = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (b) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 (c) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 Figure 15: Plots of s, T and CV in terms of zh for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='22728, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 (left) and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='22728, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='3 (middle and right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 15a, the behavior of entropy is shown in terms of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The figure shows the minimum and maximum temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As µ increases, the minimum temperature decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' This plot also shows that for 0 < µ < µc, there are minimal Tmin and maximal Tmax temperatures, between which the entropy is a function of T with three branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' When we decrease the temperature, the entropy decreases along the first branch (Tmin < T < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Then the entropy decreases along the second branch with an increase of temperature from Tmin to Tmax, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' here the black holes are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Finally the entropy increases along the third branch with an increase of temperature for 0 < T < Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Such a behavior in terms of event horizon, one can see in the 15b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In each panel, the critical point has been shown in red color curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Upon varying the Hawking temperature, a phase transition from the large black hole phase to the thermal AdS phase takes place at a critical temperature Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' This is the famous black hole- thermal AdS Hawking-Page phase transition which occurs in the presence of chemical potential µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='22728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 16 by using the approximate analysis, the phase diagram of the holographic QCD model for anisotropic background and for Einstien gravity (long dashed line), Einstien- 19 Gauss-Bonnet gravity (solid line) has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' As one can see at µ = 0 the system undergoes a black hole to thermal gas first order phase transition so that T (EGB)(µ = 0) > T (E)(µ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For 0 < µ < µc (in the transition lines), the system undergoes a large black hole to a small black hole first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For 0 < µ < µI, the temperature of the black hole to black hole transition of Einstien gravity (T (E)(µ)) is less than Einstien-Gauss-Bonnet gravity (T (EGB)(µ)) and for µI < µ < µc vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The first order phase transition stops at the critical point (µc, Tc), where the phase transition becomes second order, herewith T (EGB) c < T (E) c , µ(EGB) c < µ(E) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For µ > µc, the system has a sharp but smooth crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' These thermal AdS and black hole phases in the usual language of gauge-gravity duality are dual to the confinement and deconfinement phases in the dual boundary theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (a) ν = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5, l = 1, c = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5 Figure 16: The phase diagram in T and µ plane for anisotropic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' At small µ, the system undergoes a first order phase transition at finite T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The first order phase transition stops at the critical point (µc, Tc) ∼ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='22728, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1242), where the phase transition becomes second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The solid line is for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='1 and long dashed line is for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 3 Conclusion In this work, we extended the AdS/QCD model to quadratic gravity to gain insight into the influence of gravity on QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' To do so, we considered an anisotropic black hole metric as a solution to a system of 5D Einstein-quadratic-two Maxwell-dilaton fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The anisotropic background is specified by an arbitrary exponent, a non-zero dilaton field, a non-zero time component of the first Maxwell field, and a non-zero longitudinal magnetic component of the second Maxwell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The field equations for the considered theory are coupled and bulky differential equations for six unknown functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Therefore, obtaining the solution to such field equations is too hard, this is why we considered the special case of field equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' γ = α, β = −4α (Einstien- Gauss-Bonnet gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The differential equation for the metric function in EGB gravity is a nonlinear second-order equation that has been solved in special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' At the first, we obtained 20 the exact solutions for the differential equations with zero warp functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this case, it doesn’t occur any thermodynamical phase transitions to the black brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The second case that we have considered is the case with zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The blackening function in this case supports the Van der Waals-like phase transition between small and large black holes for suitable values of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' The third case that has been considered is nonzero warp function and chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In this case, in addition to the small/large phase transition, the blackening function supported the phase transition from the large black hole phase to the thermal AdS phase at a critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Holographically, this phase transition corresponds to the confinement- deconfinement phase transition in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In each case, we investigated the anisotropy influence and the effect of parameters of theory on the thermodynamic properties of our background, in particular, on the small/large black holes phase transition diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (16), the effect of the Gauss-Bonnet term on the phase transition has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' This figure shows that before/after the intersection point for constant chemical potential, the temperature of a black hole in EGB gravity is more/less than that of Einstien gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Clearly, before the intersection point, α has a dominant impact on the temperature (compared to the effect of µ on the temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' In the isotropic case corresponding to ν = 1 (zero magnetic fields) and , α → 0 reproduces previously known results [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' For future work, one can consider the Weyl-squared term by using the combination γ = 6α and β = −4/3γ for the parameters of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' But since in this case, the field equation for the metric has a 4th-order derivative, the differential equations should be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Also, following the paper [9], one can study the effect of the magnetic field on the system in the framework of EGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Acknowledgements I would like to thank the referee for her/his fruitful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' I also would like to thank the School of Physics of the Institute for Research in Fundamental Sciences (IPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' Constants The constants related to equation(19): E = (ν + 1)((ν + 1)(ν4l4 − 8αc2) − 4ανc1z 2ν+2 ν ), D = νl2 − ν2l2 − � ν4l4 − 8αc2, F = νl2 − ν2l2 + � ν4l4 − 8αc2, G = ν3l2 − νl2 − √ E ν + 1 , H = −l2ν(ν − 1) � ν4l4 − 8αc2 + ν4l4 − 8αc2, K = −l2ν � ν − 1 2 � √ E − 2αc1νz 2ν+2 ν + (ν + 1) � ν4l4 − 4αc2 − l4ν3 2 � , L = l2ν(ν − 1) � ν4l4 − 8αc2 + ν4l4 − 8αc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' (59) 21 The constants related to equation(21): ¯ A = (ν + 1)(−4αc1νz 2ν+2 ν + (ν + 1)(ν4l4 − 8αc2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯B = −4αν2c1z 2ν+2 ν + (ν + 1)(ν + 3 2)(ν4l4 − 8αc2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯C = −4ανc1z 2ν+2 ν + (ν + 1)(ν4l4 − 8αc2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯E = −4c2 1α2ν2(3ν3 − 4ν2 + 3ν + 2)z 2(2ν+2) ν + 2c1να(ν2 − 1)(ν4l4 + 2l4ν3 − 16ανc2 − 8αc2)z 2ν+2 ν + (ν − 1)(ν + 1)2(ν4l4 − 8αc2)(4αc2 + ν5l5 − l4ν3 − ν4l4 2 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯F = ν7l4 − 7 2ν6l4 + 2ν5l4 + 5 2ν4l4 + 4ν3αc2 + 24αν2c2 − 36ανc2 − 8αc2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯H = 4c2 1α2(5ν + 2 − 2ν2 + ν3)z 2(2ν+2) ν − 4c1α(ν2 − 1)(l4ν3 − 8ανc2 + 4αc2)z 2ν+2 ν − (ν + 1)2(ν − 1)ν2l4(ν2 − 1 2ν − 1)(ν4l4 − 8αc2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯G = ν5l4 + 2αc2 + 1 2ν4l4 − 1 2ν3l4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯K = ν5l4 + 1 3ν4l4 − 1 2l4ν3 + 4αc2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' ¯L = 576α2c2 2 ¯G ¯C + 1 2 ¯ A2 ¯H,' 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+page_content='1007/JHEP09(2011)041 [arXiv:1103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content='5389 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf'} diff --git a/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/2301.12061v1.pdf.txt b/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/2301.12061v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc426842541e6ddbbe1bc38e34ec5f13215f5743 --- /dev/null +++ b/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/2301.12061v1.pdf.txt @@ -0,0 +1,4003 @@ +(Private) Kernelized Bandits with Distributed Biased Feedback +Fengjiao Li* +Xingyu Zhou† +Bo Ji* +Abstract +In this paper, we study kernelized bandits with distributed biased feedback. This problem is motivated +by several real-world applications (such as dynamic pricing, cellular network configuration, and policy +making), where users from a large population contribute to the reward of the action chosen by a central +entity, but it is difficult to collect feedback from all users. Instead, only biased feedback (due to user +heterogeneity) from a subset of users may be available. In addition to such partial biased feedback, we +are also faced with two practical challenges due to communication cost and computation complexity. To +tackle these challenges, we carefully design a new distributed phase-then-batch-based elimination (DPBE) +algorithm, which samples users in phases for collecting feedback to reduce the bias and employs maximum +variance reduction to select actions in batches within each phase. By properly choosing the phase length, +the batch size, and the confidence width used for eliminating suboptimal actions, we show that DPBE +achieves a sublinear regret of ˜O(T 1−α/2 + √γT T), where α ∈ (0, 1) is the user-sampling parameter +one can tune. Moreover, DPBE can significantly reduce both communication cost and computation +complexity in distributed kernelized bandits, compared to some variants of the state-of-the-art algorithms +(originally developed for standard kernelized bandits). Furthermore, by incorporating various differential +privacy models (including the central, local, and shuffle models), we generalize DPBE to provide privacy +guarantees for users participating in the distributed learning process. Finally, we conduct extensive +simulations to validate our theoretical results and evaluate the empirical performance. +1 +Introduction +Bandit optimization is a popular online learning paradigm for sequential decision making and has been widely +used in a wide variety of real-world applications, including hyperparameter tuning [Li17], recommendation +systems [Li10], and dynamic pricing [MSA19]. In such problems, each decision point (called an arm or +action), if chosen, yields an unknown reward. The goal of the agent is to maximize the cumulative reward +by making proper decisions sequentially. An important way to capture general (e.g., non-linear and even +non-convex) unknown objective functions is to consider a smoothness condition specified by a small norm of +a Reproducing Kernel Hilbert Space (RKHS) associated with a kernel function. This setup is often referred +to as kernelized bandits. +Thanks to the strong link between RKHS functions and Gaussian processes (GP) [Kan18; CG17; Sri09], +an extensive line of work has exploited GP models to estimate an unknown function f given a set of (noisy) +evaluations of its values f(x) at chosen actions x. However, in many applications, the value f(x) could +represent an overall effect of action x on a large population of users where it is difficult for the learning agent +to make direct observations; yet, the agent could collect some partial feedback from the distributed users in the +*Fengjiao Li (fengjiaoli@vt.edu), Bo Ji (boji@vt.edu), Department of Computer Science, Virginia Tech. +†Xingyu Zhou (xingyu.zhou@wayne.edu), Department of Electrical and Computer Engineering, Wayne State University. +This work has been accepted by ACM SIGMETRICS’23. +1 +arXiv:2301.12061v1 [cs.LG] 28 Jan 2023 + +Local Model: +item 1 +Decision +Global Model: +Pricing +Customer +$5 +item 2 +$9 +item 3 +$8 +... +Expected total profits +Local expense +Influence +(Private) +Aggregation +Local Feedback +Figure 1: Dynamic pricing: a motivating application of our problem. +population. In addition, feedback from these users could be biased due to user heterogeneity (e.g., different +preferences). Therefore, we assume that each user u in the population is associated with a local function fu, +which is a function sampled from a GP with mean f. Consider the dynamic pricing problem [MSA19] as +an example (see Figure 1). When a company sets a pricing mechanism x, this decision influences all the +customers, and every customer, based on her individual demand and preference, makes a choice (purchase or +not), which contributes to the total profits f(x). Without knowing products’ demand curves in advance, the +company makes a sequence of pricing decisions with the goal of maximizing profits while learning. That is, +the company aims to infer the expected demand and thus the expected profits f by collecting feedback from +customers in each decision epoch. Note that it might be difficult for the company to collect feedback from all +the customers - since purchases may take place at many local stores at different locations. For example, it is +impractical for McDonald’s headquarters to collect sales information from all of the nationwide customers +within each decision epoch. Instead, the headquarter might be able to get feedback (i.e., sales information) +from a subset of the customers. However, each customer’s choice depends not only on her own preference +towards the products and their prices but also on several other factors (location, competitors, promotion +events, etc.), which is often biased feedback for the overall profits. +To that end, we study a new kernelized bandit setting where the agent could not get direct evaluations +of the unknown reward function but only distributed biased feedback. We refer to this setting as kernelized +bandits with distributed biased feedback. This bandit problem is shared by several other practical applications, +including cellular network configuration [Mah21] and public policy making [BRA20]. However, existing +learning algorithms developed for standard kernelized/GP bandits (e.g., GP-UCB [Sri09; CG17]) rarely +consider such partial biased feedback in a distributed setting. To solve this new problem, a learning algorithm +needs to be able to learn the unknown function from such biased feedback in a sample-efficient manner. +Moreover, two practical challenges naturally arise in our problem: communication cost due to distributed +learning [Che21a] and computation complexity due to GP update [Cal22]. Therefore, not only need the +learning algorithms be sample-efficient, but they must also be scalable in terms of both communication +efficiency and computation complexity. +To that end, we propose the learning with communication framework where the biased feedback is +communicated in phases, and design a new distributed phase-then-batch-based elimination algorithm that +2 + +aggregates the distributed biased feedback in a communication-efficient manner and eliminates suboptimal +actions in a computation-efficient manner while achieving a sublinear regret. Our main contributions are +summarized as follows. +• To the best of our knowledge, this is the first work that studies a new kernelized bandit setting +with distributed biased feedback, where three key challenges (user heterogeneity, communication +efficiency, and computation complexity) inherently arise in the design of sample-efficient, scalable +learning algorithms. While it is natural to consider phased elimination type of algorithms in such +settings, the standard phased elimination algorithm relies on the so-called (near-)optimal experimental +design [LSW20], which cannot be directly applied to kernelized bandits due to the possible infinite +feature dimension of RKHS functions. +• To that end, we design a new phased elimination algorithm, called distributed phase-then-batch- +based elimination (DPBE), which is carefully crafted to address all the aforementioned challenges. +In particular, DPBE adds a user-sampling process to reduce the impact of bias from each individual +user and selects actions according to maximum variance reduction within each phase. Moreover, a +batching strategy is employed to improve both communication efficiency and computation complexity. +That is, instead of selecting a new action at each round, DPBE plays the same action for a batch of +rounds before switching to the next one. Not only does it help reduce the number of times one needs to +compute the next action via GP update, but it also allows for reducing the dimensions of the vectors +and matrices involved in both communication and computation. +• We show that DPBE achieves a sublinear regret of ˜O(T 1−α/2 + √γT T)1 while incurring a communica- +tion cost of O(γT T α) and a computation complexity of O((|D|γ3 +T + γ4 +T ) log T + γT T α), where γT is +the maximum information gain associated with the kernel of the unknown function f, D is the decision +set, and α > 0 is a user-sampling parameter that we can tune. It is worth noting that DPBE with +α ∈ (0, 1) has a better computation complexity than some variants of the state-of-the-art algorithms +(originally developed for standard kernelized bandits without biased feedback). Specifically, DPBE +achieves three significant improvements compared to the state-of-the-art algorithms: (i) user-sampling +efficiency (O(T α) vs. T), (ii) communication cost (O(γT T α) vs. T), and (iii) computation complexity +(O(γT T α) vs. O(T 3)). Furthermore, we conduct extensive simulations to validate our theoretical +results and evaluate the empirical performance in terms of regret, communication cost, and running +time. +• Finally, we generalize our phase-then-batch framework to incorporate various differential privacy (DP) +models (including the central, local, and shuffle models) into DPBE, which ensures privacy guarantees +for users participating in the distributed learning process. +2 +Related Work +Kernelized bandits. Since [Sri09] studied GP in the bandit setting, kernelized bandits (also called GP +bandits) have been widely adopted to address black-box function optimization over a large or infinite +domain [CG17]. Considering different application scenarios, kernelized bandits under different settings +have recently been studied, including heavy-tailed payoffs [RG19], model misspecification [BK21], and +corrupted rewards [Bog22]. As typically considered in the literature, these works also assume that direct +1The notation ˜O(·) ignores polylog terms. Bounds on γT of different kernel functions can be found in Appendix A.2. +3 + +(noisy) feedback of the unknown function at a chosen action is available to the agent. In sharp contrast, we +study a new, practical setting where only distributed biased feedback can be obtained. Under this setting, +not only does one need to use biased feedback in a sample-efficient manner, but one also has to consider +communication efficiency, which is a common issue in distributed bandit-learning settings. +Distributed/collaborative kernelized bandits. While distributed or collaborative kernelized bandits have +been studied recently [Du21; DLJ20; Sim21], we highlight the key difference between our model and theirs +as follows: motivated by real-world applications, we aim to learn one (global) bandit while most of them also +aim to learn every local model, which results in quite different regret definitions (their group regret vs. our +standard regret defined in Section 3). Moreover, they assume that every party (corresponding to a user in our +problem) shares the same objective function. While [DLJ20] also studies similar bandit optimization with +biased feedback, they assume a fixed number of local agents and bound the regret in terms of the distance +between the target function and local functions, which could be very large. In addition, [DLJ20] does not +consider communication efficiency, which is a key challenge in distributed learning. +Recently, the work of [LZJ22] studies a similar global reward maximization problem without direct +feedback and also employs a phase-based elimination algorithm. However, the main difference is that they +only consider linear bandits by assuming a linear reward function while we study kernelized bandits that can +capture general non-linear and even non-convex functions and recover linear bandits as a special case when +choosing a linear kernel. This strict generalization introduces three unique challenges: (i) different from +the linearly parameterized bandits where the bias in the feedback can be quantified with a same-dimension +random vector (i.e., ξu = θu − θ∗ ∈ Rd at each user u), it is unclear how to make an assumption of the +bias in the non-parametric kernelized bandits setting in order to learn the unknown global reward function; +(ii) due to the possible infinite feature dimension of functions in an RKHS, the (near-)optimal experimental +design approach used in the phased-elimination algorithm for linear bandits cannot be directly adapted to +kernelized bandits. Despite some recent efforts towards extending this experimental design based approach +to kernelized bandits [Zhu21; CJK21], there still remain some key limitations (see our discussion below); (iii) +since computation complexity is a critical bottleneck in kernelized bandits, a proper computation-efficient +learning algorithm is desired when addressing our problem. +Experimental design for kernelized bandits. In [Zhu21], the authors propose to adaptively embed the +feature representation of each action into a lower-dimensional space in order to apply the (near-)optimal +experimental design for finite-dimensional actions. However, the intermediate regret due to the approximation +error over T rounds is not considered at all because their goal is to find an ε-optimal arm at the end of T (i.e., +a pure exploration problem) rather than minimizing the cumulative regret. While [CJK21] aims at minimizing +the cumulative regret, their algorithm and analysis are more complex than ours: it requires a non-standard +robust estimator, obtaining an optimal distribution on the simplex, drawing samples from this distribution, +and solving a second optimization problem. In contrast, we simply use the standard GP posterior mean and +variance estimators, which can be computed in closed-form. Moreover, our algorithm can also be easily +extended to handle infinite action sets (see Remark 4.2) rather than a finite set considered in [CJK21]. +3 +Preliminaries +Notation. Throughout this paper, we use lower-case letters (e.g., x) for scalars, lower-case bold letters (e.g., +x) for vectors, and upper-case bold letters (e.g., X) for matrices. Let [n] ≜ {1, . . . , n} denote any positive +integer up to n, let |U| denote the cardinality of set U, and let ∥x∥2 denote the ℓ2-norm of vector x. +4 + +3.1 +Problem Setting +We introduce a new kernelized bandit problem where the unknown function represents the overall reward +over a large population containing an infinite number of users. The unknown reward function f : D → R +is assumed to be fixed over a finite set of decisions D ⊆ Rd. At round t, the agent chooses an action +xt ∈ D, leading to a reward with mean f(xt). This reward is unknown to the agent but captures the overall +effectiveness of action xt over the entire population U, thus called global reward. Meanwhile, each user u +in the population observes a (noisy) local reward: yu,t = fu(xt) + ηu,t with mean fu(xt), where ηu,t is the +noise, and fu : D → R is the local reward function, assumed to be an (unknown) realization of a random +function (specified soon) with mean f. In this setting, the exact global reward corresponding to the entire +population cannot be observed; only biased local reward feedback is available to the agent. We make the +following assumptions about the unknown function f, the local function fu, and the noise in the reward +observations. +Assumption 1. We assume that function f is in the Reproducing Kernel Hilbert Spaces (RKHS), denoted +by Hk. Note that RKHS Hk is completely specified by its kernel function k(·, ·) (and vice-versa), with an +inner product ⟨·, ·⟩k obeying the reproducing property: f(x) = ⟨f(·), k(x, ·)⟩k for all f ∈ Hk [CG17]. We +list the most commonly used kernel functions (such as Squared Exponential (SE) and Mat´ern kernels) in +Appendix A. Moreover, we assume that function f has a bounded norm: ∥f∥k ≜ +� +⟨f, f⟩k ⩽ B, and that the +kernel function is also bounded: k(x, x) ⩽ κ2 for every x ∈ D, where both B and κ are positive constants. +Assumption 2. When the agent samples a user u to collect feedback, the local reward function fu at u is as- +sumed to be a function sampled from the GP with mean f and covariance2 k(·, ·), i.e., fu ∼ GP(f(·), k(·, ·)). +In addition, we assume that each user is sampled independently for collecting feedback. +Assumption 3. We assume that the observation noise ηu,t∼N(0, σ2) is Gaussian with variance σ > 0 and +that it is independent and identically distributed (i.i.d.) over time and across users. +The goal of the agent is to maximize the cumulative global reward, or equivalently, to minimize the regret +defined as follows: +R(T) ≜ +T +� +t=1 +� +max +x∈D f(x) − f(xt) +� +. +(1) +3.2 +Learning with Communication +For black-box function optimization based on noisy bandit feedback, kernelized bandit algorithms have +shown strong empirical and theoretical performance. However, the agent in our problem setting does not have +access to unbiased feedback of the object function f but has to collect biased feedback from distributed users +from a large population. This scenario leads to the following framework of learning with communication. +Communication happens when some users are selected to report their feedback to the agent based on their +biased local reward observations. By aggregating such biased feedback from the users, the agent improves +her confidence in estimating function f and adjusts her decisions in the following rounds accordingly. To +account for scalability, the agent collects distributed feedback from users periodically instead of immediately +after making each decision. We call the time duration between two communications as a phase. Consider +2Our theoretical framework is applicable to a more general setting where the covariance of the local reward function is v2k(·, ·), +i.e., fu ∼ GP(f(·), v2k(·, ·)). This scaling parameter v2 captures the variance of the bias in the local reward function fu with its +mean being the global reward function f. For this more general setting, our theoretical results still hold with only a slight adjustment +to the posterior variance in the confidence width function (12). +5 + +a particular phase l. Let Tl be the set of round indices in the l-th phase and Ul be the set of selected users, +called participants, that will report their feedback. With the actions {xt : t ∈ Tl} chosen by the agent in this +phase, each user u in Ul sends the feedback g({yu,t}t∈Tl) to the agent at the end of the phase, where g(·) is +a function (e.g., the average) of the local reward observations and is assumed to be the same for all users. +Then, by aggregating all feedback {g({yu,t}t∈Tl)}u∈Ul, the agent estimates f and decides xt for round t in +the next phase Tl+1. This learning with communication process is repeated until the end of T, with the goal +of maximizing the cumulative (global) reward. +In this framework, we assume that the agent can employ some existing incentive mechanisms [Lim20] +in order to collect enough feedback for learning, but the cost has to be considered, e.g., the communication +resources consumed for collecting feedback data. In addition, communication cost is also a critical factor in a +general distributed learning system. In this work, we use the total quantity of communicated numbers (between +the agent and all users) as another metric, in addition to the regret metric, to evaluate the communication +efficiency of learning algorithms for our problem. Let L be the total number of phases in T rounds and +Nu,l ≜ dim (g({yu,t}t∈Tl)) be the dimension of user u’s feedback (which is the number of scalars in user +u’s feedback). Then, the total communication cost, denoted by C(T), is as follows: +C(T) ≜ +L +� +l=1 +� +u∈Ul +Nu,l. +(2) +In the following, we explain the learning with GP framework for standard kernelized bandits. +3.3 +Learning with Gaussian Process +A Gaussian process (GP) over input domain D, denoted by GP(µ(·), k(·, ·)), is a collection of random +variables {f(x)}x∈D where every finite number of them {f(xi)}n +i=1, n ∈ N, is jointly Gaussian with mean +E[f(xi)] = µ(xi) and covariance E[(f(xi) − µ(xi))(f(xj) − µ(xj))] = k(xi, xj) for every 1 ⩽ i, j ⩽ n. +Hence, GP(µ(·), k(·, ·)) is specified by its mean function µ and a (bounded) covariance function k : D×D → +[0, κ2]. Assume that choosing action xt at round t reveals a noisy observation: +yt = f(xt) + ηt, +(3) +where ηt ∼ N(0, λ) is a zero-mean Gaussian noise with variance λ > 0. Standard GP algorithms implicitly +use GP(0, k(·, ·)) as the prior distribution over f. Then, given the observations yt = [y1, . . . , yt]⊤ corre- +sponding to a sequence of actions Xt = [x⊤ +1 , . . . , x⊤ +t ]⊤, the posterior distribution is also Gaussian with the +mean and variance in the following closed-form: +µt(x) ≜ k(x, Xt)⊤(KXtXt + λI)−1yt, +(4) +σ2 +t (x) ≜ k(x, x) − k(x, Xt)⊤(KXtXt + λI)−1k(x, Xt), +(5) +where k(x, Xt) = [k(x, xs)]⊤ +s=1,··· ,t ∈ Rt×1 and KXtXt = [k(x, x′)]x,x′∈Xt ∈ Rt×t is the corresponding +kernel matrix. +Next, we introduce an important kernel-dependent quantity called maximum information gain [Sri09]: +γt(k, D) ≜ +max +X⊆D:|X|=t +1 +2 log det +� +I + λ−1KXX +� +, +(6) +which is often used to derive regret bounds. In addition, we have that γt(k, D) scales sublinearly with t +for most commonly used kernels (see Appendix A). For ease of notation, we often simply use γt to denote +γt(k, D) when the kernel function k and the dataset D are clear from the context. +6 + +Thanks to the strong connection between RKHS functions and GP [Kan18] with the same kernel function +k, one can use the above GP model to approximate unknown function f ∈ Hk within a reliable confidence +interval with high probability. +4 +Algorithm Design +4.1 +New Challenges and Main Ideas +In Section 3, we describe the learning with communication framework, which requires the distributed biased +feedback to be communicated in phases and exhibits experimental scalability. This framework naturally leads +us to consider a phased elimination algorithm that gradually eliminates suboptimal actions by periodically +aggregating and analyzing the local feedback from the participants. However, several new challenges arise in +our setting compared to the standard phase elimination algorithm in linear bandits [LSW20; LS20]. +(i) How to select actions for each phase? The standard phase elimination algorithm often relies on the +so-called near-optimal experimental design (i.e., a probability distribution over the currently active set) that +minimizes the worst-case variance [LS20]. However, due to the possible infinite feature dimension of RKHS +functions, adapting this approach to kernelized bandits setting is nontrivial even with the strong assumptions, +requirements, and complicated algorithm design (e.g., [Zhu21] and [CJK21], see discussion in Section 2). We +are wondering if there is a simple and efficient method of selecting actions in each phase for our kernelized +bandits setting. (Challenge a⃝). +(ii) How to use biased feedback? In contrast to the standard phase elimination algorithm where feedback +is unbiased, in our setting the local feedback from a particular user is biased. In order to reduce the impact +of bias, an efficient user-sampling scheme is needed. However, how to incorporate this idea into the phase +elimination algorithm is unclear (Challenge b⃝). +(iii) How to deal with scalability? In our setting, scalability refers to both computation complexity and +communication cost. On the one hand, it is well-known that standard GP bandits suffer a poor computation +complexity (e.g., O(T 3)) due to the matrix inverse at each step for GP posterior update. On the other hand, +due to the communication between the agent and the users, it is imperative to ensure a low communication +cost (Challenge c⃝). +Our approach. We propose a novel phase elimination algorithm that is able to simultaneously address +all the above challenges. We highlight the main ideas as follows. (i) User-sampling for distributed biased +feedback. In each phase, a well-tuned subset of users is sampled to reduce the impact of bias from each +individual user. (ii) Maximum variance reduction for action selection. Upon selecting the next action within +each phase, it simply selects the one that has the largest posterior variance. (iii) Batching strategy for +scalability. Instead of selecting a new action at each round within a phase, it consistently plays the same +action for a batch of rounds before selecting the next one, i.e., rare-switching. By reducing the number +of times selecting a new action (which could be much smaller than the phase length), it also reduces the +number of unique actions chosen within each phase, which can be utilized to improve the scalability in +terms of both computation and communication through a proper design. Specifically, (a) Computation: via +a posterior reformulation (specified in Section 4.2), we convert the dimension of the matrix in the inverse +operation from the total rounds to the number of batches in each phase; (b) Communication: we let each +participant merge the local reward observations in each batch before sending her feedback at the end of each +phase. That is, the feedback g({yu,t}t∈Tl) from each participating user u in phase l is a vector, where each +element corresponds to the average local reward of a batch. Then, the dimension of the feedback g({yu,t}t∈Tl +becomes the number of batches. For example, consider a particular phase with a total of 10 rounds. Without +7 + +Figure 2: The phase-then-batch strategy: T rounds are divided into L phases; at the end of each phase, +participants report their feedback, which is used for deciding actions in the next phase; within each phase l, +decisions are made in a batched fashion, e.g., playing ah at all the rounds in the h-th batch. +batching strategy, one requires to select an action for each round, i.e., 10 actions for this phase. However, +the batching strategy selects an action for each batch. If each batch has size two, there are 5 batches in this +phase, and the dimension of the matrix in the inverse operation is shrunk from 10 to 5, which will reduce the +computation complexity about 103/53 = 8 times for matrix inverse operations! In addition, by merging local +observations of each unique action, only 5, instead of 10, (averaged) local rewards are communicated at each +user. +4.2 +Distributed Phase-then-Batch-based Elimination (DPBE) +Following the main ideas stated in the above section, we propose the phase-then-batch schedule strategy, +shown in Figure 2 and design the distributed phase-then-batch-based elimination (DPBE) algorithm in +Algorithm 1. +The DPBE algorithm is a phased elimination algorithm, which maintains a set Dl of active actions that +are possible to be optimal and updates the active set after aggregating the distributed feedback. +Consider a particular phase l, DPBE has three main steps: 1) action selection (Lines 5-10); 2) distributed +feedback collection (Lines 12-16); and 3) action elimination (Lines 17-21). +Before describing the details of DPBE, we explain some additional notations used in the algorithm. +Throughout this paper, we use another notation “a” to denote the specific chosen action under our algorithm +to avoid too many subscripts or superscripts for all the batch, phase, or round indices. Consider the l-th phase. +Let tl and Tl be the time index right before the l-th phase and the length of the l-th phase, respectively. Then, +the round indices in the l-th phase can be represented as Tl = {t ∈ [T] : tl + 1 ⩽ t ⩽ tl + Tl}. In addition, +Tl(a) ≜ {t ∈ Tl : xt = a} denotes the time indices when action a is selected in this phase, and Hl represents +the number of batches in the l-th phase. +1) Action selection (Lines 5-10): In the l-th phase, actions are selected from the active set Dl. As +mentioned before, each selection is based on maximum variance reduction [Vak21], and we employ batch +schedule for scalability. Specifically, in the h-th batch, we find the action ah that maximizes a reformulated +posterior variance Σh−1(·) defined in Eq. (7) after h − 1 batches (Eq. (8)). This is possible because the +posterior variance can be computed without knowing any reward observations (see Eq. (5)). Then, play +this action for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 +h−1(ah)⌋ rounds, which forms the h-th batch. Here, the batch size +schedule is inspired by the rare-switching idea in [APS11; Cal22]. This batch schedule strategy enables us to +8 + +a1 +aHi +ahmerge rounds and thus shrink the dimensions of the matrix and vectors used for computing the variance in +Eq. (5). By the end of each batch, we update the variance function by incorporating the action in the current +batch. Let Ah = [a⊤ +1 , . . . , a⊤ +h ]⊤ ∈ Rh×d be the h × d matrix that contains the h chosen actions so far. We +reformulate the standard posterior variance in Eq. (5) and update the posterior variance as follows: +Σ2 +h(x) ≜ k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 +h )−1k(x, Ah), +(7) +where Wh ∈ Rh×h is a diagonal matrix with [Wh]ii = Tl(ai) for any i ∈ [h], and λ is set to be the +noise variance of local observations, i.e., λ = σ2. Here, we reformulate the standard posterior variance in +Eq. (5) with Eq. (7) in order to save computation complexity (especially for computing matrix inverse) while +maintaining the same order of regret (sacrificing only a constant multiplier). +2) Distributed feedback collection (Lines 12-16): To reduce the impact of bias from some specific +user(s), the agent randomly samples a subset of users (called participants) Ul from U to participate in the +learning process (Line 12). We let |Ul| = ⌈2αl⌉, where the user-sampling parameter α > 0 is an input of the +algorithm. Recall that Hl denotes the number of batches in the l-th phase. Each participant u ∈ Ul collects +their local reward observations of each chosen action a ∈ AHl and send the average yu +l (a) for every chosen +action a ∈ AHl as feedback to the agent, i.e., g({yu,t}t∈Tl) = yu +l ≜ [yu +l (a)]a∈AHl. Note that the dimension +of the feedback depends on the number of batches, which is also the communication cost associated with +each participant (Eq. (2)). Therefore, by employing the idea of rare switching, we reduce both computation +complexity and communication cost ( c⃝). +3) Action elimination (Lines 17-21): Aggregate (specifically, average) the feedback from the participants +for each action a ∈ AHl (Line 17). Then, using the aggregated feedback (i.e., the averaged local reward +¯yl = [yl(a1), . . . , yl(aHl)] of the chosen actions a ∈ AHl), the agent can compute the posterior mean +function reformulated as follows (Line 19): +¯µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1¯yl. +(11) +Considering the bias in the feedback due to user heterogeneity ( b⃝), we carefully construct a confidence +width wl(·) that incorporates both the noise and bias as follows: +wl(x) ≜ +� +2k(x, x) log(1/β) +|Ul| ++ +� +2Σ2 +Hl(x) log(1/β) +|Ul| ++ BΣHl(x), +(12) +where B is the bound of f’s kernel norm, and β is the confidence level from the input. Using this confidence +width wl(·) and the mean estimator function ¯µl(·) in Eq. (11), we can identify suboptimal actions with high +probability (w.h.p.). Finally, we update the set of active actions Dl+1 by eliminating the suboptimal actions +from Dl (Line 20). +Remark 4.1 (Merge batches). For implementation, we also merge different batches with the same chosen +action in each phase. By doing this, we further shrink the dimension of the matrix in the inverse operation (thus +reducing the time complexity) and also the dimension of local feedback (thus reducing the communication +cost). +Remark 4.2 (General decision set). Following the techniques used in [LS22], DPBE can also be extended +from a finite domain to a continuous domain (e.g., D = [0, 1]d) via a simple discretization trick and Lipschitz +continuity of functions under commonly used kernels. +9 + +Algorithm 1 Distributed Phase-then-Batch-based Elimination (DPBE) +1: Input: D ⊆ Rd, parameters α > 0, β ∈ (0, 1), C, and local noise σ2 +2: Initialization: l = 1, D1 = D, t1 = 0, and T1 = 1 +3: while tl < T do +4: +Set τ = 1, h = 0, τ1 = 0 and Σ2 +0(x) = k(x, x), for all x ∈ Dl +5: +while τ ⩽ Tl do +6: +h = h + 1 +7: +Choose action +ah ∈ argmax +x∈Dl +Σ2 +h−1(x) +(8) +8: +Play action ah for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 +h−1(ah)⌋ rounds if not reaching min{T, tl + Tl} +9: +Update τ = τ + Tl(ah), and incorporate ah into the posterior variance Σ2 +h(·) (see Eq. (7)) +10: +end while +11: +Let Hl = h denote the total number of batches in this phase +12: +Randomly select ⌈2αl⌉ participants Ul +# Operations at each participant +13: +for each participant u ∈ Ul do +14: +Collect and compute local average reward for every chosen action a ∈ AHl: +yu +l (a) = +1 +Tl(a) +� +t∈Tl(a) +yu,t +15: +Send the (local) average reward for each chosen action yu +l ≜ [yu +l (a)]a∈AHl to the agent +16: +end for +17: +Aggregate local observations for each chosen action a ∈ AHl: +yl(a) = +1 +|Ul| +� +u∈Ul +yu +l (a) +(9) +18: +Let ¯yl = [yl(a1), . . . , yl(aHl)] +19: +Update ¯µl(·) according to Eq. (11) +20: +Eliminate low-rewarding actions from Dl based on the confidence width wl(·) in Eq. (12): +Dl+1 = +� +x ∈ Dl : ¯µl(x) + wl(x) ⩾ max +b∈Dl +(¯µl(b) − wl(b)) +� +(10) +21: +Tl+1 = 2Tl, t = t + Tl, l = l + 1 +22: end while +5 +Main Results +In this section, we present the performance of our proposed DPBE algorithm in terms of regret, computation +complexity, and communication cost, respectively. +10 + +First, we analyze the regret performance of DPBE and present the upper bound in Theorem 5.1. While +the DPBE algorithm uses GP tools to define and manage the uncertainty in estimating the unknown function +f, the analysis of DPBE algorithm does not rely on any Bayesian assumption about f being drawn from the +prior GP(0, k(·, ·)), and it only requires f to be bounded in the kernel norm associated with the RKHS Hk. +Theorem 5.1 (Regret). Let β = +1 +|D|T . Under Assumptions 1, 2 and 3, the DPBE algorithm achieves the +following expected regret: +E[R(T)] = O(T 1−α/2� +log(|D|T)) + O( +� +γT T) + O( +� +γT T 1−α log(|D|T)). +(13) +We provide the detailed proof of Theorem 5.1 in Appendix C. Bounds for γT of different kernels can be +found in Appendix A.2. In the following, we make two remarks about the above result. +Remark 5.2. In the above regret upper bound (RHS of Eq. (13)), the first term, O(T 1−α/2� +log(|D|T)), +is due to the bias in the feedback at heterogeneous participants, and the last two terms, O(√γT T) + +O( +� +γT T 1−α log(|D|T)), are from the noisy feedback of each action as in the standard kernelized bandits +(cf. [Sri09]). Note that the first term (i.e., the regret caused by the bias) can be improved if one increases the +number of sampled users in the learning process (i.e., choosing a larger value of α). However, this would +also result in a larger communication cost. +Remark 5.3 (Maximum Uncertainty Reduction). Recall that DPBE selects actions that have maximum +variance for each batch (Eq. (5)). Intuitively, variance at action x indicates the uncertainty about f(x), and +thus, maximum-variance selection leads to maximum uncertainty reduction, which promotes exploration. +Remark 5.4 ((Sub-)optimality). We first note that one natural lower bound for our setting is the one for +the standard setting of kernelized bandits, where the agent receives unbiased feedback after taking an +action. In this setting, the state-of-the-art lower bounds under two commonly-used kernel functions (SE and +Mat´ern)3 are summarized in Table 5 (see Appendix A.2), which can also serve as valid lower bounds for +the setting we consider. Recall that α > 0 is the user-sampling parameter that one can choose. We discuss +the (sub-)optimality of our upper bounds in two cases: α ≥ 1 (i.e., the high-communication regime) and +α ∈ (0, 1) (i.e., the low-communication regime). (i) In the high-communication regime, the upper bound in +(13) now becomes O(√γT T), which is near-optimal under both SE and Mat´ern kernels. In particular, if one +plugs the best-known bounds on γT for SE and Mat´ern kernels (as listed in the first column in Table 5; also +see [VKP21]) into the regret upper bound O(√γT T), one can now have explicit regret upper bounds (as listed +in the third column in Table 5), which match the corresponding lower bounds, up to only a logarithmic factor. +(ii) In the low-communication regime, the first term in the regret upper bound (see Eq. (13) in Theorem 5.1) +that depends on α may be dominant and cannot be ignored. On the other hand, the existing lower bounds do +not depend on α since they are derived under the standard setting of kernelized bandits, where user sampling +is irrelevant. Therefore, an important open problem is to close the gap by deriving tighter lower and/or +upper bounds that capture the effect of user sampling in the new setting with distributed biased feedback we +consider. We leave it as our future work. +As a critical bottleneck of kernelized bandits algorithms, the computation complexity of DPBE algorithm +is analyzed in the following Theorem 5.5. +3Note that even for the standard setting of kernelized bandits, there only exist lower bounds for these specific kernel functions +rather than a general one in terms of the maximum information gain γT . +11 + +Table 1: Comparison of computation complexity under DPBE and three state-of-the-art algorithms. +Algorithms +Complexity +GP-UCB [CG17] +O(|D|T 3) +BBKB [Cal20] +O(|D|Tγ2 +T ) +MINI-GP-Opt [Cal22] +O(T + |D|γ3 +T + γ4 +T ) +DPBE (this paper) +O(γT T α + (|D|γ3 +T + γ4 +T ) log T) +Theorem 5.5 (Computation complexity). The computation complexity of DPBE is at most O(γT T α+(|D|γ3 +T + +γ4 +T ) log T). +Proof. Recall that Hl is the number of batches in the l-th phase. Then, the computation complexity of the +central agent in the l-th phase is upper bounded by the following: +O(Hl · H3 +l + Hl · |Dl|H2 +l + |Ul|Hl + |Dl|H2 +l ). +Specifically, for each h ∈ [Hl] within phase l, the agent would compute the matrix inverse in Eq. (7), the +complexity of which is at most O(h3) ≤ O(H3 +l ). With this matrix inverse result ready, the agent can solve +the maximum-variance problem in Eq. (8) with at most O(|Dl|H2 +l ) for each batch and determine the batch +length Tl(ah) with O(1) after we have the posterior variance. Since there is a total of Hl batches for phase l, +the total complexity up to this stage is O(Hl · H3 +l + Hl · |Dl|H2 +l ). Finally, in the elimination stage for phase +l, the agent first loads/aggregates all the feedbacks with O(|Ul|Hl) and can again reuse the matrix inverse +result so that only O(|Dl|H2 +l ) is required to eliminate all the bad arms. +Putting the two stages together, we have the above result. Thus, it remains to bound the number of batches +Hl within each phase l. Fortunately, inspired by [Cal22], we are able to show that Hl can be upper bounded +by the maximum information gain. We state this result in Lemma 5.6 and provide the proof in Appendix D. +Lemma 5.6 (Bound on Hl). For any phase l, the number of batches Hl is at most 4σ2C2 +C2−1 γT . +We can get that the total number of phases is O(log T) and the total number of participants satisfies +O(T α). Armed with all the above results, we arrive at our final computation complexity. +Remark 5.7 (Complexity comparison). For comparison, we list the computation complexity of the state- +of-the-art algorithms for standard kernelized bandits in Table 1. As we already know, GP-UCB has a +computation complexity of O(|D|T 3), because it requires computing the posterior mean and variance using +O(T 2) and then finds the action that maximizes the UCB function per step. Recently, BBKB in [Cal20] +improves the time complexity to (|D|Tγ2 +T ), and later MINI-GP-Opt in [Cal22] further reduces computation +complexity to O(T +|D|γ3 +T +γ4 +T ), which is currently the fastest no-regret algorithm. Although more feedback +is needed to address the additional bias in our setting, our algorithm can still achieve an improvement with +the highest order term being O(γT T α). This improvement comes from the fact that the participants help +preprocess local reward observations before sending them out. +Meanwhile, the bound on Hl also allows us to achieve a meaningful communication cost. +Theorem 5.8 (Communication cost). DPBE incurs at most O(γT T α) communication cost. +The proof for Theorem 5.8 is also provided in Appendix D. +12 + +Remark 5.9 (Communication cost when merging batches). By further merging batches according to Re- +mark 4.1, the DPBE algorithm incurs O(min{γT , |D|}T α) communication cost; We highlight that the batch +schedule strategy plays a key role in obtaining the above bounds. Otherwise, even merging rounds as Re- +mark 4.1 with the reformulated representation in Eqs. (7) and (11), the dimension of the local feedback at each +participant is O(min{Tl, |Dl|}) in order to distinguish different actions, which leads to O(min{T, |D|}T α) +(vs. ours O(min{γT , |D|}T α)). +6 +Differentially Private DPBE +As privacy is also an important factor in distributed learning, it is critical to protect users’ sensitive data when +collecting and aggregating their feedback. For example, in the dynamic pricing application, it is required that +an adversary cannot infer a customer’s private information (e.g., purchase or not) by observing the pricing +mechanism set by the company. Moreover, users may require more stringent privacy protection in some +applications — users are not willing to share their perceived Quality-of-Experience (QoE) directly with +the central controller in the cellular network configuration problem; citizens are not willing to reveal the +information about their preference for a certain policy to the government. Formally, we adopt the concept of +differential privacy (DP) [DR14] as the privacy metric. Thanks to the phase-then-batch schedule strategy +in our algorithm, different DP trust models (e.g., central [DR14], local [ZT20], and shuffle [Che19]) can +be applied through proper designs. In this section, we describe how to ensure DP under DPBE with a +trusted agent (the central DP model) and also analyze the regret under such a DP model. Extensions of the +differentially private DPBE algorithms in other DP models (e.g., the stronger local DP model) are presented +in Appendix E. +6.1 +DP Definition and Algorithm +In the central DP model, we assume that each participating user trusts the agent, and hence, the agent can +collect their raw data (i.e., the local reward yu +l in our case). The privacy guarantee is that any adversary +with arbitrary auxiliary information cannot infer a particular user’s data by observing the decisions of the +agent. To achieve this privacy protection, the central DP model requires that the decisions of the agent on two +neighboring sets of users (differing in only one user) are indistinguishable [Dwo06]. Formally, we have the +following definition. +Definition 6.1. (Differential Privacy (DP)). For any ε ⩾ 0 and δ ∈ [0, 1], a randomized algorithm M is +(ε, δ)-differentially private (or (ε, δ)-DP) if for every pair of U, U′ ⊆ U differing on a single participant and +for any subset of output actions Z = [z⊤ +1 , . . . , z⊤ +T ]⊤, we have +P[M(U) = Z] ⩽ eεP[M(U) = Z] + δ. +(14) +The parameters ε and δ indicate how private M is; the smaller, the more private. According to the +post-processing property of DP (cf. Proposition 2.1 in [DR14]), it suffices to guarantee that the aggregator +(Line 17 in Algorithm 1) is (ε, δ)-DP. To achieve this, the standard Gaussian mechanism can be applied by +adding Gaussian noise to the aggregated distributed feedback. Then, the private aggregated feedback for the +chosen actions in the l-th phase becomes +˜yl = ¯yl + (ρ1, . . . , ρHl), +(15) +13 + +where ρj +i.i.d. +∼ N(0, σ2 +nc) and the variance σ2 +nc (specified in Eq. (66)) is based on the (high-probability) +sensitivity of of the average vector ¯yl. In addition, we replace ¯yl with ˜yl in Eq. (15) to obtain the private +mean estimator: +˜µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1˜yl. +(16) +The confidence width function is also updated by counting the uncertainty introduced by privacy noise as +follows: +˜wl(x) ≜ +� +2k(x, x) log(1/β) +|Ul| ++ +� +2Σ2 +Hl(x) log(1/β) +|Ul| ++ BΣHl(x) + +� +2σ2n log(1/β), +(17) +where σn is related to the overall privacy noise and will be specified in the algorithm. We present the +differentially private version of DPBE, called DP-DPBE, in Algorithm 2 (see Appendix E). +6.2 +Performance Guarantees +In the following, we provide the main results of the DP-DPBE algorithm in terms of privacy guarantee and +regret. We start by stating an additional assumption in Assumption 4. This one-time participation assumption +is commonly used in private bandits (see, e.g., [MT15; SS19; Ten21; Dub21; CZ22b; CZ22a]). To handle +multiple-times participation, one can use (adaptive) composition theorem of differential privacy or group +privacy [DR14], depending on different cases of returning users.4 +Assumption 4. Each sampled user only participates in one phase of the learning process. +Then, we present the privacy guarantee in Theorem 6.2 and provide the proof in Appendix E.2. +Theorem 6.2 (Privacy Guarantee). Under Assumptions 1, 2, 3, and 4, for any ε > 0 and δ ∈ (0, 1), the +DP-DPBE algorithm (Algorithm 2) guarantees (ε, δ)-DP. +As an additional Gaussian noise is injected to protect privacy, DP-DPBE suffers additional regret cost. +We present its regret upper bound in Theorem 6.3. +Theorem 6.3 (Regret of DP-DPBE). Under Assumptions 1, 2, and 3, the DP-DPBE algorithm (Algorithm 2) +with β = +1 +|D|T achieves the following expected regret: +E[R(T)] = O(T 1−α/2� +log(|D|T)) + O +� +ln(1/δ)γT T 1−α� +log(|D|T) +ε +� +. +(18) +The full proof of Theorem 6.3 is provided in Appendix E.3. Regarding this regret result, we make the +following remark. +Remark 6.4 (Privacy “for free”). Comparing Theorem 6.3 with Theorem 5.1, we see that the additional regret +cost introduced by privacy noise is ˜O +� +ln(1/δ)γT T 1−α +ε +� +, which is a lower order term compared to the first +non-private term. This implies that our DP-DPBE algorithm enables us to achieve a privacy guarantee “for +free” in the kernelized bandits setting. The same observation of achieving privacy “for free” is also observed +in a recent study [LZJ22] that only considers linear bandits. However, our result is a strict generalization in +the sense that it holds for general functions and recovers their result when considering a linear kernel. +4More specifically, if the same user only participates across multiple phases, one can use advanced composition; if the same user +participates multiple times in the same phase, one can carefully bound the sensitivity or use group privacy directly. +14 + +Figure 3: +Comparison of regret perfor- +mance on a synthetic function. The shaded +area represents the standard deviation +Figure 4: The regret and communication +cost under DPBE with different values of α. +7 +Numerical Experiments +We now evaluate our proposed approach empirically on three types of functions: 1) synthetic functions in +the RKHS with an SE kernel, 2) standard benchmark functions (with an unknown RKHS norm) [SB] , and +3) functions from a real-world dataset. We implement the algorithms in python and run the numerical +experiments on a Dell desktop (Processor: Intel®Core i7 CPU, 8 cores; Memory: 32GB). +7.1 +Synthetic Function +We follow [JBG20] to construct the global function f from the RKHS by sampling m = 30d inde- +pendent points, �x1, . . . , �xm, uniformly on [0, 1]d, and �a1, . . . , �am, uniformly on [−1, 1], and defining +f(x) = �m +i=1 �aik(�xi, x) for all x ∈ D, where k is SE kernel with length-scale lSE = 0.2. The RKHS +norm is ∥f∥2 +k = �m +i=1 +�m +j=1 �ai�ajk(�xi, �xj), which is assumed to be known. Each local reward function fu, a +random function sampled from a given Gaussian process, is generated by following Algorithm 1 in [Kan18]. +In the simulations, we evaluate the algorithms in a more general setting with fu ∼ GP(f(·), v2k(·, ·)), where +v2 is a scaling parameter that can be used to set a reasonable level of local bias (see Footnote 2). +7.1.1 +Ablation Studies and Analysis +First, we show that the DPBE algorithm that selects actions according to maximum variance reduction +achieves sublinear regret, as shown in Figure 3. Then, we perform numerous ablation studies to confirm the +efficacy of other two key components in our algorithm: user-sampling and batching strategy. To this end, +we consider the corresponding variants of our algorithm. In this simulation, we perform 20 runs for each +algorithm by setting |D| = 100, d = 3, C = 1.6, σ = 0.01, v = 0.1, T = 40000, α = 0.7, β = 1/(|D|T) +and λ = σ2/v2 and present the regret performance in Figure 3 and communication cost and runtime in +Table 2. +1) Importance of (exponentially-increasing) user-sampling. To this end, we consider the first variation +of DPBE with a fixed number of participants, called DPBE-Fixed, where the number of participants in each +phase is fixed at |U| = ⌊ +�L +l=1 |Ul|∗Nu,l +�L +l=1 Nu,l +⌋ so as to have the same communication cost as DPBE. From Figure 3, +we observe that DPBE with exponentially-increasing user-sampling over phases performs much better than +15 + +×103 +DPBE-Fixed +3 +DPBE +DPBE-NoBatching +7 +0 +0 +1 +2 +3 +4 +Rounds +×104×104 +×104 +1.2 +1.0 +Cumulative regret +Communication cost +0.8 +1.0 +0.6 +0.8 +0.4 +0.6 +0.2 +0.4 +0.0 +0.4 +0.6 +0.8 +αTable 2: +Comparisons of communication cost and running time under DPBE, DPBE-Fixed, and +DPBE-NoBatching on a synthetic function. +Algorithms +Communication cost +Running time (seconds) +DPBE +5.87 × 103 +0.12 +DPBE-Fixed +5.87 × 103 +0.19 +DPBE-NoBatching +1.81 × 104 +0.61 +DPBE-Fixed with the same communication cost. It demonstrates that the exponentially-increasing user- +sampling mechanism in DPBE is critical to striking a balance between regret and communication cost. +From Table 2, we observe that DPBE-Fixed takes a little longer time than DPBE. This is mainly because +DPBE-Fixed needs more phases to find the optimal action (i.e., L is larger when |DL| = 1). +2) Benefits of batching strategy. To illustrate the impact of batching schedule strategy, we consider +another variant of DPBE that does not employ batching strategy, called DPBE-NoBatching. In par- +ticular, it selects an action according to Eq. (8) for each round in any phase. Without batching strategy, +DPBE-NoBatching communicates local observations directly without merging, and computes the posterior +mean and variance according to standard update formula: Eq. (4) and Eq. (5) respectively; From Figure 3, +we observe that DPBE, similar to other rare-switching algorithms [APS11], achieves a slightly worse regret +performance than DPBE-NoBatching. However, as shown in Table 2, it significantly saves communication +cost (∼ 3×) by merging local observations in batches and reduces computation time (∼ 5×) by shrinking the +dimension of posterior reformulations. +7.1.2 +Regret-communication Tradeoff +We now turn to investigate the regret-communication tradeoff captured by the user-sampling parameter α, as +shown in Theorem 5.1. +Consider α ∈ {0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}. The cumulative regret and total communication cost of +DPBE with different values of α are presented in Figure 4. As expected, while a larger value of α yields a +lower regret, it generally results in a higher communication cost. Notice that DPBE incurs slightly higher +communication cost when α = {0.4, 0.5, 0.6} compared to α = 0.7, this is mainly because DPBE with a +smaller value of α needs more phases to find the optimal action (i.e., L is larger when |DL| = 1). One can +tune the user-sampling parameter α to achieve a better regret-communication cost accordingly, e.g., α = 0.7 +for this synthetic function setting. +7.1.3 +Regret-privacy Tradeoff +Finally, we evaluate the performance of the differentially private DPBE, i.e., DP-DPBE, and present the result +in Figure 5. Figure 5(a) shows how the cumulative regret at the end of T = 106 rounds varies with different +values of privacy parameters ε ∈ {5, 10, 15, 20, 25, 30} and δ = o(1/T) = 10−6, which reveals a tradeoff +between regret and privacy. Figures 5(b) shows the regret performance of DPBE and DP-DPBE with privacy +parameters ε = 15 and δ = 10−6. We observe that although DP-DPBE adds extra noise to protect privacy, it +can still achieve no-regret (i.e., limT→∞ +R(T) +T +→ 0). Indeed, to protect privacy, DP-DPBE requires much +more time to find the optimal action, which is the typical regret-privacy tradeoff. However, for a large T, the +gap compared to the non-private one is small, which also validates the privacy “for-free” result. +16 + +(a) +(b) +Figure 5: Performance of DP-DPBE. (a) Final cumulative regret vs. the privacy budget ε with δ = 10−6; (b) +Per-round regret vs. time with parameters ε = 15 and δ = 10−6. +7.2 +Standard Benchmark Functions +In addition, we study the performance of DPBE on standard optimization benchmark functions. This +corresponds to a more realistic setting where the RKHS norm of the target function is unknown in advance. +In particular, we use three common functions in global optimization problems [SB]: (a) Sphere function, +(b) Six-hump Camel function, and (c) Michalewicz function, and provide the performance comparison of +DPBE-Fixed, DPBE, and DPBE-NoBatching in Figure 6 and Table 3. In the simulations, we scale +the range of the function values to [−1, 1] and use RKHS norm B = 1 in the algorithms as in [JBG20]. +Without knowing the exact kernel of the target function, each local reward function fu is constructed by +sampling a function from the GP GP(f(·), v2k(·, ·)), where we choose v2 = 0.001 and use the SE kernel +with lSE = 0.2. In addition, we set T = 4 × 104 and |D| = 100 and run each algorithm on each function for +20 times. +From Figure 6 and Table 3, we observe similar results to those of the synthetic function with the same +kernel. First, compared to DPBE-Fixed that incurs the same communication cost, DPBE might perform +slightly worse at the very beginning (e.g., Figure 6(a)) but eventually achieves a much smaller regret. +Note that DPBE-Fixed may not be able to find the optimal action by the end of T (e.g., Figure 6(b)). +This phenomenon strengthens our argument on the exponentially-increasing user-sampling mechanism in +DPBE. While DPBE-NoBatching has slightly better regret performance than DPBE, it incurs much higher +communication cost (5 ∼ 13×) and requires a much longer time (6 ∼ 23×, see running time column in +Table 3), which demonstrates the key benefits of the batching strategy in improving communication efficiency +and computation complexity. +In addition, we also evaluate the regret-privacy tradeoff under DP-DPBE. Due to space limitations, we +present the numerical results in Appendix F (see Figures 8 and 9). +7.3 +Functions from Real-World Data +We also evaluate the performance of DPBE on a function from a real-world dataset, where there is no explicit +closed-form expression. +Light Sensor Data. We use the light sensor data collected from the CMU Intelligent Workplace in +November 2005, which is available online [Sch]. It contains locations of 41 sensors, 601 training samples, +17 + +x105 +1.5 +1.0 +0.5 +0.0 +T +5 +10 +15 +20 +25 +30 +Pivacy budget 0.6 +DP-DPBE +DPBE +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rounds +×106(a) Sphere function +(b) Six-Hump Camel function +(c) Michalewicz function +(d) Light sensor data +Figure 6: Comparison of regret performance under DPBE, DPBE-Fixed, and DPBE-NoBatching on +four functions. (a) Sphere function. Settings: d = 3, C = 1.6, σ = 0.01, λ = σ2/v2, α = 0.7; (b) Six-Hump +Camel function. Settings: d = 2, C = 1.6, σ = 0.01, λ = σ2/v2, α = 0.7; (c) Michalewicz function. +Settings: d = 2, C = 1.6, σ = 0.1, λ = σ2/v2, α = 0.6; (d) Function from light sensor data. Settings: +d = 2, C = 1.42, σ = 0.01, λ = σ2/v2, α = 0.8. +and 192 testing samples. Following [Sri09; CG17; ZJ22], we compute the empirical covariance matrix of +the training samples and use it as the kernel matrix in the algorithm. Here, for each location x, we let f(x) +be the average of the normalized sample readings at x and set B = maxx f(x) in the algorithm. For this +function (from real data), we construct each local function fu by sampling a function from a Gaussian process +with mean f and the kernel constructed above, and set the noise in the local feedback as σ = 0.01 and +the bias in each local feedback as v = 0.1. We run DPBE with input parameters α = 0.7, β = 1/(|D|T), +and λ = σ2/v2, and present the regret performance in Figure 6(d) and communication cost and running +time in Table 3. The observations are qualitatively similar to those made in simulations on other functions: +DPBE outperforms DPBE-Fixed in regret given the same communication cost and achieves a regret close +to DPBE-NoBatching, which has much longer running time. Besides, we also run DP-DPBE on this +real-world dataset and present the results in Appendix F (see Figures 8(d) and 9(d)), which validates the +regret-privacy tradeoff. +18 + +×103 +2.5 +DPBE-Fixed +DPBE +2.0 +DPBE-NoBatching +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +Rounds +×104×103 +DPBE-Fixed +1.5 +DPBE +DPBE-NoBatching +1.0 +Cumulative! +0.5 +0.0 +0 +1 +2 +3 +4 +Rounds +×104x103 +5 +DPBE-Fixed +DPBE +4 +DPBE-NoBatching +m +1 +0 +0 +2 +3 +4 +Rounds +×104×103 +3.0 +DPBE-Fixed +2.5 +DPBE +ret +DPBE-NoBatching + 2.0 +Re +Cumulative +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +Rounds +×104Table 3: Communication cost and running time under DPBE, DPBE-Fixed, and DPBE-NoBatching +Function +Algorithm +Communication cost +Running time (seconds) +Sphere +DPBE +1.49 × 103 +0.07 +DPBE-Fixed +1.49 × 103 +0.12 +DPBE-NoBatching +6.16 × 103 +0.69 +Six-Hump Camel +DPBE +1.26 × 103 +0.03 +DPBE-Fixed +1.26 × 103 +0.12 +DPBE-NoBatching +1.45 × 104 +0.17 +Michalewicz +DPBE +2.06 × 103 +0.06 +DPBE-Fixed +2.06 × 103 +0.14 +DPBE-NoBatching +2.73 × 104 +0.49 +Light Sensor Data +DPBE +5.17 × 103 +0.22 +DPBE-Fixed +5.17 × 103 +0.28 +DPBE-NoBatching +2.73 × 104 +5.20 +8 +Comparison with the State-of-the-Arts +8.1 +Discussion +We now consider an alternative way of addressing kernelized bandits with distributed biased feedback. One +may incorporate the local bias as another level of noise added to the noise in the rewards as a new noisy +measurement of the global function f with a larger variance. In this case, the state-of-the-art algorithms for +the traditional kernelized bandits [Sri09; CG17] may be adapted to our setting. However, they have some key +limitations. +Consider two representative state-of-the-art algorithms: GP-UCB [CG17] and BPE [LS22]. GP-UCB +is one of the most commonly used algorithms for standard kernelized bandits, It was proposed in [Sri09] +and improved in [CG17]. By resorting to the Gaussian process surrogate model (see Section 3.3), GP-UCB +adaptively selects the action with the maximal upper confidence bound in each round based on historical +observations up to the current round. BPE is a batch-based algorithm that eliminates suboptimal actions +batch by batch, and within each batch, actions are chosen independently from reward observations. In the +following, we compare our proposed DPBE algorithm with GP-UCB and BPE (adapted to our setting) and +show their limitations in user-sampling, communication cost, and computation complexity. +First, both GP-UCB and BPE require to collect feedback from one user per step, which results in T +users involved in the learning process. In practice, even though there is a large population, not all users are +willing to send their feedback. Hence, it may not be feasible to collect feedback from too many users. In our +algorithm, instead of sampling more users to reduce the overall uncertainty, we ask each sampled user (who +is more willing to participate) to participate in more rounds and send their feedback. In this way, we alleviate +the user-sampling burden by letting the participating users collect more reward samples of the chosen actions. +However, due to the bias in the feedback of each user, we could not just sample one user and then let her +report the feedback during the entire horizon. We need to balance the tradeoff between sampling more users +and letting the users participate in more rounds. +Second, by collecting feedback in each round, both GP-UCB and BPE incur a very high communication +cost of T. Instead, we employ a phase-based communication protocol where feedback corresponding to any +particular action at each participant is averaged and only communicated at the end of each phase. Then, the +19 + +total communication cost depends on the number of phases, the number of distinct actions in each phase, and +the number of sampled users. The smaller each of these three factors, the smaller the communication cost. +By carefully designing the algorithm, we can reduce the communication cost to O(min{γT , |D|}T α), where +α ∈ (0, 1) is the user-sampling parameter one can choose. +Finally, at each round t, GP-UCB finds the decision action xt that maximizes an acquisition function +(specifically, the UCB index, which is the sum of the posterior mean and variance). Note that obtaining the +posterior mean and variance requires computing matrix inverse (see Eqs. (4) and (5)), which still has a com- +putation complexity of O(t2) even using rank-one recursive updates [CG17, Appendix 7]. Hence, the overall +computation complexity of GP-UCB is O(|D|T 3). Similarly, BPE may also compute the posterior variance +using the rank-one recursive update within each batch, and then the total computation complexity depends +on the batch size and the number of batches. As in [LS22], the batch size is updated as Ni = +� +T√Ni−1, +initialized with N0 = 1, which results in ⌈log log(T)⌉ batches in total. Therefore, the computation complexity +of BPE is O(|D|T 3). In our design, we employ the batch schedule strategy and reformulate the posterior +mean and variance as Eqs. (11) and (7), where the dimension of the matrix becomes much smaller. This leads +to a much smaller overall computation complexity of O(γT T α + (|D|γ3 +T + γ4 +T ) log T). +8.2 +Empirical Performance +In this subsection, we evaluate the empirical performance of DPBE with different values of user-sampling +parameter α compared to GP-UCB and BPE. +The simulations are run on the same three types of functions as in the preceding section: the synthetic +function in Section 7.1, the standard benchmark functions in Section 7.2, and the function from light sensor +data in Section 7.3. Due to space limitation, we only present the results of the synthetic function here and put +the results of the latter two types of functions in Appendix F. +Consider5 α ∈ {0.4, 0.5, 0.6, 0.7, 0.8, 0.9} for DPBE. We show the empirical regret performance of all +algorithms in Figure 7 and the running time in Table 4. From Figure 7, we observe that the empirical regret +performance of DPBE can be fairly close to or even better than that of GP-UCB and BPE via properly choosing +parameter α. However, it consumes much less time for DPBE with each α ∈ {0.4, 0.5, 0.6, 0.7, 0.8, 0.9} +than both GP-UCB and BPE. For example, while DPBE takes about 0.15 second in most scenarios, GP-UCB +takes more than 5 seconds, which is more than 30 times slower. BPE takes around 27 seconds, which is even +slower. +Recall the empirical communication cost of DPBE with different values of α shown in Figure 4. While +the communication cost of GP-UCB and BPE is 4 × 104 (specifically, one feedback per round), DPBE incurs +a much smaller communication cost even when α = 0.9 (4 × 104 vs. 1.19 × 104). +In summary, the comparison of empirical performance under DPBE with GP-UCB and BPE demonstrates +the significant improvements of DPBE in terms of communication cost and computation complexity, although +little regret performance is sacrificed when α is not big enough. +9 +Conclusion +In this paper, we studied a new kernelized bandit problem with distributed biased feedback, where the +feedback of the unknown objective function is biased due to user heterogeneity. To learn and optimize +the unknown function using distributed biased feedback, we proposed the learning with communication +5Note that the smaller the value of α, the larger the cumulative regret. In Figure 7, we omit the regret performance when α < 0.4 +since they are much larger than others. +20 + +Table 4: Comparison of running time (seconds) under GP-UCB, BPE, and DPBE with different values of α. +Algorithms +DPBE +GP-UCB +BPE +α = 0.4 +α = 0.5 +α = 0.6 +α = 0.7 +α = 0.8 +α = 0.9 +Running time +0.24 +0.19 +0.14 +0.12 +0.13 +0.17 +5.32 +27.49 +Figure 7: Regret performance comparison of GP-UCB, BPE, and DPBE with different values of α. +framework. Considering the communication cost for collecting feedback and the computational bottleneck +of kernelized bandits, we carefully designed the distributed phase-then-batch-based elimination (DPBE) +algorithm to address all the new challenges. Specifically, DPBE selects actions according to maximum +variance reduction, reduces bias via user-sampling, and improves communication efficiency and computation +complexity via the batching strategy. Furthermore, we showed that DPBE achieves a sublinear regret while +being scalable in terms of communication efficiency and computation complexity. Finally, we generalized +DPBE to incorporate various differential privacy models to ensure privacy guarantees for participating users. +Future work. While we proposed a new DPBE algorithm to address the new challenges that arise in our +problem setup, it would be worthwhile to explore other batch-based algorithms and investigate whether one +can further improve the tradeoff among regret, communication efficiency, and computation complexity. In +addition, as discussed in Remark 5.4, the lower bound derived for the standard kernelized bandits is also a +valid lower bound for our problem. We show that our algorithm, if sampling a sufficient number of users, can +achieve this lower bound. In general, however, it is an important open problem to close the gap by deriving +tighter lower and/or upper bounds that capture the effect of user sampling in our new setting. We leave it as +our future work. +10 +ACKNOWLEDGMENTS +We thank our shepherd, Giulia Fanti, and the anonymous paper reviewers for their insightful feedback. We +also thank Duo Cheng for fruitful discussions. This work is supported in part by the NSF grants under +CNS-2112694 and CNS-2153220. +21 + +×103 +DPBEα=0.4 +DPBEα=0.8 +5 +DPBEα=0.5 +DPBEα=0.9 +DPBEα=0.6 +BPE +4 +DPBEα=0.7 +GP-UCB +Regret +3 +2 +1 +0 +0 +1 +2 +3 +4 +Rounds +×104References +[APS11] +Yasin Abbasi-Yadkori, D´avid P´al, and Csaba Szepesv´ari. “Improved algorithms for linear stochas- +tic bandits”. In: Advances in neural information processing systems 24 (2011). +[Bit17] +Andrea Bittau et al. “Prochlo: Strong privacy for analytics in the crowd”. 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In: arXiv +preprint arXiv:2010.06709 (2020). +24 + +Table 5: Bounds on γT and Regret under Two Common Kernels [VKP21] +Kernel +Upper Bound on γT +Regret Lower Bound +Regret Upper Bound O(√γT T) +SE +O +� +logd+1(T) +� +Ω +�� +T log +d +2 (T) +� +O +�� +T logd+1(T) +� +Mat´ern-ν +O +� +T +d +2ν+d log +2ν +2ν+d (T) +� +Ω +� +T +ν+d +2ν+d +� +O +� +T +ν+d +2ν+d log +ν +2ν+d (T) +� +A +Kernelized Bandits: Useful Definitions and Useful Results +A.1 +Example Kernel Functions +In the following, we list some commonly used kernel functions k : D × D → R: +• Linear kernel: klin(x, x′) = x⊤x′, +• Squared exponential kernel: kSE(x, x′) = exp +� +− ∥x−x′∥ +2l2 +� +, +• Mat´ern kernel: kMat(x, x′) = 21−ν +Γ(ν) +� √ +2ν∥x−x′∥ +l +� +Jν +� √ +2ν∥x−x′∥ +l +� +, +where l denotes the length-scale hyperparameter, ν > 0 is an additional hyperparameter that dictates the +smoothness, and Jν and Γν denote the modified Bessel function and the Gamma function, respectively +[RW06]. +A.2 +Maximum Information Gain for Different Kernels +We present the bounds on γT and regret under two common kernels below in Table 5. +A.3 +Useful Results +Lemma A.1 (Sum of variance. Lemma 6 in [RG19]). Let Xt = [x⊤ +1 , . . . , x⊤ +t ]⊤, and σ2 +t (x) ≜ k(x, x) − +k(x, Xt)⊤(KXtXt + λI)−1k(x, Xt) for any x ∈ D. Then, we have +t +� +s=1 +σ2 +s(xs) ⩽ λ ln |λ−1KXtXt + I| ⩽ 2λγt. +(19) +Lemma A.2 (Proposition A.1 in [Cal22]/Lemma 4 in [Cal20]). For any kernel k, set of points Xτ, x ∈ D, +and τ ′ < τ, we have +1 ⩽ σ2 +τ ′(x) +σ2τ(x) ⩽ 1 + +τ +� +s=τ ′+1 +σ2 +τ ′(xs). +(20) +A.4 +Formulation in Feature Space +For several of the proofs, it will be useful to introduce the so-called feature space (RKHS) formulation of any +point in the primal space Rd. In particular, we define a feature map ϕ(x) = k(x, ·) where ϕ : D → Hk with +Hk being the reproducing kernel Hilbert space (RKHS) associated with kernel function k. According to the +properties of RKHS, we have the following observations: +25 + +• For any x, x′, k(x, x′) = ϕ(x)⊤ϕ(x′). +• For any function f ∈ H, f(x) = ⟨f, ϕ(x)⟩ = ϕ(x)⊤f. +• Fundamental linear algebra equality +(BB⊤ + λI)−1B = B(B⊤B + λI)−1. +(21) +• Define Φh ≜ [ϕ(a1)⊤, . . . , ϕ(ah)⊤]⊤. Then, the kernel matrix KAhAh = ΦhΦ⊤ +h and k(x, Ah) = +Φhϕ(x), and the variance function Σ2 +h(·) represented in the feature space is the following: +Σ2 +h(x) = k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 +h )−1k(x, Ah) += ϕ(x)⊤ϕ(x) − ϕ(x)⊤Φ⊤ +h (ΦhΦ⊤ +h + λW−1 +h )−1Φhϕ(x). +(22) +• Consider any phase l. Recall that Hl is the number of batches in the l-th phase. Define ΦHl ≜ +[ϕ(a1)⊤, . . . , ϕ(aHl)⊤]⊤. Then, the kernel matrix KAHlAHl = ΦHlΦ⊤ +Hl and k(x, AHl) = ΦHlϕ(x). +• Define Φτ ≜ [ϕ(xtl+1)⊤, . . . , ϕ(xtl+τ)⊤]⊤. Then, the kernel matrix KXτXτ = ΦτΦ⊤ +τ , k(x, Xτ) = +Φτϕ(x), and the variance function σ2 +τ(·) represented in the feature space is the following: +σ2 +τ(x) = k(x, x) − k(x, Xτ)⊤(KXτXτ + λI)−1k(x, Xτ) += ϕ(x)⊤ϕ(x) − ϕ(x)⊤Φ⊤ +τ (ΦτΦ⊤ +τ + λI)−1Φτϕ(x) += ϕ(x)⊤ϕ(x) − ϕ(x)⊤(Φ⊤ +τ Φτ + λI)−1Φ⊤ +τ Φτϕ(x) += ϕ(x)⊤(Φ⊤ +τ Φτ + λI)−1(Φ⊤ +τ Φτ + λI)ϕ(x) − ϕ(x)⊤(Φ⊤ +τ Φτ + λI)−1Φ⊤ +τ Φτϕ(x) += λϕ(x)⊤(Φ⊤ +τ Φτ + λI)−1ϕ(x). +(23) +• Define ΦTl ≜ [ϕ(xtl+1)⊤, . . . , ϕ(xtl+Tl)⊤]⊤. Then, the kernel matrix KXTlXTl = ΦTlΦ⊤ +Tl, k(x, XTl) = +ΦTlϕ(x), and f(XTl) = ΦTlf. +B +Auxiliary Results and Proofs for Regret Analysis +B.1 +Equivalent Representations +Consider any phase l. We use τ to denote the within-phase time index, i.e., τ ∈ {1, · · · , Tl}. Define τh as the +last within-phase time index of the h-th batch, i.e., τh ≜ max{τ : tl + τ ∈ Tl(ah)}. Then, after playing τh +actions, the posterior variance in the traditional GP model is the following: +σ2 +τh(x) = k(x, x) − k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh). +(24) +For the posterior mean, without the observations yTl = [ytl+1, . . . , ytl+Tl]⊤ corresponding to the actions +XTl = [x⊤ +tl+1 . . . , x⊤ +tl+Tl]⊤, we replace yTl with +1 +|Ul| +� +u∈Ul yl,u where yl,u = [yu,tl+1, . . . , yu,tl+Tl]⊤ in the +traditional GP model. Then, the posterior mean becomes the following: +µTl(x) = +1 +|Ul| +� +u∈Ul +k(x, XTl)⊤(KXTlXTl + λI)−1yl,u. +(25) +In our algorithm, in order to save computation complexity and communication cost, we use Eq. (7) and +Eq. (11) instead of the above formula. In the following lemma, we show that they are equivalent. +26 + +Lemma B.1 (Equivalent representations). Consider any phase l. By the end of the h-th phase, the posterior +variance Eq. (24) in the traditional GP model is equivalent to Eq. (7) used in our DPBE algorithm. That is, +for any x ∈ D, we have +σ2 +τh(x) = Σ2 +h(x), +∀h = 1, . . . , Hl. +(26) +Moreover, we have the two representations (Eq. (25) and Eq. (11)) for the posterior mean function are +equivalent. That is, for any x ∈ D, we have +µTl(x) = ¯µl(x). +(27) +Proof. First, we have the following result, which helps connect the two representations of mean and variance +functions: +Φ⊤ +τhΦτh = +tl+τh +� +t=tl+1 +ϕ(xt)ϕ(xt)⊤ +(a) += +h +� +i=1 +Tl(ai)ϕ(ai)ϕ(ai)⊤ += Φ⊤ +h WhΦh, +(28) +where (a) is due to our algorithm decisions: xt = ai for any t ∈ Tl(ai) = {tl + τi−1 + 1, tl + τi−1 + Tl(ai)} +and the last step holds because Wh is a diagonal matrix with (Wh)ii = Tl(ai) for any i ∈ [h]. +Then, we are ready to derive the equivalence of two representations of the mean function. +1) Variance representation equivalence: σ2 +τh(x) = Σ2 +h(x) for h = 1, . . . , Hl. This implies +k(x, x) − k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh) += k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 +h )−1k(x, Ah). +It remains to show the following: +k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh) = k(x, Ah)⊤(KAhAh + λW−1 +h )−1k(x, Ah). +(29) +Using the feature space formulations, we have +k(x, Ah)⊤(KAhAh + λW−1 +h )−1k(x, Ah) +=ϕ(x)⊤Φ⊤ +h (ΦhΦ⊤ +h + λW−1 +h )−1Φhϕ(x) +=ϕ(x)⊤Φ⊤ +h W1/2 +h (W1/2 +h ΦhΦ⊤ +h W1/2 +h ++ λI)−1W1/2 +h Φhϕ(x) +=ϕ(x)⊤(Φ⊤ +h W1/2 +h W1/2 +h Φh + λI)−1Φ⊤ +h W1/2 +h W1/2 +h Φhϕ(x) +=ϕ(x)⊤(Φ⊤ +h WhΦh + λI)−1Φ⊤ +h WhΦhϕ(x) +(a) +=ϕ(x)⊤(Φ⊤ +τhΦτh + λI)−1Φ⊤ +τhΦτhϕ(x) +=ϕ(x)⊤Φ⊤ +τh(ΦτhΦ⊤ +τh + λI)−1Φτhϕ(x) +=k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh), +(30) +where (a) is from Eq. (28). Then, we have σ2 +τh(x) = Σ2 +h(x). +27 + +2) Mean representation equivalence: µTl(x) = ¯µl(x), i.e., +1 +|Ul| +� +u∈Ul +k(x, XTl)⊤(KXTlXTl + λI)−1yl,u = k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1¯yl. +(31) +For the last within-phase index τHl = Tl, we also have the following: +1 +|Ul| +� +u∈Ul +Φ⊤ +Tlyl,u = +1 +|Ul| +� +u∈Ul +tl+Tl +� +t=tl+1 +yu,tϕ(xt) += +1 +|Ul| +� +u∈Ul +Hl +� +h=1 +� +t∈Tl(ah) +yu,tϕ(xt) += +1 +|Ul| +� +u∈Ul +Hl +� +h=1 +ϕ(ah) +� +t∈Tl(ah) +yu,t += +1 +|Ul| +� +u∈Ul +Hl +� +h=1 +ϕ(ah)Tl(ah)yu +l (ah) += +Hl +� +h=1 +Tl(ah)yl(ah)ϕ(ah) += Φ⊤ +HlWHl ¯yl. +(32) +Then, we are ready to derive the equivalence of two representations of the mean function: +1 +|Ul| +� +u∈Ul +k(x, XTl)⊤(KXTlXTl + λI)−1yl,u += +1 +|Ul| +� +u∈Ul +ϕ(x)⊤Φ⊤ +Tl(ΦTlΦ⊤ +Tl + λI)−1yl,u += +1 +|Ul| +� +u∈Ul +ϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1Φ⊤ +Tlyl,u += ϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1 · +1 +|Ul| +� +u∈Ul +Φ⊤ +Tlyl,u +(a) += ϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1Φ⊤ +HlWHl ¯yl += ϕ(x)⊤(Φ⊤ +HlW1/2 +Hl W1/2 +Hl ΦHl + λI)−1Φ⊤ +HlW1/2 +Hl W1/2 +Hl ¯yl += ϕ(x)⊤((W1/2 +Hl ΦHl)⊤(W1/2 +Hl ΦHl) + λI)−1(W1/2 +Hl ΦHl)⊤W1/2 +Hl ¯yl += ϕ(x)⊤Φ⊤ +HlW1/2 +Hl (W1/2 +Hl ΦHlΦ⊤ +HlW1/2 +Hl + λI)−1W1/2 +Hl ¯yl += ϕ(x)⊤Φ⊤ +Hl(ΦHlΦ⊤ +Hl + λW−1 +Hl )−1¯yl += k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1¯yl = ¯µl(x), +(33) +where (a) is from Eq. (28) with τHl = Tl and the result in Eq. (32). +28 + +B.2 +Impact of Batch Schedule Strategy on Posterior Variance +In our batch schedule strategy, the decision xt does not change for Tl(ah) rounds when starting choosing ah +after τh−1 rounds within the l-th phase. Applying Lemma A.2 to our setting with τ ′ = τh−1, we obtain the +following corollary. +Corollary B.2. Consider any phase l. Recall that τh−1 is the within-phase time index before starting +choosing ah. Then, give any set of chosen actions Ah−1 for the first h − 1 batches, for any kernel k, any +x ∈ D, and any τ ∈ [τh−1 + 1, τh−1 + Tl(ah)], we have +1 ⩽ Σh−1(x) +στ(x) +⩽ C. +(34) +Proof. Applying Lemma A.2 to our setting, we have +1 ⩽ +σ2 +τh−1(x) +σ2τ(x) +⩽ 1 + Tl(ah)σ2 +τh(ah). +(35) +Moreover, by selecting Tl(ah) = ⌊(C2 − 1)/Σ2 +h−1(ah)⌋ = ⌊(C2 − 1)/σ2 +τh−1(ah)⌋ (Lemma B.1) in our +algorithm, we derive the result in Eq. (34). +One key step to getting the regret upper bound is to bound the confidence width, which is related to the +maximal value of the posterior variance by the end of each phase. (See Eq. (12). In the following, we provide +a bound for the maximal value of the posterior variance. +Lemma B.3. The posterior variance after Hl batches (decisions) in the l-th phase satisfies +max +x∈Dl ΣHl(x) ⩽ +� +2σ2C2γTl +Tl +. +(36) +Proof. Recall that DPBE plays action ah when τ ∈ [τh−1 + 1, τh−1 + Tl(ah)] within the l-th phase. First, +we have for any x ∈ Dl, any h ⩽ Hl, +ΣHl(x) +(a) +⩽ Σh−1(x) +(b) +⩽ Σh−1(ah) = στh−1(ah), +(37) +where (a) holds because Σh(·) is non-increasing in h, (b) is based on our decision, and the last step is due to +29 + +the equivalent representation result. Then, we have the following: +max +x∈Dl ΣHl(x) ⩽ 1 +Tl +Hl +� +h=1 +Tl(ah)Σh−1(ah) += 1 +Tl +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +Σh−1(ah) += 1 +Tl +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +Σh−1(ah) +στ(ah) +· στ(ah) +(a) +⩽ 1 +Tl +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +Cστ(ah) +(b) += C +Tl +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +στ(xtl+τ) += C +Tl +Tl +� +τ=1 +στ(xtl+τ) +(c) +⩽ C +Tl +� +� +� +�Tl +Tl +� +τ=1 +σ2τ(xtl+τ) +(d) +⩽ C +Tl +� +Tl · 2λγTl = +� +2λC2γTl +Tl +, +(38) +where the inequality (a) is from Corollary B.2, (b) is based on our algorithm decision: xtl+τ = ah for any +τ ∈ [τh−1 + 1, τh−1 + Tl(ah)], (c) is by Cauchy-Schwartz inequality, and (d) is from Lemma A.1. +B.3 +Other Useful Results +Lemma B.4. Consider any particular phase l. In the traditional GP models, without noise in the reward +observations, the difference between the ground truth and regression estimator satisfies +���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) +��� ⩽ BσTl(x). +(39) +30 + +Proof. Representing f(x) in the feature space, we have +���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) +��� += +���ϕ(x)⊤f − ϕ(x)⊤Φ⊤ +Tl(ΦTlΦ⊤ +Tl + λI)−1ΦTlf +��� += +���ϕ(x)⊤f − ϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1Φ⊤ +TlΦTlf +��� += +���λϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1f +��� +⩽ ∥f∥k∥λ(Φ⊤ +TlΦTl + λI)−1ϕ(x)∥k +⩽ B +� +λϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1λI(Φ⊤ +TlΦTl + λI)−1ϕ(x) +⩽ B +� +λϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1(Φ⊤ +TlΦTl + λI)(Φ⊤ +TlΦTl + λI)−1ϕ(x) +⩽ B +� +λϕ(x)⊤(Φ⊤ +TlΦTl + λI)−1ϕ(x) += BσTl(x), +(40) +where the last step is from Eq. (23). +C +Proofs of Theorem 5.1 +Before proving Theorem 5.1, we first provide the key concentration inequality under DPBE in Theorem C.1. +Theorem C.1. For any particular phase l, with probability at least 1 − 4β, the following holds +|f(x) − ¯µl(x)| ⩽ wl(x), +(41) +where mean function ¯µl(x) and confidence width function wl(x) are defined in Eq. (11) and Eq. (12). +Proof. In this proof, we will show the following concentration inequality holds for any x ∈ D +P[|f(x) − ¯µl(x)| ⩾ wl(x)] ⩽ 4β. +(42) +For any x ∈ D, we let wl(x) = wl,1(x) + wl,2(x), where +wl,1(x) ≜ +� +2k(x, x) log(1/β) +|Ul| +and wl,2(x) ≜ ΣHl(x) +�� +2 log(1/β) +|Ul| ++ B +� +. +First, for any x ∈ D, we have the following inequality: +|f(x) − ¯µl(x)| ⩽ +������ +f(x) − +1 +|Ul| +� +u∈Ul +fu(x) +������ ++ +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +. +31 + +Then, we have +P [|f(x) − ¯µl(x)| ⩾ wl(x)] +⩽P +� +� +������ +f(x) − +1 +|Ul| +� +u∈Ul +fu(x) +������ ++ +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,1(x) + wl,2(x) +� +� +⩽P +� +� +������ +f(x) − +1 +|Ul| +� +u∈Ul +fu(x) +������ +⩾ wl,1(x) +� +� + P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +� +� , +(43) +where the last inequality is from union bound. +In the following, we try to bound the above two terms, respectively. +i) Recall that each user u is associated with a local reward function fu ∼ GP(f(·), k(·)). Hence, +fu(x) ∼ N(f(x), k(x, x)), +∀x ∈ D. +(44) +Note that Ul is a set of ⌈2αl⌉ independently sampled random users. Then, we have +1 +|Ul| +� +u∈Ul +fu(x) ∼ N +� +f(x), k(x, x) +|Ul| +� +, +∀x ∈ D. +Combining the concentration inequality for Gaussian random variables, we have +P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − f(x) +������ +⩾ wl,1(x) +� +� ⩽ 2 exp +� +− +|Ul|w2 +l,1(x) +2k(x, x)) +� += 2β. +(45) +ii) Then, we want to bound the second term in Eq. (43): +P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +� +� += +� +Λ +P[Λ = {yl,u}u∈Ul] · P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +������ +{yl,u}u∈Ul +� +� , +where yl,u = [yu,tl+1, . . . , yu,tl+Tl]⊤ denotes the realization of the local reward observations at user u in the +l-th phase. According to our assumption, the participant user u is associated with a local reward function +fu sampled from Gaussian Process GP(f(·), k(·, ·)). Given the points XTl = [x⊤ +tl+1, . . . , x⊤ +tl+Tl]⊤ in D, +the corresponding vector of local rewards yl,u = [yu,tl+1, . . . , yu,tl+Tl]⊤ has the multivariate Gaussian +distribution N(f(XTl), (KXTlXTl + λI)) where f(XTl) = [f(xtl+1), · · · , f(xtl+Tl)]⊤ and KXTlXTl = +[k(x, x′)]x,x′∈XTl is the kernel matrix for the Tl selected actions in the l-th phase. Due to the properties of +GPs, we have that yl,u and fu(x) are jointly Gaussian given XTl: +� +fu(x) +yl,u +� +∼ N +�� +f(x) +f(XTl) +� +, +� +k(x, x) +k(x, XTl)⊤ +k(x, XTl) +KXTlXTl + λI +�� +, +(46) +32 + +where k(x, XTl) = [k(x, xtl+1), . . . , k(x, xtl+Tl)]⊤. According to the basic formula for conditional distri- +butions of Gaussian random vectors (see [Ras03, Appendix A.2] or [Kan18, Proposition 3.2]), we have that +conditioned on yl,u (corresponding to the points XTl), the following holds: +fu(x)|yl,u ∼ N(mu(x), σ2 +Tl(x)), +where we have +mu(x) ≜ f(x) + k(x, XTl)⊤(KXTlXTl + λI)−1(yl,u − f(XTl)), +(47) +σ2 +Tl(x) = k(x, x) − k(x, XTl)⊤(KXTlXTl + λI)−1k(x, XTl). +(48) +Note that we sample the participants Ul independently and that the local reward noise is also independent +across participants. Then, we have the following result: +� +� 1 +|Ul| +� +u∈Ul +fu(x) +� +� +��� {yl,u}u∈Ul = +1 +|Ul| +� +u∈Ul +(fu(x) | yl,u) ∼ N +� +� 1 +|Ul| +� +u∈Ul +mu(x), +σ2 +Tl(x) +|Ul| +� +� . +Combining the Gaussian concentration inequality, we have the following result +P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − +1 +|Ul| +� +u∈Ul +mu(x) +������ +⩾ +� +2σ2 +Tl(x) log(1/β) +|Ul| +������ +{yl,u}u∈Ul +� +� ⩽ 2β. +(49) +From Lemma B.1, we have the following equation: +1 +|Ul| +� +u∈Ul +k(x, XTl)⊤(KXTlXTl + λI)−1yl,u = k(x, Ah)⊤(KAhAh + λW−1 +h )−1¯yl = ¯µl(x), +(50) +which implies +1 +|Ul| +� +u∈Ul +mu(x) = ¯µl(x) + f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl). +(51) +Then, the gap between the average local function +1 +|Ul| +� +u∈Ul fu(·) and the estimator ¯µl(·) satisfies +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩽ +������ +1 +|Ul| +� +u∈Ul +fu(x) − +1 +|Ul| +� +u∈Ul +mu(x) +������ ++ +���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) +��� +(a) +⩽ +������ +1 +|Ul| +� +u∈Ul +fu(x) − +1 +|Ul| +� +u∈Ul +mu(x) +������ ++ BσTl(x), +(52) +33 + +where (a) is from Lemma B.4. Combining the result in Eq. (49), we have +P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +������ +{yl,u}u∈Ul +� +� +⩽P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − +1 +|Ul| +� +u∈Ul +mu(x) +������ ++ BσTl(x0) ⩾ wl,2(x) +������ +{yl,u}u∈Ul +� +� +(a) +=P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − +1 +|Ul| +� +u∈Ul +mu(x) +������ +⩾ +� +2σ2 +Tl(x) log(1/β) +|Ul| +������ +{yl,u}u∈Ul +� +� ⩽ 2β, +(53) +where (a) is from σ2 +Tl(x) = Σ2 +Hl(x) according to Lemma B.1. Therefore, we derive the desired result: +P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +� +� += +� +Λ +P[Λ = {yl,u}u∈Ul] · P +� +� +������ +1 +|Ul| +� +u∈Ul +fu(x) − ¯µl(x) +������ +⩾ wl,2(x) +������ +{yl,u}u∈Ul +� +� +⩽ +� +Λ +P[Λ = {yl,u}u∈Ul] · 2β = 2β. +(54) +To prove Theorem 5.1, we first present three main conclusions when the concentration inequality in +Theorem C.1 holds, then get an upper bound for the regret incurred in a particular phase l with high probability, +and finally sum up the regret over all phases. +Define a “good” event when Eq. (41) holds in the l-th phase as: +El ≜ {∀x ∈ Dl, |f(x) − ¯µl(x)| ⩽ wl(x)} . +We have P[El] ⩾ 1−4|D|β via the union bound. Then, under event El in the l-th phase, we have the following +three observations: +1. For any optimal action x∗ ∈ argmaxx∈D f(x), if x∗ ∈ Dl, then x∗ ∈ Dl+1. +2. Let f∗ = maxx∈D f(x). Supposed that x∗ ∈ Dl. For any x ∈ Dl+1, its reward gap from the optimal +reward is bounded by 4 maxx∈Dl wl(x), i.e., +f∗ − f(x) ⩽ 4 max +x∈Dl wl(x). +3. The confidence width function satisfies +max +x∈Dl wl(x) ⩽ +� +2κ2 log(1/β) +|Ul| ++ +� +4σ2C2γTl log(1/β) +Tl|Ul| ++ +� +2σ2B2C2γTl +Tl +. +34 + +Proof. Observation 1: Let b ∈ argmaxx∈Dl(¯µl(x) − wl(x)). Then under event El, we have +¯µl(x∗) + wl(x∗) ⩾ f(x∗) ⩾ f(b) ⩾ ¯µl(b) − wl(b), +(55) +which indicates x∗ ∈ Dl+1 according to Eq. (10). +Observation 2: For any x ∈ Dl+1, we have x ∈ Dl and +¯µl(x) + wl(x) ⩾ ¯µl(b) − wl(b) ⩾ ¯µl(x∗) − wl(x∗). +(56) +Then, we have the regret of choosing any action x ∈ Dl+1 satisfying +f(x∗) − f(x) +(a) +⩽ ¯µl(x∗) + wl(x∗) − ¯µl(x) + wl(x) +(b) +⩽ 2(wl(x) + wl(x∗)) +⩽ 4 max +x∈Dl wl(x), +(57) +where (a) holds under event El and the second inequality (b) is from Eq. (56). Then, we derive Observation +2. +Observation 3: Based on the result in Lemma B.3, we have +max +x∈Dl wl(x) = max +x∈Dl +�� +2k(x, x) log(1/β) +|Ul| ++ ΣHl(x) +�� +2 log(1/β) +|Ul| ++ B +�� +⩽ +� +2κ2 log(1/β) +|Ul| ++ max +x∈Dl ΣHl(x) +�� +2 log(1/β) +|Ul| ++ B +� +⩽ +� +2κ2 log(1/β) +|Ul| ++ +� +4λC2γTl log(1/β) +Tl|Ul| ++ +� +2λB2C2γTl +Tl += +� +2κ2 log(1/β) +|Ul| ++ +� +4σ2C2γTl log(1/β) +Tl|Ul| ++ +� +2σ2B2C2γTl +Tl +. +(58) +Then, we are ready to prove Theorem 5.1. +Proof of Theorem 5.1. Let the regret in the l-th phase be rl ≜ � +t∈Tl(maxx∈D f(x)−f(xt)). For any l ⩾ 2, +35 + +we assume event El−1 holds. Then, we have the following result +rl = +� +t∈Tl +(max +x∈D f(x) − f(x)) +⩽ +� +t∈Tl +4 max +x∈Dl−1 wl−1(x) +⩽ 4Tl max +x∈Dl−1 wl−1(x) +⩽ 4Tl +� +� +� +2κ2 log(1/β) +|Ul−1| ++ +� +4σ2C2γTl−1 log(1/β) +Tl−1|Ul−1| ++ +� +2σ2B2C2γTl−1 +Tl−1 +� +� +(a) +⩽ 4 · 2l−1 +�� +2κ2 log(1/β) +2α(l−1) ++ +� +4σ2C2γT log(1/β) +2(1+α)(l−1)−1 ++ +� +2σ2B2C2γT +2l−2 +� +⩽ 4 +� +2κ2 log(1/β) +� +2(2−α)(l−1) + 8σC +� +2γT log(1/β) +� +2(1−α)(l−1) + 8σBC +� +γT 2l−1, +(59) +where (a) is from γTl−1 ⩽ γT and |Ul| ⩾ 2αl. +Define Eg as the event where the “good” event occurs in every phase, i.e., Eg ≜ �L +l=1 El. It is not +difficult to obtain P[Eg] ⩾ 1 − 4|D|βL by applying union bound. At the same time, let Rg be the regret +under event Eg, and Rb be the regret if event Eg does not hold. Then, the expected total regret in T is +E[R(T)] = P[Eg]Rg + (1 − P[Eg])Rb. +Under event Eg, the regret in the l-th phase rl satisfies Eq. (59) for any l ⩾ 2. Note that r1 ⩽ 2T1Bκ ⩽ +2Bκ since T1 = 1 and for any x ∈ D, +|f(x)| = |⟨f, k(x, ·)⟩k| ⩽ ∥f∥k⟨k(x, ·), k(x, ·)⟩1/2 +k +⩽ Bk(x, x)1/2 ⩽ Bκ. +Then, we have +Rg = +L +� +l=1 +rl +⩽ 2Bκ + +L +� +l=2 +4 +� +2κ2 log(1/β) +� +2(2−α)(l−1) ++ +L +� +l=2 +8σC +� +2γT log(1/β) +� +2(1−α)(l−1) ++ +L +� +l=2 +8σBC +� +γT 2l−1 +⩽ 2Bκ + 4 +� +2κ2 log(1/β) · 4 +� +2(L−1)(2−α) ++ 8σC +� +2γT log(1/β) · C1 +� +2(1−α)(L−1) +� +C1 = +√ +21−α/( +√ +21−α − 1) +� ++ 8σBC√γT · 4 +√ +2L−1 +⩽ 2Bκ + 16 +� +2κ2 log(1/β)T 1−α/2 + 8σC1C +� +2γT log(1/β)T 1−α + 32σBC +� +γT T, +(60) +where the last step is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 +l=1 Tl + 1 ⩽ T. +36 + +On the other hand, Rb ⩽ 2BκT since | maxx∈D f(x) − f(x)| ⩽ 2Bκ for all x ∈ D. Choose β = +1/(|D|T) in Algorithm 1. Finally, we have the following results: +E[R(T)] += P[Eg]Rg + (1 − P[Eg])Rb +⩽ Rg + 4|D|βL · 2BκT +⩽ 2Bκ + 16 +� +2κ2 log(1/β)T 1−α/2 + 8σC1C +� +2γT log(1/β)T 1−α + 32σBC +� +γT T ++ 8Bκ|D|βLT += 2Bκ + 16T 1−α/2� +2κ2 log(|D|T) + 8σC1C +� +2γT T 1−α log(|D|T) ++ 32σBC +� +γT T + 8Bκ log(2T) += O(T 1−α/2� +log(|D|T)) + O( +� +γT T 1−α log(|D)T) + O( +� +γT T). +(61) +D +Proofs for Communication and Computation Results +The results regarding computation complexity and communication cost highly depend on the number of +batches Hl in each phase l. Hence, we first provide the proof for Lemma 5.6. +Proof of Lemma 5.6. To bound the number of batches in the l-th phase, we follow a similar line to the proof +of Lemma 4.3 in [Cal22]. For any 1 ⩽ h ⩽ Hl, we have +Tl(ah) = +� +C2 − 1 +Σ2 +h−1(ah) +� +⩾ +C2 − 1 +Σ2 +h−1(ah) − 1 +⇒ +Σ2 +h−1(ah)(Tl(ah) + 1) ⩾ C2 − 1 +⇒ +2Σ2 +h−1(ah)Tl(ah) ⩾ C2 − 1. +(62) +Recall that we use τh to denote the last within-phase time index in the h-th batch. Then, summing the above +37 + +inequality across all batches up to Hl, we have +Hl(C2 − 1) ⩽ +Hl +� +h=1 +2Σ2 +h−1(ah)Tl(ah) +⩽ 2 +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +Σ2 +h−1(ah) += 2 +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +Σ2 +h−1(ah) +σ2τ(ah) +· σ2 +τ(ah) +(a) +⩽ 2 +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +C2 · σ2 +τ(ah) +(b) += 2C2 +Hl +� +h=1 +τh−1+Tl(ah) +� +τ=τh−1+1 +σ2 +τ(xtl+τ) += 2C2 +Tl +� +τ=1 +σ2 +τ(xtl+τ) +(c) +⩽ 4σ2C2γTl, +(63) +where (a) is from Corollary B.2, (b) is based on our algorithm decision: xtl+τ = ah for any τ ∈ [τh−1 + +1, τh−1 + Tl(ah)], (c) is from Lemma A.1 where XTl = [x⊤ +tl+1, . . . , x⊤ +tl+Tl]⊤ for any phase l and λ = σ2. +Hence, we derive +Hl ⩽ 4σ2C2 +C2 − 1γTl. +(64) +We already analyze how to derive the computation complexity for DPBE in Remark 5.7. In the following, +we prove Theorem 5.8, which tells the result regarding communication cost: O(γT T α). +Proof of Theorem 5.8. Note that the communicating data in each phase between participants and the agent is +the local average performance yu +l (a) for each action a chosen in the corresponding batch. That is, Nu,l ⩽ Hl +for every participant u. (Here, the inequality holds when merging batches as Remark 4.1). Combining the +bound of Hl in Lemma 5.6, we derive the total communication cost satisfying +L +� +l=1 +|Ul|Hl ⩽ +L +� +l=1 +4σ2C2 +C2 − 1γTl · (2αl + 1) = O +� σ2C2 +C2 − 1 · γT T α +� +, +(65) +where the last step is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 +l=1 Tl + 1 ⩽ T. +E +Differentially Private DPBE Extensions +In this section, we extend the differentially private DPBE in Section 6 to two other celebrated DP models: the +local model and the shuffle model. +38 + +To begin with, we present the details of the DP-DPBE algorithm (see Algorithm 2) in the central DP +model discussed in Section 6. Recall that in the central DP model, with a trusted agent, data privacy is +protected by privatizing the aggregated feedback so that the output of the algorithm is indistinguishable +between any two users. In a particular phase l, the aggregated feedback for each chosen action becomes +˜yl = ¯yl + (ρ1, . . . , ρHl) (see Eq. (15)), where ρj +i.i.d. +∼ N(0, σ2 +nc) is the injected Gaussian noise for ensuring +the required (ε, δ)-DP and is chosen according to the (high-probability) sensitivity of ¯yl. Specifically, we set +the variance of the injected Gaussian noise to the following: +σnc = 2 +� +2(κ2 + σ2)Hl log(2Hl/δ1) ln(1.25/δ2) +ε|Ul| +, +(66) +where δ1 ∈ (0, δ) is the probability of sensitivity concentration of ¯yl (i.e., Eq. (77) holds with probability at +least 1 − δ1) and δ2 = δ − δ1. By accounting for privacy noise, we update the confidence width function in +Eq. (17) with σn = σnc +� +2C2γT , where C is the rare-switching parameter. +Differentially Private DPBE in the Local DP Model. In the local model, the users do not trust the +agent, and thus, each is equipped with a local randomizer R to protect its own local reward. Let Y be the set +of all possible values of the local reward. Formally, a local randomizer R is (ε, δ)-local differentially private +(or (ε, δ)-LDP) if for any two user inputs, the probability that R outputs two values in Y that are not different +by more than a multiplicative factor of eε and an additive factor of δ. To guarantee LDP, the local randomizer +R at each user u injects Gaussian noise before sending the local reward observations out to the central agent. +That is, +R(yu +l ) = yu +l + (ρu,1, . . . , ρu,Hl), +(70) +where ρu,j∼N(0, σ2 +nl) is i.i.d. across both users and actions and the variance σ2 +nl is chosen according to the +(high-probability) sensitivity of yu +l (see Eq. (80)). Then, the private aggregated feedback for the chosen +actions in the l-th phase in the local DP model becomes +˜yl = +1 +|Ul| +� +u∈Ul +R(yu +l ) = +1 +|Ul| +� +u∈Ul +(yu +l + (ρu,1, . . . , ρu,Hl)) . +(71) +We call the differentially private version of DPBE in the local DP model LDP-DPBE. Specifically, +we extend the DPBE algorithm (Algorithm 2) to LDP-DPBE by employing a local randomizer R as in +Eq. (70) at each participant in the l-th phase and then using the privately aggregated feedback in Eq. (71) +to estimate the mean function ˜µl(·) in Eq. (16). The injected Gaussian noise at each participant is σnl = +2√ +2(κ2+σ2)Hl log(2Hl/δ1) ln(1.25/δ2)) +ε +, where δ1 ∈ (0, δ) is the probability of sensitivity concentration of ¯yl +(i.e., Eq. (80) holds with probability at least 1−δ1) and δ2 = δ−δ1. In LDP-DPBE, we update the confidence +width function in Eq. (17) with σn = +� +2C2σ2 +nlγT +|Ul| +, where C is the rare-switching parameter. +Differentially Private DPBE in the Shuffle DP Model. While local DP provides a more stringent +privacy guarantee, it usually incurs larger regret cost [ZT20]. The shuffle model is recently proposed to +achieve a better tradeoff between regret and privacy [Che19]. In the shuffle model, between the users and the +agent, there exists a shuffler that permutes the local feedback from the participants before they are observed +by the agent so that the agent cannot distinguish between two users’ feedback. Thus, an additional layer +of randomness is introduced via shuffling, which can often be easily implemented using Cryptographic +primitives (e.g., mixnets) due to its simple operation [Bit17]. Specifically, the shuffle DP model consists +of three components: a local randomizer R at each user side, a shuffler S between the users and the agent, +and an analyzer A at the agent side. Let UT ≜ (Ul, · · · , Ul) be the participants throughout the T rounds. +39 + +Define the (composite) mechanism Ms(UT ) ≜ ((S ◦ R)(U1), (S ◦ R)(U2), . . . , (S ◦ R)(UL)), where +(S ◦ R)(Ul) ≜ S({R(yu +l )}u∈Ul). Formally, We say the DP-DPBE algorithm satisfies the shuffle differential +privacy (SDP) if the composite mechanism Ms is DP, which leads to the following formal definition. +Definition E.1. (Shuffle Differential Privacy (SDP)). For any ε ⩾ 0 and δ ∈ [0, 1], the DP-DPBE is (ε, δ)- +shuffle differential privacy (or (ε, δ)-SDP) if for any pair UT and U +′ +T that differ by one user, and for any +Z ∈ Range(Ms)6: +P[Ms(UT ) ∈ Z] ⩽ eεP[Ms(U +′ +T ) ∈ Z] + δ. +(72) +In our case, we apply a shuffle model to the feedback from participants of every particular phase. That is, +the private aggregated feedback for the chosen actions in the l-th phase in the shuffle DP model becomes +˜yl = A (S ({R(yu +l }u∈Ul))) , +(73) +where the local randomizer injects a sub-Gaussian noise with variance σ2 +ns, which is i.i.d. across both users +and actions. Thanks to our phase-then-batch strategy, the recently proposed vector summation protocol +[Che21b] can be extended to our algorithm as [LZJ22]. We present the concrete pseudocodes of R, S, and A +in Algorithm 3. +We call the differentially private version of DPBE in the shuffle model SDP-DPBE, which is extended +from DP-DPBE by using the privately aggregated feedback in Eq. (73), where R, S, and A are specified +in Algorithm 3. For any δ1 ∈ (0, δ), let ∆ ≜ Bκ√Hl + +� +2(κ2 + σ2)Hl log(2Hl/δ1). It is not difficult to +show that ∥yu +l ∥2 ⩽ ∆ with probability at least 1 − δ1. SDP-DPBE employs Algorithm 3 in each phase l +with input {yu +l }u∈Ul, ∆, and privacy parameters ε and δ2 = δ − δ1. According to [LZJ22], the introduced +error for privacy is sub-Gaussian with variance σ2 +ns = O +� +Hl(κ2+σ2) log(Hl/δ1) ln(Hl/δ2)2 +ε2|Ul|2 +� +. In SDP-DPBE, +we update the confidence width function in Eq. (17) with σn = σns +� +2C2γT , where C is the rare-switching +parameter. +E.1 +Performance Guarantee +For the DP-DPBE algorithm incorporated with the above local and shuffle DP models, we provide the DP +guarantee and regret in the following. +Theorem E.2 (DP guarantee). Under Assumptions 1, 2, 3, and 4, for any ε > 0 and δ ∈ (0, 1), +i) LDP-DPBE guarantees (ε, δ)-LDP; +ii) SDP-DPBE guarantees (ε, δ)-SDP. +We achieve the above LDP guarantee of i) directly by employing the Gaussian mechanism given the +(high-probability) sensitivity of yu +l . In the shuffle model, we follow the shuffle protocol for each phase in +[LZJ22] and derive the corresponding SDP guarantee from Theorem A.2 therein. +From the above results, we derive that compared to the local model the shuffle model injects much less +noise (σ2 +ns vs. σ2 +nl) without requiring a trusted agent. In the following, we present the regret performance of +DP-DPBE in these two DP models. +6Rang(M) denotes the range of the output of the mechanism M. +40 + +Table 6: Regret of DP-DPBE in Different DP Models +Algorithms +Regret +DPBE +O(T 1−α/2� +log(|D|T)) +CDP-DPBE +O(T 1−α/2� +log(|D|T)) + O +� +ln(1/δ)γT T 1−α√ +log(|D|T) +ε +� +LDP-DPBE +O(T 1−α/2� +log(|D|T)) + O +� +ln(1/δ)γT T 1−α/2√ +log(|D|T) +ε +� +SDP-DPBE +O(T 1−α/2� +log(|D|T)) + O +� +ln3/2(γT /δ)γT T 1−α√ +log(|D|T) +ε +� +Notes: CDP-DPBE, LDP-DPBE, and SDP-DPBE represent the DP-DPBE algorithm in the central, local, +and shuffle models, respectively, which guarantee (ε, δ)-DP, (ε, δ)-LDP, and (ε, δ)-SDP, respectively. +Theorem E.3 (LDP-DPBE). Under Assumptions 1, 2, and 3, the LDP-DPBE algorithm with β = +1 +|D|T +achieves the following expected regret: +E[R(T)] = O(T 1−α/2� +log(|D|T)) + O +� +ln(1/δ)γT T 1−α/2� +log(|D|T) +ε +� +. +(75) +Theorem E.4 (SDP-DPBE). Under Assumptions 1, 2, and 3, the SDP-DPBE algorithm with β = +1 +|D|T +achieves the following expected regret: +E[R(T)] = O(T 1−α/2� +log(|D|T)) + O +� +ln3/2(γT /δ)γT T 1−α� +log(|D|T) +ε +� +. +(76) +We omit the proofs for the above two theorems because they can be derived by directly replacing σn of +the central model with σn = +� +2C2σ2 +nlγT +|Ul| +of the local model and σn = σns +� +2C2γT of the shuffle model. +See Appendix E.3. +E.2 +Proofs for DP Guarantees +Before providing the DP guarantee of the DPBE algorithm in the three DP models, we first show the ℓ2 +sensitivity of ¯yl, which is a key parameter to decide the Gaussian noise. +Lemma E.5. Let UT , U′ +T ⊆ U be two sets of participants in DPBE differing on a single user that is +participating in the l-th phase, and let ¯yl and ¯y′ +l be the corresponding average local reward. For any +δ1 ∈ (0, 1), we have that with probability at least 1 − δ1, the maximal ℓ2 distance between ¯yl and ¯y′ +l is +bounded by +max |¯y′ +l − ¯yl| ⩽ 2 +� +(κ2 + σ2)Hl log(2Hl/δ1) +|Ul| +, +(77) +where Hl denotes the dimension of ¯yl and σ2 is the variance of the noisy observations. +41 + +Proof. Let Ul, U ′ +l be the sets of participating users in l-th phase corresponding to UT and U′ +T respectively. +We have |Ul| = |U ′ +l| and the maximal ℓ2 distance between ¯yl, ¯y′ +l is the following: +max |¯y′ +l − ¯yl| = max +UT ,U′ +T +������ +1 +|Ul| +� +u∈U′ +l +yu +l − +1 +|Ul| +� +u∈Ul +yu +l +������ +2 += +1 +|Ul| max +u,u′∈U ∥yu′ +l − yu +l ∥2. +(78) +For any chosen action a ∈ AHl, we have the following result: +|yu′ +l (a) − yu +l (a)| = +������ +1 +Tl(a) +� +t∈Tl(a) +yu′,t − +1 +Tl(a) +� +t∈Tl(a) +yu,t +������ += +������ +1 +Tl(a) +� +t∈Tl(a) +(yu′,t − yu,t) +������ += +������ +1 +Tl(a) +� +t∈Tl(a) +(fu′(xt) + ηu′,t − fu(xt) − ηu,t) +������ +⩽ +1 +Tl(a) +� +t∈Tl(a) +��fu′(xt) + ηu′,t − fu(xt) − ηu,t +�� . +Note that fu(x) ∼ N(f(x), k(x, x)), ηu,t ∼ N(0, σ2), and the participating users are independent from each +other. We have (fu′(xt) + ηu′,t − fu(xt) − ηu,t) ∼ N(0, 2(k(xt, xt) + σ2)). According to the concentration +property of Gaussian distribution, we have with probability at least 1 − δ1, +|fu′(xt) + ηu′,t − fu(xt) + ηu,t)| ⩽ 2 +� +(k(xt, xt) + σ2) log(2/δ1) ⩽ 2 +� +(κ2 + σ2) log(2/δ1), +(79) +which results in |yu′ +l (a) − yu +l (a)| ⩽ 2 +� +(κ2 + σ2) log(2/δ1) for any particular a ∈ AHl with probability at +least 1 − δ1. By substituting the above result into Eq. (78) and applying union bound, we have that with +probability at least 1 − δ1, the following is satisfied: +max +u,u′∈U ∥yu′ +l − yu +l ∥2 ⩽ 2 +� +Hl(κ2 + σ2) log(2Hl/δ1), +(80) +and then with probability at least 1 − δ1, the ℓ2 distance between ¯yl and ¯y′ +l is bounded by +max |¯y′ +l − ¯yl| ⩽ maxu,u′∈U ∥yu′ +l − yu +l ∥2 +|Ul| +⩽ 2 +� +Hl(κ2 + σ2) log(2Hl/δ1) +|Ul| +, +(81) +where the last step is because Hl is the dimension of yu +l and also the number of actions in AHl. +For both the central model and the local model, we employ the Gaussian mechanism in the differential +privacy literature, which is described in the following. +Theorem E.6. (Gaussian Mechanism [DR14]). Given any vector-valued function7 f : U∗ → Rs, define +∆2 ≜ maxU1,U′ +2⊆U ∥f(U1) − f(U2)∥2. Let σ = ∆2 +� +2 ln(1.25/δ)/ε. The Gaussian mechanism, which adds +independently drawn random noise from N(0, σ2) to each output of f(·), i.e. returning f(U) + (ρ1, . . . , ρs) +with ρj +i.i.d. +∼ N(0, σ2), ensures (ε, δ)-DP. +7We use the superscript ∗ to indicate that the length could be varying. +42 + +Proof of Theorem 6.2. Let E denote the event that Eq. (77) holds, and thus, P[E] ⩾ 1 − δ1. Let ∆2 ≜ +max |¯y′ +l − ¯yl|. If E holds, adding independently drawn noise from N +� +0, 2∆2 +2 ln(1.25/δ2) +ε +� +to each element of +¯yl, i.e., returning ¯yl + (ρ1, · · · , ρHl) with ρj +i.i.d. +∼ N +� +0, 2∆2 +2 ln(1.25/δ2) +ε +� +, ensures (ε, δ2)-DP . Specifically, +the following inequality holds +P[M(UT ) ∈ Z|E] ⩽ eεP[M(U′ +T ) ∈ Z|E] + δ2. +(82) +Then, we have +P[M(UT ) ∈ Z] ⩽ P[M(UT ) ∈ Z|E]P[E] + 1 − P[E] +⩽ (eεP[M(U′ +T ) ∈ Z|E] + δ2)P[E] + δ1 +⩽ eεP[M(U′ +T ) ∈ Z|E]P[E] + δ2 + δ1 +⩽ eεP[M(U′ +T ) ∈ Z|E]P[E] + δ2 + δ1 +⩽ eεP[M(U′ +T ) ∈ Z, E] + δ2 + δ1 +⩽ eεP[M(U′ +T ) ∈ Z] + δ, +(83) +where δ = δ1 + δ2. +Similarly, we can derive the (ε, δ)-LDP. Meanwhile, we can achieve (ε, δ)-SDP by combining the analysis +in Eq. (83) and the proof for Theorem A.2 in [LZJ22]. +E.3 +Proof of Theorem 6.3 +Following a similar line to the proof for Theorem 5.1, we first provide the key concentration inequality under +DP-DPBE in Theorem E.7. +Theorem E.7. For any particular phase l, with probability at least 1 − 6β, the following holds +|f(x) − ˜µl(x)| ⩽ ˜wl(x), +(84) +where mean function ˜µl(x) and confidence width function ˜wl(x) are defined in Eq. (16) and Eq. (17). +Proof. In this proof, we will show the following concentration inequality holds for any x ∈ D +P[|f(x) − ˜µl(x)| ⩾ ˜wl(x)] ⩽ 6β. +(85) +Let ρ ≜ (ρ1, . . . , ρHl). Note that +˜µl(x) = k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1˜yl += k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1(¯yl + ρ) += ¯µl(x) + k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ. +(86) +Then, we have +|f(x) − ˜µl(x)| ⩽ |f(x) − ¯µl(x)| + |k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ|. +(87) +43 + +For any x ∈ D, we have +P [|f(x) − ˜µl(x)| ⩾ ˜wl(x)] +⩽P +� +|f(x) − ¯µl(x)| + +���k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ +��� ⩾ wl(x) + 2C +� +γT σ2n log(1/β) +� +⩽P [|f(x) − ¯µl(x)| ⩾ wl(x)] + P +����k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ +��� ⩾ 2C +� +γT σ2n log(1/β) +� +⩽4β + P +����k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ +��� ⩾ 2C +� +γT σ2n log(1/β) +� +, +(88) +where the first inequality is due to ˜wl(x) = wl(x) + +� +2σ2n log(1/β) from Eq. (17), the second inequality is +from union bound, and the last one is from Theorem C.1. Hence, it remains to bound the second probability +in Eq. (88). +Recall that ρ = (ρ1, . . . , ρHl) where ρj +i.i.d. +∼ N(0, σ2 +nc). Then, k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ is +the sum of Hl i.i.d. Gaussian variables, and the total variance (denoted by σ2 +sum) is +σ2 +sum = k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1(KAHlAHl + λW−1 +Hl )−1k(x, AHl)σ2 +nc. +(89) +Notice that +k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1(KAHlAHl + λW−1 +Hl )−1k(x, AHl) +=ϕ(x)⊤Φ⊤ +Hl(ΦHlΦ⊤ +Hl + λW−1 +Hl )−1(ΦHlΦ⊤ +Hl + λW−1 +Hl )−1ΦHlϕ(x) +=ϕ(x)⊤Φ⊤ +HlW1/2 +Hl (W1/2 +Hl ΦHlΦ⊤ +HlW1/2 +Hl + λI)−1W1/2 +Hl · W1/2 +Hl (W1/2 +Hl ΦHlΦ⊤ +HlW1/2 +Hl + λI)−1W1/2 +Hl ΦHlϕ(x) +=ϕ(x)⊤(Φ⊤ +HlW1/2 +Hl W1/2 +Hl ΦHl + λI)−1Φ⊤ +HlW1/2 +Hl · WHl · W1/2 +Hl ΦHl(Φ⊤ +HlW1/2 +Hl W1/2 +Hl ΦHl + λI)−1ϕ(x) +=ϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1Φ⊤ +HlW2 +HlΦHl(Φ⊤ +HlWHlΦHl + λI)−1ϕ(x) +(a) +⩽Tlϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1Φ⊤ +HlWHlΦHl(Φ⊤ +HlWHlΦHl + λI)−1ϕ(x) +=Tlϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1(Φ⊤ +HlWHlΦHl + σ2 +nI)(Φ⊤ +HlWHlΦHl + λI)−1ϕ(x) +− λTlϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1(Φ⊤ +HlWHlΦHl + λI)−1ϕ(x) +⩽Tlϕ(x)⊤(Φ⊤ +HlWHlΦHl + λI)−1ϕ(x) +(b) +=Tlϕ(x)⊤(Φ⊤ +τHlΦτHl + λI)−1ϕ(x) +(c) += +Tlσ2 +τHl(x) +λ +(d) += +TlΣ2 +Hl(x) +λ += +TlΣ2 +Hl(x) +σ2 +⩽ 2C2γTl, +(90) +where (a) is from Φ⊤ +HlW2 +HlΦHl ⩽ Φ⊤ +Hl(TlI)WHlΦHl = TlΦ⊤ +HlWHlΦHl because each diagonal entry of +WHl satisfies [WHl]hh = Tl(ah) ⩽ Tl, (b) is based on Eq. (28), (c) is from Eq. (23), and (d) is according to +the equivalence representation in Lemma B.1. The last step is from the result in Lemma B.3. +Substituting the above result into Eq. (89), we have +σ2 +sum ⩽ 2C2γTlσ2 +nc = σ2 +n. +(91) +According to the tail bound of Gaussian variables, we have +P +����k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1ρ +��� ⩾ +� +2σ2n log(1/β) +� +⩽ 2 exp +� +−4C2γT σ2 +nc log(1/β) +2σ2sum +� +⩽ 2β. +44 + +Proof of Theorem 6.3. Similar to the proof of Theorem 5.1, we, to prove Theorem 6.3, first present three +results when the concentration inequality in Theorem E.7 holds, then obtain an upper bound for the regret +incurred in a particular phase l > 2 with high probability, and finally sum up the regret over all phases. +1) Three observations when Eq. (84) holds +Define a “good” event when Eq. (84) holds in the l-th phase as: +˜El ≜ {∀x ∈ Dl, |f(x) − ˜µl(x)| ⩽ ˜wl(x)} . +We have P[ ˜El] ⩾ 1 − 6|D|β via the union bound. Then, similar to the non-private case, under event ˜El in the +l-th phase, we have the following three observations: +1. For any optimal action x∗ ∈ argmaxx∈D f(x), if x∗ ∈ Dl, then x∗ ∈ Dl+1. +2. Let f∗ = maxx∈D f(x). Supposed that x∗ ∈ Dl. For any x ∈ Dl+1, its reward gap from the optimal +reward is bounded by 4 maxx∈Dl ˜wl(x), i.e., +f∗ − f(x) ⩽ 4 max +x∈Dl ˜wl(x). +3. The confidence width function in the private setting satisfies +max +x∈Dl ˜wl(x) ⩽ max +x∈Dl wl(x) + G1γT +� +2 log(1/β) +|Ul| +, +(92) +where G1 ≜ 8C2√ +2(κ2+σ2)σ2 log(1/δ1) ln(1.25/δ2) +ε +√ +C2−1 +. +The first two observations can be derived similar to the non-private case. Regarding the third observation, we +have the confidence width function in the private setting ˜wl(x) = wl(x) + +� +2σ2n log(1/β) and +� +2σ2n log(1/β) += 2C +� +γT σ2nc log(1/β) += 4C +� +2(κ2 + σ2)HlγT log(1/δ1) ln(1.25/δ2) log(1/β) +ε|Ul| +(a) +⩽ 8C2γT +� +2(κ2 + σ2)σ2 log(1/δ1) ln(1.25/δ2) log(1/β) +ε|Ul| +√ +C2 − 1 +⩽ 8C2� +2(κ2 + σ2)σ2 log(1/δ1) ln(1.25/δ2) +ε +√ +C2 − 1 +� +�� +� +G1 +·γT +� +2 log(1/β) +|Ul| +, +where (a) is from Lemma 5.6. +2) Regret in a specific phase l > 2. +45 + +Under event ˜El−1, the regret incurred in the l-th phase is +� +t∈Tl +f∗ − f(xt) +⩽ +� +t∈Tl +4 max +x∈Dl−1 ˜wl−1(x) +⩽4Tl max +x∈Dl−1 wl−1(x) +(a) +⩽4Tl max +x∈Dl−1 wl−1(x) + 4Tl · G1γT +� +2 log(1/β) +|Ul−1| +⩽4Tl max +x∈Dl−1 wl−1(x) + 4G1γT +� +2 log(1/β)2(1−α)(l−1) +⩽4 +� +2κ2 log(1/β) +� +2(2−α)(l−1) + 8σC +� +2γT log(1/β) +� +2(1−α)(l−1) + 8σBC +� +γT 2l−1 ++ 4G1γT +� +2 log(1/β)2(1−α)(l−1), +where (a) is from Observation 3 and the last step is from Eq. (59). +3) Total regret. +Define ˜Eg as the event where the “good” event occurs in every phase in the private setting, i.e., ˜Eg ≜ +�L +l=1 ˜El. It is not difficult to obtain P[Eg] ⩾ 1 − 6|D|βL by applying union bound. At the same time, the +total regret under event ˜Eg becomes +Rg = +L +� +l=1 +� +t∈Tl +(f∗ − f(xt)) +⩽ 2Bκ + +L +� +l=2 +4 +� +2κ2 log(1/β) +� +2(2−α)(l−1) ++ +L +� +l=2 +8σC +� +2γT log(1/β) +� +2(1−α)(l−1) ++ +L +� +l=2 +8σBC +� +γT 2l−1 + +L +� +l=2 +4G1γT +� +2 log(1/β)2(1−α)(l−1) +⩽ 2Bκ + 4 +� +2κ2 log(1/β) · 4 +� +2(L−1)(2−α) ++ 8σC +� +2γT log(1/β) · C1 +� +2(1−α)(L−1) +� +C1 ≜ +√ +21−α +√ +21−α − 1 +� ++ 8σBC√γT · 4 +√ +2L−1 ++ 4G1γT +� +2 log(1/β) · C22(1−α)(L−1) +� +C2 ≜ +21−α +21−α − 1 +� +⩽2Bκ + 16 +� +2κ2 log(1/β)T 1−α/2 + 8σC1C +� +2γT log(1/β)T 1−α ++ 32σBC +� +γT T + 4C2G1γT +� +2 log(1/β)T 1−α, +(93) +where the last step is is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 +l=1 Tl + 1 ⩽ T. On the other hand, +Rb ⩽ 2BκT since | maxx∈D f(x) − f(x)| ⩽ 2Bκ for all x ∈ D. Choose β = 1/(|D|T) in Algorithm 2. +46 + +Then, the expected regret is: +E[R(T)] = P[ ˜Eg]Rg + (1 − P[ ˜Eg])Rb +⩽ Rg + 6|D|βL · 2BκT +⩽ 2Bκ + 16 +� +2κ2 log(1/β)T 1−α/2 + 8σC1C +� +2γT log(1/β)T 1−α + 32σBC +� +γT T ++ 4C2G1γT +� +2 log(1/β)T 1−α + 12Bκ|D|βLT += 2Bκ + 16T 1−α/2� +2κ2 log(|D|T) + 8σC1C +� +2γT T 1−α log(|D|T) + 32σBC +� +γT T ++ 4C2G1γT +� +2 log(|D|T)T 1−α + 12Bκ log(2T) += O(T 1−α/2� +log(|D|T)) + O( +� +γT T 1−α log(|D)T) + O(G1γT T 1−α� +log(|D|T)) + O( +� +γT T). +(94) +Finally, substituting G1 with δ1 = δ2 = δ/2, we have the total expected regret under the DP-DPBE with the +central model is +E[R(T)] = O(T 1−α/2� +log(|D|T)) + O +� +ln(1/δ)γT T 1−α� +log(kT) +ε +� ++ O( +� +γT T 1−α log(|D)T) + O( +� +γT T). +(95) +While the DPBE algorithm uses GP tools to define and manage the uncertainty in estimating the unknown +function f, the analysis of DPBE algorithm does not rely on any Bayesian assumption about f being actually +drawn from the prior GP(0, k), and it only requires f to be bounded in the kernel norm associated with the +RKHS Hk. +F +Additional Numerical Results +F.1 +Evaluation of DP-DPBE +In Section 7, we evaluated DP-DPBE on the synthetic function. In this subsection, we present additional +numerical results for DP-DPBE on the standard benchmark functions and the function from real-world +(light-sensor) data. By considering the same setting as for the synthetic function, we run T = 106 rounds and +present how the cumulative regret at the end of T varies with privacy budget ε ∈ {5, 10, 15, 20, 25, 30} and +δ = 10−6 in Figure 8. Then, by choosing privacy parameters δ = 10−6 and ε = 15, we also compare the +per-round regret of DP-DPBE and DPBE for the three benchmark functions and the real-world (light-sensor) +data and present the results in Figure 9. We perform 20 runs for each simulation. From these results, we make +similar observations to those for the synthetic function: the privacy-regret tradeoff and achieving privacy “for +free”. +F.2 +Comparison with State-of-the-Art +In Section 8, we provide simulation results on the regret performance and running time of GP-UCB, BPE, +and our algorithm DPBE with different values of α on the synthetic data generated in Section 7.1 In this +section, we add additional numerical results on three benchmark functions (Sphere, Six-hump Camel, +Michalewicz) and one function from real-world data– Light sensor data [Sch]. The parameters of the +47 + +(a) Sphere function +(b) Six-Hump Camel function +(c) Michalewicz function +(d) Function from light sensor data +Figure 8: Performance of DP-DPBE: Final cumulative regret vs. privacy budget ε with δ = 10−6. +Table 7: Comparison of running time (seconds) under GP-UCB, BPE, and DPBE with different values of α. +Algorithms +DPBE +GP-UCB +BPE +α = 0.4 +α = 0.5 +α = 0.6 +α = 0.7 +α = 0.8 +α = 0.9 +Sphere +0.08 +0.07 +0.07 +0.07 +0.09 +0.13 +4.68 +37.87 +Six-Hump Camel +0.04 +0.03 +0.04 +0.03 +0.04 +0.04 +4.79 +10.43 +Michalewicz +0.04 +0.04 +0.05 +0.06 +0.07 +0.11 +4.95 +4.48 +Light Sensor Data +0.04 +0.06 +0.07 +0.03 +0.06 +0.05 +3.22 +82.08 +problem setting and the algorithms are as follows: T = 4 × 104, |D| = 100, and k = kSE with lSE = 0.2; +(a) Sphere function. Settings: d = 3, C = 1.5, σ = 0.01, v2 = 0.001, λ = σ2/v2; (b) Six-Hump +Camel function. Settings: d = 2, C = 1.5, σ = 0.01, v2 = 0.01, λ = σ2/v2; (c) Michalewicz function. +Settings: d = 2, C = 1.5, σ = 0.01, v2 = 0.01, λ = σ2/v2; (d) Functions from real-world data. Settings: +d = 2, C = 1.42, σ = 0.01, v2 = 0.01, λ = σ2/v2. We plot the cumulative regret for all the algorithms in +Figure 10 and present the running time in Table 7. +48 + +×105 +1.5 +1.0 +0.5 +0.0 +5 +10 +15 +20 +25 +30 +Pivacy budget x105 +1.5 +1.0 +0.5 +0.0 +5 +10 +15 +20 +25 +30 +Pivacy budget x105 +1.5 +1.0 +0.5 +0.0 +5 +10 +15 +20 +25 +30 +Pivacy budget ×104 +2.5 +2.0 +1.5 +1.0 +0.5 +0.0 +5 +10 +15 +20 +25 +30 +Pivacy budget (a) Sphere function +(b) Six-Hump Camel function +(c) Michalewicz function +(d) Function from light sensor data +Figure 9: Performance of DP-DPBE: Per-round regret vs. time with parameters ε = 15 and δ = 10−6. +49 + +1.2 +DP-DPBE +1.0 +DPBE +egret +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rounds +×1061.0 +DP-DPBE +0.8 +DPBE +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rounds +×1061.0 +DP-DPBE +DPBE +0.8 +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rounds +×106DP-DPBE +0.8 +DPBE +0.6 +0.4 +0.2 +0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rounds +×106Algorithm 2 Differentially Private Distributed Phase-then-Batch-based Elimination (DP-DPBE) +1: Input: D ⊆ Rd, α ∈ (0, 1), β ∈ (0, 1), rare-switching parameter C, local noise σ2, privacy parameters +ε and δ +2: Initialization: l = 1, D1 = D, t1 = 0, and T1 = 1 +3: while tl < T do +4: +Set τ = 1, h = 0, τ1 = 0 and Σ2 +0(x) = k(x, x), for all x ∈ Dl +5: +while τ ⩽ Tl do +6: +h = h + 1 +7: +Choose +ah ∈ argmax +x∈Dl +Σ2 +h−1(x) +(67) +8: +Play action ah for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 +h−1(ah)⌋ times if not reaching min{T, tl + Tl} +9: +Update τ = τ + Tl(ah), and the posterior variance Σ2 +h(·) by including ah according to Eq. (7). +10: +end while +11: +Let Hl = h denote the total number of batches in this phase. +12: +Randomly select ⌈2αl⌉ participants Ul +# Operations at each participant +13: +for each participant u ∈ Ul do +14: +Collect and compute local average reward for every chosen action a ∈ AHl: +yu +l (a) = +1 +Tl(a) +� +t∈Tl(a) +yu,t +15: +Send the local average reward for every chosen action yu +l ≜ [yu +l (a)]a∈AHl to the agent +16: +end for +17: +Aggregate local observations for each chosen action a ∈ AHl: +yl(a) = +1 +|Ul| +� +u∈Ul +yu +l (a) +18: +Let ¯yl = [yl(a1), . . . , yl(aHl)] and +˜yl = ¯yl + (ρ1, . . . , ρHl), +where ρj +i.i.d. +∼ N(0, σ2 +nc) and σnc is specified in Eq. (66). +19: +Update ˜µl(·): +˜µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 +Hl )−1˜yl +(68) +20: +Eliminate low-rewarding actions from Dl based on the confidence width function ˜wl(·) in Eq. (17) +with σn = σnc +� +2C2γT : +Dl+1 = +� +x ∈ Dl : ˜µl(x) + ˜wl(x) ⩾ max +b∈Dl +(˜µl(b) − ˜wl(b)) +� +. +(69) +21: +Tl+1 = 2Tl, t = t + Tl; l = l + 1 +22: end while +50 + +Algorithm 3 M : Shuffle Protocol for a Set of Vectors with Users U [LZJ22] +1: Input: {yu}u∈U, where each yu ∈ Rs, ∥yu∥2 ⩽ ∆, privacy parameters ε, δ2 ∈ (0, 1) +2: Let +� +� +� +� +� +� +� +� +� +� +� +� +� +�ε = +ε +18√ +log(2/δ2) +g ≜ max{�ε +� +|U|/(6 +� +5 ln ((4s)/δ2)), √s, 10} +b ≜ ⌈ 180g2 ln (4s/δ2) +�ε2|U| +⌉ +p ≜ 90g2 ln (4s/δ2) +b�ε2|U| +(74) +// Local Randomizer +function R(yu) +3: +for coordinate j ∈ [s] do +4: +Shift data to enforce non-negativity: wu,j = (yu)j + ∆, ∀u ∈ U +//randomizer for each entry +5: +Set ¯wu,j ← ⌊wu,jg/(2∆)⌋ +//max |(yu)j + ∆| ⩽ 2∆ +6: +Sample rounding value γ1 ∼ Ber(wu,jg/(2∆) − ¯wu,j) +7: +Sample privacy noise value γ2 ∼ Bin(b, p) +8: +Let φu +j be a multi-set of (g + b) bits associated with the j-th coordinate of user u, where φu +j consists +of ¯wu,j + γ1 + γ2 copies of 1 and g + b − ( ¯wi,j + γ1 + γ2) copies of 0 +9: +end for +10: +Report {(j, φu +j )}j∈[s] to the shuffler +end function +// Shuffler +function S({(j, φj)}j∈[s]) +//φj = (φu +j )u∈U +11: +for each coordinate j ∈ [s] do +12: +Shuffle and output all (g + b)|U| bits in φj +13: +end for +end function +// Analyzer +function A(S({(j, φj)}j∈[s]) +14: +for coordinate j ∈ [s] do +15: +Compute zj ← +2∆ +g|U|((�(g+b)|U| +i=1 +(φj)i) − b|U|p) +// (φj)i denotes the i-th bit in φj +16: +Re-center: oj ← zj − ∆ +17: +end for +18: +Output the estimator of vector average o = (oj)j∈[s] +end function +51 + +(a) Sphere function +(b) Six-Hump Camel function +(c) Michalewicz function +(d) Function from light sensor data +Figure 10: Comparison of regret performance under DPBE, GP-UCB, and BPE on three benchmark functions +and one function from real-world dataset. The shaded area represents the standard deviation. +52 + +x103 +3.0 +DPBEα=0.4 +DPBEα=0.8 +2.5 +DPBE α =0.5 +中 +DPBEα=0.9 +4 +DPBEα=0.6 +BPE +2.0 +DPBEα=0.7 +GP-UCB +1.5 +1.0 +0.5 +0.0 +0 +1 +2 +3 +4 +Rounds +×104×103 +1.4 +DPBEα=0.4 +DPBEα=0.8 +DPBE α=0.5 +1.2 +DPBE α =0.9 +4 +DPBE α =0.6 +BPE +1.0 +DPBEα=0.7 +GP-UCB +gret +0.8 +0.6 +0.4 +0.2 +0.0 +0 +1 +2 +3 +4 +Rounds +×104×103 +5 +DPBEα=0.4 +DPBEα=0.8 +DPBEα=0.5 +中 +DPBEα=0.9 +4 +DPBEα=0.6 +BPE +4 +DPBE α = 0.7 +GP-UCB +2 +1 +0 +0 +1 +2 +3 +4 +Rounds +×104×103 +DPBEα=0.4 +DPBEα=0.8 +DPBEα=0.5 +DPBE α= 0.9 +1.5 +DPBEα=0.6 +BPE +DPBE α=0.7 +GP-UCB +Regr +1.0 +0.5 +0.0 +0 +L +2 +3 +4 +Rounds +×104 \ No newline at end of file diff --git a/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/load_file.txt b/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa78b0cb16c76d942a20edac3579b5fba0af61ab --- /dev/null +++ b/a9FLT4oBgHgl3EQfXi8l/content/tmp_files/load_file.txt @@ -0,0 +1,2003 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf,len=2002 +page_content='(Private) Kernelized Bandits with Distributed Biased Feedback Fengjiao Li* Xingyu Zhou† Bo Ji* Abstract In this paper, we study kernelized bandits with distributed biased feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This problem is motivated by several real-world applications (such as dynamic pricing, cellular network configuration, and policy making), where users from a large population contribute to the reward of the action chosen by a central entity, but it is difficult to collect feedback from all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Instead, only biased feedback (due to user heterogeneity) from a subset of users may be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition to such partial biased feedback, we are also faced with two practical challenges due to communication cost and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To tackle these challenges, we carefully design a new distributed phase-then-batch-based elimination (DPBE) algorithm, which samples users in phases for collecting feedback to reduce the bias and employs maximum variance reduction to select actions in batches within each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By properly choosing the phase length, the batch size, and the confidence width used for eliminating suboptimal actions, we show that DPBE achieves a sublinear regret of ˜O(T 1−α/2 + √γT T), where α ∈ (0, 1) is the user-sampling parameter one can tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, DPBE can significantly reduce both communication cost and computation complexity in distributed kernelized bandits, compared to some variants of the state-of-the-art algorithms (originally developed for standard kernelized bandits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Furthermore, by incorporating various differential privacy models (including the central, local, and shuffle models), we generalize DPBE to provide privacy guarantees for users participating in the distributed learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, we conduct extensive simulations to validate our theoretical results and evaluate the empirical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1 Introduction Bandit optimization is a popular online learning paradigm for sequential decision making and has been widely used in a wide variety of real-world applications, including hyperparameter tuning [Li17], recommendation systems [Li10], and dynamic pricing [MSA19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In such problems, each decision point (called an arm or action), if chosen, yields an unknown reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The goal of the agent is to maximize the cumulative reward by making proper decisions sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' An important way to capture general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', non-linear and even non-convex) unknown objective functions is to consider a smoothness condition specified by a small norm of a Reproducing Kernel Hilbert Space (RKHS) associated with a kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This setup is often referred to as kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Thanks to the strong link between RKHS functions and Gaussian processes (GP) [Kan18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CG17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Sri09], an extensive line of work has exploited GP models to estimate an unknown function f given a set of (noisy) evaluations of its values f(x) at chosen actions x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, in many applications, the value f(x) could represent an overall effect of action x on a large population of users where it is difficult for the learning agent to make direct observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' yet, the agent could collect some partial feedback from the distributed users in the Fengjiao Li (fengjiaoli@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='edu), Bo Ji (boji@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='edu), Department of Computer Science, Virginia Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' †Xingyu Zhou (xingyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='zhou@wayne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='edu), Department of Electrical and Computer Engineering, Wayne State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This work has been accepted by ACM SIGMETRICS’23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='12061v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='LG] 28 Jan 2023 Local Model: item 1 Decision Global Model: Pricing Customer $5 item 2 $9 item 3 $8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Expected total profits Local expense Influence (Private) Aggregation Local Feedback Figure 1: Dynamic pricing: a motivating application of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, feedback from these users could be biased due to user heterogeneity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', different preferences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, we assume that each user u in the population is associated with a local function fu, which is a function sampled from a GP with mean f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider the dynamic pricing problem [MSA19] as an example (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' When a company sets a pricing mechanism x, this decision influences all the customers, and every customer, based on her individual demand and preference, makes a choice (purchase or not), which contributes to the total profits f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Without knowing products’ demand curves in advance, the company makes a sequence of pricing decisions with the goal of maximizing profits while learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, the company aims to infer the expected demand and thus the expected profits f by collecting feedback from customers in each decision epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that it might be difficult for the company to collect feedback from all the customers - since purchases may take place at many local stores at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For example, it is impractical for McDonald’s headquarters to collect sales information from all of the nationwide customers within each decision epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Instead, the headquarter might be able to get feedback (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', sales information) from a subset of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, each customer’s choice depends not only on her own preference towards the products and their prices but also on several other factors (location, competitors, promotion events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ), which is often biased feedback for the overall profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To that end, we study a new kernelized bandit setting where the agent could not get direct evaluations of the unknown reward function but only distributed biased feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We refer to this setting as kernelized bandits with distributed biased feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This bandit problem is shared by several other practical applications, including cellular network configuration [Mah21] and public policy making [BRA20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, existing learning algorithms developed for standard kernelized/GP bandits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', GP-UCB [Sri09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CG17]) rarely consider such partial biased feedback in a distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To solve this new problem, a learning algorithm needs to be able to learn the unknown function from such biased feedback in a sample-efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, two practical challenges naturally arise in our problem: communication cost due to distributed learning [Che21a] and computation complexity due to GP update [Cal22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, not only need the learning algorithms be sample-efficient, but they must also be scalable in terms of both communication efficiency and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To that end, we propose the learning with communication framework where the biased feedback is communicated in phases, and design a new distributed phase-then-batch-based elimination algorithm that 2 aggregates the distributed biased feedback in a communication-efficient manner and eliminates suboptimal actions in a computation-efficient manner while achieving a sublinear regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Our main contributions are summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To the best of our knowledge, this is the first work that studies a new kernelized bandit setting with distributed biased feedback, where three key challenges (user heterogeneity, communication efficiency, and computation complexity) inherently arise in the design of sample-efficient, scalable learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While it is natural to consider phased elimination type of algorithms in such settings, the standard phased elimination algorithm relies on the so-called (near-)optimal experimental design [LSW20], which cannot be directly applied to kernelized bandits due to the possible infinite feature dimension of RKHS functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To that end, we design a new phased elimination algorithm, called distributed phase-then-batch- based elimination (DPBE), which is carefully crafted to address all the aforementioned challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In particular, DPBE adds a user-sampling process to reduce the impact of bias from each individual user and selects actions according to maximum variance reduction within each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, a batching strategy is employed to improve both communication efficiency and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, instead of selecting a new action at each round, DPBE plays the same action for a batch of rounds before switching to the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Not only does it help reduce the number of times one needs to compute the next action via GP update, but it also allows for reducing the dimensions of the vectors and matrices involved in both communication and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We show that DPBE achieves a sublinear regret of ˜O(T 1−α/2 + √γT T)1 while incurring a communica- tion cost of O(γT T α) and a computation complexity of O((|D|γ3 T + γ4 T ) log T + γT T α), where γT is the maximum information gain associated with the kernel of the unknown function f, D is the decision set, and α > 0 is a user-sampling parameter that we can tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It is worth noting that DPBE with α ∈ (0, 1) has a better computation complexity than some variants of the state-of-the-art algorithms (originally developed for standard kernelized bandits without biased feedback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, DPBE achieves three significant improvements compared to the state-of-the-art algorithms: (i) user-sampling efficiency (O(T α) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' T), (ii) communication cost (O(γT T α) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' T), and (iii) computation complexity (O(γT T α) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' O(T 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Furthermore, we conduct extensive simulations to validate our theoretical results and evaluate the empirical performance in terms of regret, communication cost, and running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, we generalize our phase-then-batch framework to incorporate various differential privacy (DP) models (including the central, local, and shuffle models) into DPBE, which ensures privacy guarantees for users participating in the distributed learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2 Related Work Kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Since [Sri09] studied GP in the bandit setting, kernelized bandits (also called GP bandits) have been widely adopted to address black-box function optimization over a large or infinite domain [CG17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Considering different application scenarios, kernelized bandits under different settings have recently been studied, including heavy-tailed payoffs [RG19], model misspecification [BK21], and corrupted rewards [Bog22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As typically considered in the literature, these works also assume that direct 1The notation ˜O(·) ignores polylog terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Bounds on γT of different kernel functions can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3 (noisy) feedback of the unknown function at a chosen action is available to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In sharp contrast, we study a new, practical setting where only distributed biased feedback can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under this setting, not only does one need to use biased feedback in a sample-efficient manner, but one also has to consider communication efficiency, which is a common issue in distributed bandit-learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Distributed/collaborative kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While distributed or collaborative kernelized bandits have been studied recently [Du21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' DLJ20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Sim21], we highlight the key difference between our model and theirs as follows: motivated by real-world applications, we aim to learn one (global) bandit while most of them also aim to learn every local model, which results in quite different regret definitions (their group regret vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' our standard regret defined in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, they assume that every party (corresponding to a user in our problem) shares the same objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While [DLJ20] also studies similar bandit optimization with biased feedback, they assume a fixed number of local agents and bound the regret in terms of the distance between the target function and local functions, which could be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, [DLJ20] does not consider communication efficiency, which is a key challenge in distributed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recently, the work of [LZJ22] studies a similar global reward maximization problem without direct feedback and also employs a phase-based elimination algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, the main difference is that they only consider linear bandits by assuming a linear reward function while we study kernelized bandits that can capture general non-linear and even non-convex functions and recover linear bandits as a special case when choosing a linear kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This strict generalization introduces three unique challenges: (i) different from the linearly parameterized bandits where the bias in the feedback can be quantified with a same-dimension random vector (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', ξu = θu − θ∗ ∈ Rd at each user u), it is unclear how to make an assumption of the bias in the non-parametric kernelized bandits setting in order to learn the unknown global reward function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (ii) due to the possible infinite feature dimension of functions in an RKHS, the (near-)optimal experimental design approach used in the phased-elimination algorithm for linear bandits cannot be directly adapted to kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Despite some recent efforts towards extending this experimental design based approach to kernelized bandits [Zhu21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CJK21], there still remain some key limitations (see our discussion below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (iii) since computation complexity is a critical bottleneck in kernelized bandits, a proper computation-efficient learning algorithm is desired when addressing our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Experimental design for kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In [Zhu21], the authors propose to adaptively embed the feature representation of each action into a lower-dimensional space in order to apply the (near-)optimal experimental design for finite-dimensional actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, the intermediate regret due to the approximation error over T rounds is not considered at all because their goal is to find an ε-optimal arm at the end of T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', a pure exploration problem) rather than minimizing the cumulative regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While [CJK21] aims at minimizing the cumulative regret, their algorithm and analysis are more complex than ours: it requires a non-standard robust estimator, obtaining an optimal distribution on the simplex, drawing samples from this distribution, and solving a second optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In contrast, we simply use the standard GP posterior mean and variance estimators, which can be computed in closed-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, our algorithm can also be easily extended to handle infinite action sets (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2) rather than a finite set considered in [CJK21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3 Preliminaries Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Throughout this paper, we use lower-case letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', x) for scalars, lower-case bold letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', x) for vectors, and upper-case bold letters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', X) for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let [n] ≜ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , n} denote any positive integer up to n, let |U| denote the cardinality of set U, and let ∥x∥2 denote the ℓ2-norm of vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Problem Setting We introduce a new kernelized bandit problem where the unknown function represents the overall reward over a large population containing an infinite number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The unknown reward function f : D → R is assumed to be fixed over a finite set of decisions D ⊆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' At round t, the agent chooses an action xt ∈ D, leading to a reward with mean f(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This reward is unknown to the agent but captures the overall effectiveness of action xt over the entire population U, thus called global reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Meanwhile, each user u in the population observes a (noisy) local reward: yu,t = fu(xt) + ηu,t with mean fu(xt), where ηu,t is the noise, and fu : D → R is the local reward function, assumed to be an (unknown) realization of a random function (specified soon) with mean f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this setting, the exact global reward corresponding to the entire population cannot be observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' only biased local reward feedback is available to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We make the following assumptions about the unknown function f, the local function fu, and the noise in the reward observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We assume that function f is in the Reproducing Kernel Hilbert Spaces (RKHS), denoted by Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that RKHS Hk is completely specified by its kernel function k(·, ·) (and vice-versa), with an inner product ⟨·, ·⟩k obeying the reproducing property: f(x) = ⟨f(·), k(x, ·)⟩k for all f ∈ Hk [CG17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We list the most commonly used kernel functions (such as Squared Exponential (SE) and Mat´ern kernels) in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, we assume that function f has a bounded norm: ∥f∥k ≜ � ⟨f, f⟩k ⩽ B, and that the kernel function is also bounded: k(x, x) ⩽ κ2 for every x ∈ D, where both B and κ are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' When the agent samples a user u to collect feedback, the local reward function fu at u is as- sumed to be a function sampled from the GP with mean f and covariance2 k(·, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', fu ∼ GP(f(·), k(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, we assume that each user is sampled independently for collecting feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We assume that the observation noise ηu,t∼N(0, σ2) is Gaussian with variance σ > 0 and that it is independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=') over time and across users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The goal of the agent is to maximize the cumulative global reward, or equivalently, to minimize the regret defined as follows: R(T) ≜ T � t=1 � max x∈D f(x) − f(xt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Learning with Communication For black-box function optimization based on noisy bandit feedback, kernelized bandit algorithms have shown strong empirical and theoretical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, the agent in our problem setting does not have access to unbiased feedback of the object function f but has to collect biased feedback from distributed users from a large population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This scenario leads to the following framework of learning with communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Communication happens when some users are selected to report their feedback to the agent based on their biased local reward observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By aggregating such biased feedback from the users, the agent improves her confidence in estimating function f and adjusts her decisions in the following rounds accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To account for scalability, the agent collects distributed feedback from users periodically instead of immediately after making each decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We call the time duration between two communications as a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider 2Our theoretical framework is applicable to a more general setting where the covariance of the local reward function is v2k(·, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', fu ∼ GP(f(·), v2k(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This scaling parameter v2 captures the variance of the bias in the local reward function fu with its mean being the global reward function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For this more general setting, our theoretical results still hold with only a slight adjustment to the posterior variance in the confidence width function (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 5 a particular phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let Tl be the set of round indices in the l-th phase and Ul be the set of selected users, called participants, that will report their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' With the actions {xt : t ∈ Tl} chosen by the agent in this phase, each user u in Ul sends the feedback g({yu,t}t∈Tl) to the agent at the end of the phase, where g(·) is a function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the average) of the local reward observations and is assumed to be the same for all users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, by aggregating all feedback {g({yu,t}t∈Tl)}u∈Ul, the agent estimates f and decides xt for round t in the next phase Tl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This learning with communication process is repeated until the end of T, with the goal of maximizing the cumulative (global) reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this framework, we assume that the agent can employ some existing incentive mechanisms [Lim20] in order to collect enough feedback for learning, but the cost has to be considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the communication resources consumed for collecting feedback data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, communication cost is also a critical factor in a general distributed learning system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this work, we use the total quantity of communicated numbers (between the agent and all users) as another metric, in addition to the regret metric, to evaluate the communication efficiency of learning algorithms for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let L be the total number of phases in T rounds and Nu,l ≜ dim (g({yu,t}t∈Tl)) be the dimension of user u’s feedback (which is the number of scalars in user u’s feedback).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the total communication cost, denoted by C(T), is as follows: C(T) ≜ L � l=1 � u∈Ul Nu,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (2) In the following, we explain the learning with GP framework for standard kernelized bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Learning with Gaussian Process A Gaussian process (GP) over input domain D, denoted by GP(µ(·), k(·, ·)), is a collection of random variables {f(x)}x∈D where every finite number of them {f(xi)}n i=1, n ∈ N, is jointly Gaussian with mean E[f(xi)] = µ(xi) and covariance E[(f(xi) − µ(xi))(f(xj) − µ(xj))] = k(xi, xj) for every 1 ⩽ i, j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, GP(µ(·), k(·, ·)) is specified by its mean function µ and a (bounded) covariance function k : D×D → [0, κ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Assume that choosing action xt at round t reveals a noisy observation: yt = f(xt) + ηt, (3) where ηt ∼ N(0, λ) is a zero-mean Gaussian noise with variance λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Standard GP algorithms implicitly use GP(0, k(·, ·)) as the prior distribution over f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, given the observations yt = [y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yt]⊤ corre- sponding to a sequence of actions Xt = [x⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , x⊤ t ]⊤, the posterior distribution is also Gaussian with the mean and variance in the following closed-form: µt(x) ≜ k(x, Xt)⊤(KXtXt + λI)−1yt, (4) σ2 t (x) ≜ k(x, x) − k(x, Xt)⊤(KXtXt + λI)−1k(x, Xt), (5) where k(x, Xt) = [k(x, xs)]⊤ s=1,··· ,t ∈ Rt×1 and KXtXt = [k(x, x′)]x,x′∈Xt ∈ Rt×t is the corresponding kernel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Next, we introduce an important kernel-dependent quantity called maximum information gain [Sri09]: γt(k, D) ≜ max X⊆D:|X|=t 1 2 log det � I + λ−1KXX � , (6) which is often used to derive regret bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, we have that γt(k, D) scales sublinearly with t for most commonly used kernels (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For ease of notation, we often simply use γt to denote γt(k, D) when the kernel function k and the dataset D are clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 6 Thanks to the strong connection between RKHS functions and GP [Kan18] with the same kernel function k, one can use the above GP model to approximate unknown function f ∈ Hk within a reliable confidence interval with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 4 Algorithm Design 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 New Challenges and Main Ideas In Section 3, we describe the learning with communication framework, which requires the distributed biased feedback to be communicated in phases and exhibits experimental scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This framework naturally leads us to consider a phased elimination algorithm that gradually eliminates suboptimal actions by periodically aggregating and analyzing the local feedback from the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, several new challenges arise in our setting compared to the standard phase elimination algorithm in linear bandits [LSW20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' LS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (i) How to select actions for each phase?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The standard phase elimination algorithm often relies on the so-called near-optimal experimental design (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', a probability distribution over the currently active set) that minimizes the worst-case variance [LS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, due to the possible infinite feature dimension of RKHS functions, adapting this approach to kernelized bandits setting is nontrivial even with the strong assumptions, requirements, and complicated algorithm design (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', [Zhu21] and [CJK21], see discussion in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We are wondering if there is a simple and efficient method of selecting actions in each phase for our kernelized bandits setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (Challenge a⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (ii) How to use biased feedback?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In contrast to the standard phase elimination algorithm where feedback is unbiased, in our setting the local feedback from a particular user is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In order to reduce the impact of bias, an efficient user-sampling scheme is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, how to incorporate this idea into the phase elimination algorithm is unclear (Challenge b⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (iii) How to deal with scalability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In our setting, scalability refers to both computation complexity and communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' On the one hand, it is well-known that standard GP bandits suffer a poor computation complexity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', O(T 3)) due to the matrix inverse at each step for GP posterior update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' On the other hand, due to the communication between the agent and the users, it is imperative to ensure a low communication cost (Challenge c⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We propose a novel phase elimination algorithm that is able to simultaneously address all the above challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We highlight the main ideas as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (i) User-sampling for distributed biased feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In each phase, a well-tuned subset of users is sampled to reduce the impact of bias from each individual user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (ii) Maximum variance reduction for action selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Upon selecting the next action within each phase, it simply selects the one that has the largest posterior variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (iii) Batching strategy for scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Instead of selecting a new action at each round within a phase, it consistently plays the same action for a batch of rounds before selecting the next one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', rare-switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By reducing the number of times selecting a new action (which could be much smaller than the phase length), it also reduces the number of unique actions chosen within each phase, which can be utilized to improve the scalability in terms of both computation and communication through a proper design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, (a) Computation: via a posterior reformulation (specified in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2), we convert the dimension of the matrix in the inverse operation from the total rounds to the number of batches in each phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (b) Communication: we let each participant merge the local reward observations in each batch before sending her feedback at the end of each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, the feedback g({yu,t}t∈Tl) from each participating user u in phase l is a vector, where each element corresponds to the average local reward of a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the dimension of the feedback g({yu,t}t∈Tl becomes the number of batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For example, consider a particular phase with a total of 10 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Without 7 Figure 2: The phase-then-batch strategy: T rounds are divided into L phases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' at the end of each phase, participants report their feedback, which is used for deciding actions in the next phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' within each phase l, decisions are made in a batched fashion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', playing ah at all the rounds in the h-th batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' batching strategy, one requires to select an action for each round, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', 10 actions for this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, the batching strategy selects an action for each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' If each batch has size two, there are 5 batches in this phase, and the dimension of the matrix in the inverse operation is shrunk from 10 to 5, which will reduce the computation complexity about 103/53 = 8 times for matrix inverse operations!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, by merging local observations of each unique action, only 5, instead of 10, (averaged) local rewards are communicated at each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Distributed Phase-then-Batch-based Elimination (DPBE) Following the main ideas stated in the above section, we propose the phase-then-batch schedule strategy, shown in Figure 2 and design the distributed phase-then-batch-based elimination (DPBE) algorithm in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The DPBE algorithm is a phased elimination algorithm, which maintains a set Dl of active actions that are possible to be optimal and updates the active set after aggregating the distributed feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider a particular phase l, DPBE has three main steps: 1) action selection (Lines 5-10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2) distributed feedback collection (Lines 12-16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and 3) action elimination (Lines 17-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Before describing the details of DPBE, we explain some additional notations used in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Throughout this paper, we use another notation “a” to denote the specific chosen action under our algorithm to avoid too many subscripts or superscripts for all the batch, phase, or round indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let tl and Tl be the time index right before the l-th phase and the length of the l-th phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the round indices in the l-th phase can be represented as Tl = {t ∈ [T] : tl + 1 ⩽ t ⩽ tl + Tl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, Tl(a) ≜ {t ∈ Tl : xt = a} denotes the time indices when action a is selected in this phase, and Hl represents the number of batches in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1) Action selection (Lines 5-10): In the l-th phase, actions are selected from the active set Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As mentioned before, each selection is based on maximum variance reduction [Vak21], and we employ batch schedule for scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, in the h-th batch, we find the action ah that maximizes a reformulated posterior variance Σh−1(·) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7) after h − 1 batches (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This is possible because the posterior variance can be computed without knowing any reward observations (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, play this action for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 h−1(ah)⌋ rounds, which forms the h-th batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Here, the batch size schedule is inspired by the rare-switching idea in [APS11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Cal22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This batch schedule strategy enables us to 8 a1 aHi ahmerge rounds and thus shrink the dimensions of the matrix and vectors used for computing the variance in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By the end of each batch, we update the variance function by incorporating the action in the current batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let Ah = [a⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , a⊤ h ]⊤ ∈ Rh×d be the h × d matrix that contains the h chosen actions so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We reformulate the standard posterior variance in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5) and update the posterior variance as follows: Σ2 h(x) ≜ k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 h )−1k(x, Ah), (7) where Wh ∈ Rh×h is a diagonal matrix with [Wh]ii = Tl(ai) for any i ∈ [h], and λ is set to be the noise variance of local observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', λ = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Here, we reformulate the standard posterior variance in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7) in order to save computation complexity (especially for computing matrix inverse) while maintaining the same order of regret (sacrificing only a constant multiplier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2) Distributed feedback collection (Lines 12-16): To reduce the impact of bias from some specific user(s), the agent randomly samples a subset of users (called participants) Ul from U to participate in the learning process (Line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We let |Ul| = ⌈2αl⌉, where the user-sampling parameter α > 0 is an input of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that Hl denotes the number of batches in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Each participant u ∈ Ul collects their local reward observations of each chosen action a ∈ AHl and send the average yu l (a) for every chosen action a ∈ AHl as feedback to the agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', g({yu,t}t∈Tl) = yu l ≜ [yu l (a)]a∈AHl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that the dimension of the feedback depends on the number of batches, which is also the communication cost associated with each participant (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, by employing the idea of rare switching, we reduce both computation complexity and communication cost ( c⃝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3) Action elimination (Lines 17-21): Aggregate (specifically, average) the feedback from the participants for each action a ∈ AHl (Line 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, using the aggregated feedback (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the averaged local reward ¯yl = [yl(a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yl(aHl)] of the chosen actions a ∈ AHl), the agent can compute the posterior mean function reformulated as follows (Line 19): ¯µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1¯yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11) Considering the bias in the feedback due to user heterogeneity ( b⃝), we carefully construct a confidence width wl(·) that incorporates both the noise and bias as follows: wl(x) ≜ � 2k(x, x) log(1/β) |Ul| + � 2Σ2 Hl(x) log(1/β) |Ul| + BΣHl(x), (12) where B is the bound of f’s kernel norm, and β is the confidence level from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Using this confidence width wl(·) and the mean estimator function ¯µl(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11), we can identify suboptimal actions with high probability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, we update the set of active actions Dl+1 by eliminating the suboptimal actions from Dl (Line 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 (Merge batches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For implementation, we also merge different batches with the same chosen action in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By doing this, we further shrink the dimension of the matrix in the inverse operation (thus reducing the time complexity) and also the dimension of local feedback (thus reducing the communication cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 (General decision set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Following the techniques used in [LS22], DPBE can also be extended from a finite domain to a continuous domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', D = [0, 1]d) via a simple discretization trick and Lipschitz continuity of functions under commonly used kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 9 Algorithm 1 Distributed Phase-then-Batch-based Elimination (DPBE) 1: Input: D ⊆ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' parameters α > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' β ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and local noise σ2 2: Initialization: l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' D1 = D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' t1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and T1 = 1 3: while tl < T do 4: Set τ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' h = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' τ1 = 0 and Σ2 0(x) = k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' for all x ∈ Dl 5: while τ ⩽ Tl do 6: h = h + 1 7: Choose action ah ∈ argmax x∈Dl Σ2 h−1(x) (8) 8: Play action ah for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 h−1(ah)⌋ rounds if not reaching min{T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' tl + Tl} 9: Update τ = τ + Tl(ah),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and incorporate ah into the posterior variance Σ2 h(·) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7)) 10: end while 11: Let Hl = h denote the total number of batches in this phase 12: Randomly select ⌈2αl⌉ participants Ul # Operations at each participant 13: for each participant u ∈ Ul do 14: Collect and compute local average reward for every chosen action a ∈ AHl: yu l (a) = 1 Tl(a) � t∈Tl(a) yu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='t 15: Send the (local) average reward for each chosen action yu l ≜ [yu l (a)]a∈AHl to the agent 16: end for 17: Aggregate local observations for each chosen action a ∈ AHl: yl(a) = 1 |Ul| � u∈Ul yu l (a) (9) 18: Let ¯yl = [yl(a1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yl(aHl)] 19: Update ¯µl(·) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11) 20: Eliminate low-rewarding actions from Dl based on the confidence width wl(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (12): Dl+1 = � x ∈ Dl : ¯µl(x) + wl(x) ⩾ max b∈Dl (¯µl(b) − wl(b)) � (10) 21: Tl+1 = 2Tl, t = t + Tl, l = l + 1 22: end while 5 Main Results In this section, we present the performance of our proposed DPBE algorithm in terms of regret, computation complexity, and communication cost, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 10 First, we analyze the regret performance of DPBE and present the upper bound in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While the DPBE algorithm uses GP tools to define and manage the uncertainty in estimating the unknown function f, the analysis of DPBE algorithm does not rely on any Bayesian assumption about f being drawn from the prior GP(0, k(·, ·)), and it only requires f to be bounded in the kernel norm associated with the RKHS Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 (Regret).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let β = 1 |D|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2 and 3, the DPBE algorithm achieves the following expected regret: E[R(T)] = O(T 1−α/2� log(|D|T)) + O( � γT T) + O( � γT T 1−α log(|D|T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (13) We provide the detailed proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Bounds for γT of different kernels can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we make two remarks about the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the above regret upper bound (RHS of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (13)), the first term, O(T 1−α/2� log(|D|T)), is due to the bias in the feedback at heterogeneous participants, and the last two terms, O(√γT T) + O( � γT T 1−α log(|D|T)), are from the noisy feedback of each action as in the standard kernelized bandits (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Sri09]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that the first term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the regret caused by the bias) can be improved if one increases the number of sampled users in the learning process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', choosing a larger value of α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, this would also result in a larger communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 (Maximum Uncertainty Reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that DPBE selects actions that have maximum variance for each batch (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Intuitively, variance at action x indicates the uncertainty about f(x), and thus, maximum-variance selection leads to maximum uncertainty reduction, which promotes exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 ((Sub-)optimality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We first note that one natural lower bound for our setting is the one for the standard setting of kernelized bandits, where the agent receives unbiased feedback after taking an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this setting, the state-of-the-art lower bounds under two commonly-used kernel functions (SE and Mat´ern)3 are summarized in Table 5 (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2), which can also serve as valid lower bounds for the setting we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that α > 0 is the user-sampling parameter that one can choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We discuss the (sub-)optimality of our upper bounds in two cases: α ≥ 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the high-communication regime) and α ∈ (0, 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the low-communication regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (i) In the high-communication regime, the upper bound in (13) now becomes O(√γT T), which is near-optimal under both SE and Mat´ern kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In particular, if one plugs the best-known bounds on γT for SE and Mat´ern kernels (as listed in the first column in Table 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' also see [VKP21]) into the regret upper bound O(√γT T), one can now have explicit regret upper bounds (as listed in the third column in Table 5), which match the corresponding lower bounds, up to only a logarithmic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (ii) In the low-communication regime, the first term in the regret upper bound (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (13) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1) that depends on α may be dominant and cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' On the other hand, the existing lower bounds do not depend on α since they are derived under the standard setting of kernelized bandits, where user sampling is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, an important open problem is to close the gap by deriving tighter lower and/or upper bounds that capture the effect of user sampling in the new setting with distributed biased feedback we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We leave it as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As a critical bottleneck of kernelized bandits algorithms, the computation complexity of DPBE algorithm is analyzed in the following Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3Note that even for the standard setting of kernelized bandits, there only exist lower bounds for these specific kernel functions rather than a general one in terms of the maximum information gain γT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 11 Table 1: Comparison of computation complexity under DPBE and three state-of-the-art algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Algorithms Complexity GP-UCB [CG17] O(|D|T 3) BBKB [Cal20] O(|D|Tγ2 T ) MINI-GP-Opt [Cal22] O(T + |D|γ3 T + γ4 T ) DPBE (this paper) O(γT T α + (|D|γ3 T + γ4 T ) log T) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 (Computation complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The computation complexity of DPBE is at most O(γT T α+(|D|γ3 T + γ4 T ) log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that Hl is the number of batches in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the computation complexity of the central agent in the l-th phase is upper bounded by the following: O(Hl · H3 l + Hl · |Dl|H2 l + |Ul|Hl + |Dl|H2 l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, for each h ∈ [Hl] within phase l, the agent would compute the matrix inverse in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7), the complexity of which is at most O(h3) ≤ O(H3 l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' With this matrix inverse result ready, the agent can solve the maximum-variance problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (8) with at most O(|Dl|H2 l ) for each batch and determine the batch length Tl(ah) with O(1) after we have the posterior variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Since there is a total of Hl batches for phase l, the total complexity up to this stage is O(Hl · H3 l + Hl · |Dl|H2 l ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, in the elimination stage for phase l, the agent first loads/aggregates all the feedbacks with O(|Ul|Hl) and can again reuse the matrix inverse result so that only O(|Dl|H2 l ) is required to eliminate all the bad arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Putting the two stages together, we have the above result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Thus, it remains to bound the number of batches Hl within each phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Fortunately, inspired by [Cal22], we are able to show that Hl can be upper bounded by the maximum information gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We state this result in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 and provide the proof in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 (Bound on Hl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any phase l, the number of batches Hl is at most 4σ2C2 C2−1 γT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We can get that the total number of phases is O(log T) and the total number of participants satisfies O(T α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Armed with all the above results, we arrive at our final computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 (Complexity comparison).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For comparison, we list the computation complexity of the state- of-the-art algorithms for standard kernelized bandits in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As we already know, GP-UCB has a computation complexity of O(|D|T 3), because it requires computing the posterior mean and variance using O(T 2) and then finds the action that maximizes the UCB function per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recently, BBKB in [Cal20] improves the time complexity to (|D|Tγ2 T ), and later MINI-GP-Opt in [Cal22] further reduces computation complexity to O(T +|D|γ3 T +γ4 T ), which is currently the fastest no-regret algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Although more feedback is needed to address the additional bias in our setting, our algorithm can still achieve an improvement with the highest order term being O(γT T α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This improvement comes from the fact that the participants help preprocess local reward observations before sending them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Meanwhile, the bound on Hl also allows us to achieve a meaningful communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 (Communication cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' DPBE incurs at most O(γT T α) communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 is also provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 12 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 (Communication cost when merging batches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By further merging batches according to Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, the DPBE algorithm incurs O(min{γT , |D|}T α) communication cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We highlight that the batch schedule strategy plays a key role in obtaining the above bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Otherwise, even merging rounds as Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 with the reformulated representation in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7) and (11), the dimension of the local feedback at each participant is O(min{Tl, |Dl|}) in order to distinguish different actions, which leads to O(min{T, |D|}T α) (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ours O(min{γT , |D|}T α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 6 Differentially Private DPBE As privacy is also an important factor in distributed learning, it is critical to protect users’ sensitive data when collecting and aggregating their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For example, in the dynamic pricing application, it is required that an adversary cannot infer a customer’s private information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', purchase or not) by observing the pricing mechanism set by the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Moreover, users may require more stringent privacy protection in some applications — users are not willing to share their perceived Quality-of-Experience (QoE) directly with the central controller in the cellular network configuration problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' citizens are not willing to reveal the information about their preference for a certain policy to the government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Formally, we adopt the concept of differential privacy (DP) [DR14] as the privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Thanks to the phase-then-batch schedule strategy in our algorithm, different DP trust models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', central [DR14], local [ZT20], and shuffle [Che19]) can be applied through proper designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this section, we describe how to ensure DP under DPBE with a trusted agent (the central DP model) and also analyze the regret under such a DP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Extensions of the differentially private DPBE algorithms in other DP models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the stronger local DP model) are presented in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 DP Definition and Algorithm In the central DP model, we assume that each participating user trusts the agent, and hence, the agent can collect their raw data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', the local reward yu l in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The privacy guarantee is that any adversary with arbitrary auxiliary information cannot infer a particular user’s data by observing the decisions of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To achieve this privacy protection, the central DP model requires that the decisions of the agent on two neighboring sets of users (differing in only one user) are indistinguishable [Dwo06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Formally, we have the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (Differential Privacy (DP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any ε ⩾ 0 and δ ∈ [0, 1], a randomized algorithm M is (ε, δ)-differentially private (or (ε, δ)-DP) if for every pair of U, U′ ⊆ U differing on a single participant and for any subset of output actions Z = [z⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , z⊤ T ]⊤, we have P[M(U) = Z] ⩽ eεP[M(U) = Z] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (14) The parameters ε and δ indicate how private M is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' the smaller, the more private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to the post-processing property of DP (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 in [DR14]), it suffices to guarantee that the aggregator (Line 17 in Algorithm 1) is (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To achieve this, the standard Gaussian mechanism can be applied by adding Gaussian noise to the aggregated distributed feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the private aggregated feedback for the chosen actions in the l-th phase becomes ˜yl = ¯yl + (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρHl), (15) 13 where ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N(0, σ2 nc) and the variance σ2 nc (specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (66)) is based on the (high-probability) sensitivity of of the average vector ¯yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, we replace ¯yl with ˜yl in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (15) to obtain the private mean estimator: ˜µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1˜yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (16) The confidence width function is also updated by counting the uncertainty introduced by privacy noise as follows: ˜wl(x) ≜ � 2k(x, x) log(1/β) |Ul| + � 2Σ2 Hl(x) log(1/β) |Ul| + BΣHl(x) + � 2σ2n log(1/β), (17) where σn is related to the overall privacy noise and will be specified in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We present the differentially private version of DPBE, called DP-DPBE, in Algorithm 2 (see Appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Performance Guarantees In the following, we provide the main results of the DP-DPBE algorithm in terms of privacy guarantee and regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We start by stating an additional assumption in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This one-time participation assumption is commonly used in private bandits (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', [MT15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' SS19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Ten21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Dub21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CZ22b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CZ22a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To handle multiple-times participation, one can use (adaptive) composition theorem of differential privacy or group privacy [DR14], depending on different cases of returning users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Each sampled user only participates in one phase of the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we present the privacy guarantee in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 and provide the proof in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 (Privacy Guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2, 3, and 4, for any ε > 0 and δ ∈ (0, 1), the DP-DPBE algorithm (Algorithm 2) guarantees (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As an additional Gaussian noise is injected to protect privacy, DP-DPBE suffers additional regret cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We present its regret upper bound in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 (Regret of DP-DPBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2, and 3, the DP-DPBE algorithm (Algorithm 2) with β = 1 |D|T achieves the following expected regret: E[R(T)] = O(T 1−α/2� log(|D|T)) + O � ln(1/δ)γT T 1−α� log(|D|T) ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (18) The full proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 is provided in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Regarding this regret result, we make the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 (Privacy “for free”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Comparing Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we see that the additional regret cost introduced by privacy noise is ˜O � ln(1/δ)γT T 1−α ε � , which is a lower order term compared to the first non-private term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This implies that our DP-DPBE algorithm enables us to achieve a privacy guarantee “for free” in the kernelized bandits setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The same observation of achieving privacy “for free” is also observed in a recent study [LZJ22] that only considers linear bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, our result is a strict generalization in the sense that it holds for general functions and recovers their result when considering a linear kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 4More specifically, if the same user only participates across multiple phases, one can use advanced composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' if the same user participates multiple times in the same phase, one can carefully bound the sensitivity or use group privacy directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 14 Figure 3: Comparison of regret perfor- mance on a synthetic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The shaded area represents the standard deviation Figure 4: The regret and communication cost under DPBE with different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7 Numerical Experiments We now evaluate our proposed approach empirically on three types of functions: 1) synthetic functions in the RKHS with an SE kernel, 2) standard benchmark functions (with an unknown RKHS norm) [SB] , and 3) functions from a real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We implement the algorithms in python and run the numerical experiments on a Dell desktop (Processor: Intel®Core i7 CPU, 8 cores;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Memory: 32GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Synthetic Function We follow [JBG20] to construct the global function f from the RKHS by sampling m = 30d inde- pendent points, �x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , �xm, uniformly on [0, 1]d, and �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , �am, uniformly on [−1, 1], and defining f(x) = �m i=1 �aik(�xi, x) for all x ∈ D, where k is SE kernel with length-scale lSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The RKHS norm is ∥f∥2 k = �m i=1 �m j=1 �ai�ajk(�xi, �xj), which is assumed to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Each local reward function fu, a random function sampled from a given Gaussian process, is generated by following Algorithm 1 in [Kan18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the simulations, we evaluate the algorithms in a more general setting with fu ∼ GP(f(·), v2k(·, ·)), where v2 is a scaling parameter that can be used to set a reasonable level of local bias (see Footnote 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Ablation Studies and Analysis First, we show that the DPBE algorithm that selects actions according to maximum variance reduction achieves sublinear regret, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we perform numerous ablation studies to confirm the efficacy of other two key components in our algorithm: user-sampling and batching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To this end, we consider the corresponding variants of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this simulation, we perform 20 runs for each algorithm by setting |D| = 100, d = 3, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, T = 40000, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, β = 1/(|D|T) and λ = σ2/v2 and present the regret performance in Figure 3 and communication cost and runtime in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1) Importance of (exponentially-increasing) user-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To this end, we consider the first variation of DPBE with a fixed number of participants, called DPBE-Fixed, where the number of participants in each phase is fixed at |U| = ⌊ �L l=1 |Ul|∗Nu,l �L l=1 Nu,l ⌋ so as to have the same communication cost as DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From Figure 3, we observe that DPBE with exponentially-increasing user-sampling over phases performs much better than 15 ×103 DPBE-Fixed 3 DPBE DPBE-NoBatching 7 0 0 1 2 3 4 Rounds ×104×104 ×104 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Cumulative regret Communication cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 αTable 2: Comparisons of communication cost and running time under DPBE, DPBE-Fixed, and DPBE-NoBatching on a synthetic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Algorithms Communication cost Running time (seconds) DPBE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='87 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='12 DPBE-Fixed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='87 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='19 DPBE-NoBatching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='81 × 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='61 DPBE-Fixed with the same communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It demonstrates that the exponentially-increasing user- sampling mechanism in DPBE is critical to striking a balance between regret and communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From Table 2, we observe that DPBE-Fixed takes a little longer time than DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This is mainly because DPBE-Fixed needs more phases to find the optimal action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', L is larger when |DL| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2) Benefits of batching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To illustrate the impact of batching schedule strategy, we consider another variant of DPBE that does not employ batching strategy, called DPBE-NoBatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In par- ticular, it selects an action according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (8) for each round in any phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Without batching strategy, DPBE-NoBatching communicates local observations directly without merging, and computes the posterior mean and variance according to standard update formula: Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (5) respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From Figure 3, we observe that DPBE, similar to other rare-switching algorithms [APS11], achieves a slightly worse regret performance than DPBE-NoBatching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, as shown in Table 2, it significantly saves communication cost (∼ 3×) by merging local observations in batches and reduces computation time (∼ 5×) by shrinking the dimension of posterior reformulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Regret-communication Tradeoff We now turn to investigate the regret-communication tradeoff captured by the user-sampling parameter α, as shown in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The cumulative regret and total communication cost of DPBE with different values of α are presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As expected, while a larger value of α yields a lower regret, it generally results in a higher communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Notice that DPBE incurs slightly higher communication cost when α = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6} compared to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, this is mainly because DPBE with a smaller value of α needs more phases to find the optimal action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', L is larger when |DL| = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' One can tune the user-sampling parameter α to achieve a better regret-communication cost accordingly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 for this synthetic function setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Regret-privacy Tradeoff Finally, we evaluate the performance of the differentially private DPBE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', DP-DPBE, and present the result in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Figure 5(a) shows how the cumulative regret at the end of T = 106 rounds varies with different values of privacy parameters ε ∈ {5, 10, 15, 20, 25, 30} and δ = o(1/T) = 10−6, which reveals a tradeoff between regret and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Figures 5(b) shows the regret performance of DPBE and DP-DPBE with privacy parameters ε = 15 and δ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We observe that although DP-DPBE adds extra noise to protect privacy, it can still achieve no-regret (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', limT→∞ R(T) T → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Indeed, to protect privacy, DP-DPBE requires much more time to find the optimal action, which is the typical regret-privacy tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, for a large T, the gap compared to the non-private one is small, which also validates the privacy “for-free” result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 16 (a) (b) Figure 5: Performance of DP-DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (a) Final cumulative regret vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' the privacy budget ε with δ = 10−6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (b) Per-round regret vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' time with parameters ε = 15 and δ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Standard Benchmark Functions In addition, we study the performance of DPBE on standard optimization benchmark functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This corresponds to a more realistic setting where the RKHS norm of the target function is unknown in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In particular, we use three common functions in global optimization problems [SB]: (a) Sphere function, (b) Six-hump Camel function, and (c) Michalewicz function, and provide the performance comparison of DPBE-Fixed, DPBE, and DPBE-NoBatching in Figure 6 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the simulations, we scale the range of the function values to [−1, 1] and use RKHS norm B = 1 in the algorithms as in [JBG20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Without knowing the exact kernel of the target function, each local reward function fu is constructed by sampling a function from the GP GP(f(·), v2k(·, ·)), where we choose v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='001 and use the SE kernel with lSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, we set T = 4 × 104 and |D| = 100 and run each algorithm on each function for 20 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From Figure 6 and Table 3, we observe similar results to those of the synthetic function with the same kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' First, compared to DPBE-Fixed that incurs the same communication cost, DPBE might perform slightly worse at the very beginning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', Figure 6(a)) but eventually achieves a much smaller regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that DPBE-Fixed may not be able to find the optimal action by the end of T (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', Figure 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This phenomenon strengthens our argument on the exponentially-increasing user-sampling mechanism in DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While DPBE-NoBatching has slightly better regret performance than DPBE, it incurs much higher communication cost (5 ∼ 13×) and requires a much longer time (6 ∼ 23×, see running time column in Table 3), which demonstrates the key benefits of the batching strategy in improving communication efficiency and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, we also evaluate the regret-privacy tradeoff under DP-DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Due to space limitations, we present the numerical results in Appendix F (see Figures 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Functions from Real-World Data We also evaluate the performance of DPBE on a function from a real-world dataset, where there is no explicit closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Light Sensor Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We use the light sensor data collected from the CMU Intelligent Workplace in November 2005, which is available online [Sch].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It contains locations of 41 sensors, 601 training samples, 17 x105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 T 5 10 15 20 25 30 Pivacy budget 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 DP-DPBE DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Rounds ×106(a) Sphere function (b) Six-Hump Camel function (c) Michalewicz function (d) Light sensor data Figure 6: Comparison of regret performance under DPBE, DPBE-Fixed, and DPBE-NoBatching on four functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (a) Sphere function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 3, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (b) Six-Hump Camel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (c) Michalewicz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, λ = σ2/v2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (d) Function from light sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='42, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and 192 testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Following [Sri09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CG17;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ZJ22], we compute the empirical covariance matrix of the training samples and use it as the kernel matrix in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Here, for each location x, we let f(x) be the average of the normalized sample readings at x and set B = maxx f(x) in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For this function (from real data), we construct each local function fu by sampling a function from a Gaussian process with mean f and the kernel constructed above, and set the noise in the local feedback as σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01 and the bias in each local feedback as v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We run DPBE with input parameters α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, β = 1/(|D|T), and λ = σ2/v2, and present the regret performance in Figure 6(d) and communication cost and running time in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The observations are qualitatively similar to those made in simulations on other functions: DPBE outperforms DPBE-Fixed in regret given the same communication cost and achieves a regret close to DPBE-NoBatching, which has much longer running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Besides, we also run DP-DPBE on this real-world dataset and present the results in Appendix F (see Figures 8(d) and 9(d)), which validates the regret-privacy tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 18 ×103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBE-Fixed DPBE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBE-NoBatching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0 1 2 3 4 Rounds ×104×103 DPBE-Fixed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBE DPBE-NoBatching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Cumulative!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0 1 2 3 4 Rounds ×104x103 5 DPBE-Fixed DPBE 4 DPBE-NoBatching m 1 0 0 2 3 4 Rounds ×104×103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBE-Fixed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBE ret DPBE-NoBatching 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Re Cumulative 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0 1 2 3 4 Rounds ×104Table 3: Communication cost and running time under DPBE, DPBE-Fixed, and DPBE-NoBatching Function Algorithm Communication cost Running time (seconds) Sphere DPBE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='49 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 DPBE-Fixed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='49 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='12 DPBE-NoBatching 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='16 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='69 Six-Hump Camel DPBE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='26 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='03 DPBE-Fixed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='26 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='12 DPBE-NoBatching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='45 × 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='17 Michalewicz DPBE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 DPBE-Fixed 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='14 DPBE-NoBatching 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='73 × 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='49 Light Sensor Data DPBE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='17 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='22 DPBE-Fixed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='17 × 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='28 DPBE-NoBatching 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='73 × 104 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='20 8 Comparison with the State-of-the-Arts 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Discussion We now consider an alternative way of addressing kernelized bandits with distributed biased feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' One may incorporate the local bias as another level of noise added to the noise in the rewards as a new noisy measurement of the global function f with a larger variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this case, the state-of-the-art algorithms for the traditional kernelized bandits [Sri09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' CG17] may be adapted to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, they have some key limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider two representative state-of-the-art algorithms: GP-UCB [CG17] and BPE [LS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' GP-UCB is one of the most commonly used algorithms for standard kernelized bandits, It was proposed in [Sri09] and improved in [CG17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By resorting to the Gaussian process surrogate model (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3), GP-UCB adaptively selects the action with the maximal upper confidence bound in each round based on historical observations up to the current round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' BPE is a batch-based algorithm that eliminates suboptimal actions batch by batch, and within each batch, actions are chosen independently from reward observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we compare our proposed DPBE algorithm with GP-UCB and BPE (adapted to our setting) and show their limitations in user-sampling, communication cost, and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' First, both GP-UCB and BPE require to collect feedback from one user per step, which results in T users involved in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In practice, even though there is a large population, not all users are willing to send their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, it may not be feasible to collect feedback from too many users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In our algorithm, instead of sampling more users to reduce the overall uncertainty, we ask each sampled user (who is more willing to participate) to participate in more rounds and send their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this way, we alleviate the user-sampling burden by letting the participating users collect more reward samples of the chosen actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, due to the bias in the feedback of each user, we could not just sample one user and then let her report the feedback during the entire horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We need to balance the tradeoff between sampling more users and letting the users participate in more rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Second, by collecting feedback in each round, both GP-UCB and BPE incur a very high communication cost of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Instead, we employ a phase-based communication protocol where feedback corresponding to any particular action at each participant is averaged and only communicated at the end of each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the 19 total communication cost depends on the number of phases, the number of distinct actions in each phase, and the number of sampled users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The smaller each of these three factors, the smaller the communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By carefully designing the algorithm, we can reduce the communication cost to O(min{γT , |D|}T α), where α ∈ (0, 1) is the user-sampling parameter one can choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, at each round t, GP-UCB finds the decision action xt that maximizes an acquisition function (specifically, the UCB index, which is the sum of the posterior mean and variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that obtaining the posterior mean and variance requires computing matrix inverse (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (4) and (5)), which still has a com- putation complexity of O(t2) even using rank-one recursive updates [CG17, Appendix 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, the overall computation complexity of GP-UCB is O(|D|T 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Similarly, BPE may also compute the posterior variance using the rank-one recursive update within each batch, and then the total computation complexity depends on the batch size and the number of batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' As in [LS22], the batch size is updated as Ni = � T√Ni−1, initialized with N0 = 1, which results in ⌈log log(T)⌉ batches in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, the computation complexity of BPE is O(|D|T 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In our design, we employ the batch schedule strategy and reformulate the posterior mean and variance as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11) and (7), where the dimension of the matrix becomes much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This leads to a much smaller overall computation complexity of O(γT T α + (|D|γ3 T + γ4 T ) log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Empirical Performance In this subsection, we evaluate the empirical performance of DPBE with different values of user-sampling parameter α compared to GP-UCB and BPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The simulations are run on the same three types of functions as in the preceding section: the synthetic function in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, the standard benchmark functions in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2, and the function from light sensor data in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Due to space limitation, we only present the results of the synthetic function here and put the results of the latter two types of functions in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider5 α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9} for DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We show the empirical regret performance of all algorithms in Figure 7 and the running time in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From Figure 7, we observe that the empirical regret performance of DPBE can be fairly close to or even better than that of GP-UCB and BPE via properly choosing parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' However, it consumes much less time for DPBE with each α ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9} than both GP-UCB and BPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For example, while DPBE takes about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='15 second in most scenarios, GP-UCB takes more than 5 seconds, which is more than 30 times slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' BPE takes around 27 seconds, which is even slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall the empirical communication cost of DPBE with different values of α shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While the communication cost of GP-UCB and BPE is 4 × 104 (specifically, one feedback per round), DPBE incurs a much smaller communication cost even when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 (4 × 104 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='19 × 104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In summary, the comparison of empirical performance under DPBE with GP-UCB and BPE demonstrates the significant improvements of DPBE in terms of communication cost and computation complexity, although little regret performance is sacrificed when α is not big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 9 Conclusion In this paper, we studied a new kernelized bandit problem with distributed biased feedback, where the feedback of the unknown objective function is biased due to user heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To learn and optimize the unknown function using distributed biased feedback, we proposed the learning with communication 5Note that the smaller the value of α, the larger the cumulative regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In Figure 7, we omit the regret performance when α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 since they are much larger than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 20 Table 4: Comparison of running time (seconds) under GP-UCB, BPE, and DPBE with different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Algorithms DPBE GP-UCB BPE α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 Running time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='32 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='49 Figure 7: Regret performance comparison of GP-UCB, BPE, and DPBE with different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Considering the communication cost for collecting feedback and the computational bottleneck of kernelized bandits, we carefully designed the distributed phase-then-batch-based elimination (DPBE) algorithm to address all the new challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, DPBE selects actions according to maximum variance reduction, reduces bias via user-sampling, and improves communication efficiency and computation complexity via the batching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Furthermore, we showed that DPBE achieves a sublinear regret while being scalable in terms of communication efficiency and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, we generalized DPBE to incorporate various differential privacy models to ensure privacy guarantees for participating users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While we proposed a new DPBE algorithm to address the new challenges that arise in our problem setup, it would be worthwhile to explore other batch-based algorithms and investigate whether one can further improve the tradeoff among regret, communication efficiency, and computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In addition, as discussed in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4, the lower bound derived for the standard kernelized bandits is also a valid lower bound for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We show that our algorithm, if sampling a sufficient number of users, can achieve this lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In general, however, it is an important open problem to close the gap by deriving tighter lower and/or upper bounds that capture the effect of user sampling in our new setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We leave it as our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 10 ACKNOWLEDGMENTS We thank our shepherd, Giulia Fanti, and the anonymous paper reviewers for their insightful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We also thank Duo Cheng for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This work is supported in part by the NSF grants under CNS-2112694 and CNS-2153220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 21 ×103 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 5 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 BPE 4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 GP-UCB Regret 3 2 1 0 0 1 2 3 4 Rounds ×104References [APS11] Yasin Abbasi-Yadkori, D´avid P´al, and Csaba Szepesv´ari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “Improved algorithms for linear stochas- tic bandits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: Advances in neural information processing systems 24 (2011).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Sch] Carnegie Mellon University School of Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Light sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Retrieved October 05, 2022, from http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='edu/˜guestrin/Class/10708- F08/ projects/lightsensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='zip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Sim21] Rachael Hwee Ling Sim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “Collaborative Bayesian optimization with fair regret”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: Inter- national Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 9691–9701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Sri09] Niranjan Srinivas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “Gaussian process optimization in the bandit setting: No regret and experimental design”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: arXiv preprint arXiv:0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3995 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [SS19] Touqir Sajed and Or Sheffet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “An optimal private stochastic-mab algorithm based on optimal pri- vate stopping rule”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 5579– 5588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Ten21] Jay Tenenbaum et al.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: Advances in Neural Information Processing Systems 34 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [VKP21] Sattar Vakili, Kia Khezeli, and Victor Picheny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “On information gain and regret bounds in gaussian process bandits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 82–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [Zhu21] Yinglun Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “Pure exploration in kernel and neural bandits”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: Advances in Neural Information Processing Systems 34 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 11618–11630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [ZJ22] Xingyu Zhou and Bo Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “On Kernelized Multi-Armed Bandits with Constraints”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='15589 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' [ZT20] Xingyu Zhou and Jian Tan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' “Local differential privacy for bayesian optimization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In: arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06709 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 24 Table 5: Bounds on γT and Regret under Two Common Kernels [VKP21] Kernel Upper Bound on γT Regret Lower Bound Regret Upper Bound O(√γT T) SE O � logd+1(T) � Ω �� T log d 2 (T) � O �� T logd+1(T) � Mat´ern-ν O � T d 2ν+d log 2ν 2ν+d (T) � Ω � T ν+d 2ν+d � O � T ν+d 2ν+d log ν 2ν+d (T) � A Kernelized Bandits: Useful Definitions and Useful Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Example Kernel Functions In the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' we list some commonly used kernel functions k : D × D → R: Linear kernel: klin(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' x′) = x⊤x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Squared exponential kernel: kSE(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' x′) = exp � − ∥x−x′∥ 2l2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Mat´ern kernel: kMat(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' x′) = 21−ν Γ(ν) � √ 2ν∥x−x′∥ l � Jν � √ 2ν∥x−x′∥ l � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' where l denotes the length-scale hyperparameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ν > 0 is an additional hyperparameter that dictates the smoothness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and Jν and Γν denote the modified Bessel function and the Gamma function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' respectively [RW06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Maximum Information Gain for Different Kernels We present the bounds on γT and regret under two common kernels below in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Useful Results Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 (Sum of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Lemma 6 in [RG19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let Xt = [x⊤ 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , x⊤ t ]⊤, and σ2 t (x) ≜ k(x, x) − k(x, Xt)⊤(KXtXt + λI)−1k(x, Xt) for any x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we have t � s=1 σ2 s(xs) ⩽ λ ln |λ−1KXtXt + I| ⩽ 2λγt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (19) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 (Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 in [Cal22]/Lemma 4 in [Cal20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any kernel k, set of points Xτ, x ∈ D, and τ ′ < τ, we have 1 ⩽ σ2 τ ′(x) σ2τ(x) ⩽ 1 + τ � s=τ ′+1 σ2 τ ′(xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (20) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 Formulation in Feature Space For several of the proofs, it will be useful to introduce the so-called feature space (RKHS) formulation of any point in the primal space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In particular, we define a feature map ϕ(x) = k(x, ·) where ϕ : D → Hk with Hk being the reproducing kernel Hilbert space (RKHS) associated with kernel function k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to the properties of RKHS, we have the following observations: 25 For any x, x′, k(x, x′) = ϕ(x)⊤ϕ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any function f ∈ H, f(x) = ⟨f, ϕ(x)⟩ = ϕ(x)⊤f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Fundamental linear algebra equality (BB⊤ + λI)−1B = B(B⊤B + λI)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (21) Define Φh ≜ [ϕ(a1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ϕ(ah)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the kernel matrix KAhAh = ΦhΦ⊤ h and k(x, Ah) = Φhϕ(x), and the variance function Σ2 h(·) represented in the feature space is the following: Σ2 h(x) = k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 h )−1k(x, Ah) = ϕ(x)⊤ϕ(x) − ϕ(x)⊤Φ⊤ h (ΦhΦ⊤ h + λW−1 h )−1Φhϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (22) Consider any phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that Hl is the number of batches in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define ΦHl ≜ [ϕ(a1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ϕ(aHl)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the kernel matrix KAHlAHl = ΦHlΦ⊤ Hl and k(x, AHl) = ΦHlϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define Φτ ≜ [ϕ(xtl+1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ϕ(xtl+τ)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the kernel matrix KXτXτ = ΦτΦ⊤ τ , k(x, Xτ) = Φτϕ(x), and the variance function σ2 τ(·) represented in the feature space is the following: σ2 τ(x) = k(x, x) − k(x, Xτ)⊤(KXτXτ + λI)−1k(x, Xτ) = ϕ(x)⊤ϕ(x) − ϕ(x)⊤Φ⊤ τ (ΦτΦ⊤ τ + λI)−1Φτϕ(x) = ϕ(x)⊤ϕ(x) − ϕ(x)⊤(Φ⊤ τ Φτ + λI)−1Φ⊤ τ Φτϕ(x) = ϕ(x)⊤(Φ⊤ τ Φτ + λI)−1(Φ⊤ τ Φτ + λI)ϕ(x) − ϕ(x)⊤(Φ⊤ τ Φτ + λI)−1Φ⊤ τ Φτϕ(x) = λϕ(x)⊤(Φ⊤ τ Φτ + λI)−1ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (23) Define ΦTl ≜ [ϕ(xtl+1)⊤, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ϕ(xtl+Tl)⊤]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the kernel matrix KXTlXTl = ΦTlΦ⊤ Tl, k(x, XTl) = ΦTlϕ(x), and f(XTl) = ΦTlf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' B Auxiliary Results and Proofs for Regret Analysis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Equivalent Representations Consider any phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We use τ to denote the within-phase time index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', τ ∈ {1, · · · , Tl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define τh as the last within-phase time index of the h-th batch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', τh ≜ max{τ : tl + τ ∈ Tl(ah)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, after playing τh actions, the posterior variance in the traditional GP model is the following: σ2 τh(x) = k(x, x) − k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (24) For the posterior mean, without the observations yTl = [ytl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ytl+Tl]⊤ corresponding to the actions XTl = [x⊤ tl+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , x⊤ tl+Tl]⊤, we replace yTl with 1 |Ul| � u∈Ul yl,u where yl,u = [yu,tl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yu,tl+Tl]⊤ in the traditional GP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the posterior mean becomes the following: µTl(x) = 1 |Ul| � u∈Ul k(x, XTl)⊤(KXTlXTl + λI)−1yl,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (25) In our algorithm, in order to save computation complexity and communication cost, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11) instead of the above formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following lemma, we show that they are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 26 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 (Equivalent representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider any phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By the end of the h-th phase, the posterior variance Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (24) in the traditional GP model is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7) used in our DPBE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, for any x ∈ D, we have σ2 τh(x) = Σ2 h(x), ∀h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , Hl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (26) Moreover, we have the two representations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (25) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11)) for the posterior mean function are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, for any x ∈ D, we have µTl(x) = ¯µl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' First, we have the following result, which helps connect the two representations of mean and variance functions: Φ⊤ τhΦτh = tl+τh � t=tl+1 ϕ(xt)ϕ(xt)⊤ (a) = h � i=1 Tl(ai)ϕ(ai)ϕ(ai)⊤ = Φ⊤ h WhΦh, (28) where (a) is due to our algorithm decisions: xt = ai for any t ∈ Tl(ai) = {tl + τi−1 + 1, tl + τi−1 + Tl(ai)} and the last step holds because Wh is a diagonal matrix with (Wh)ii = Tl(ai) for any i ∈ [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we are ready to derive the equivalence of two representations of the mean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1) Variance representation equivalence: σ2 τh(x) = Σ2 h(x) for h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , Hl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' This implies k(x, x) − k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh) = k(x, x) − k(x, Ah)⊤(KAhAh + λW−1 h )−1k(x, Ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It remains to show the following: k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh) = k(x, Ah)⊤(KAhAh + λW−1 h )−1k(x, Ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (29) Using the feature space formulations, we have k(x, Ah)⊤(KAhAh + λW−1 h )−1k(x, Ah) =ϕ(x)⊤Φ⊤ h (ΦhΦ⊤ h + λW−1 h )−1Φhϕ(x) =ϕ(x)⊤Φ⊤ h W1/2 h (W1/2 h ΦhΦ⊤ h W1/2 h + λI)−1W1/2 h Φhϕ(x) =ϕ(x)⊤(Φ⊤ h W1/2 h W1/2 h Φh + λI)−1Φ⊤ h W1/2 h W1/2 h Φhϕ(x) =ϕ(x)⊤(Φ⊤ h WhΦh + λI)−1Φ⊤ h WhΦhϕ(x) (a) =ϕ(x)⊤(Φ⊤ τhΦτh + λI)−1Φ⊤ τhΦτhϕ(x) =ϕ(x)⊤Φ⊤ τh(ΦτhΦ⊤ τh + λI)−1Φτhϕ(x) =k(x, Xτh)⊤(KXτhXτh + λI)−1k(x, Xτh), (30) where (a) is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we have σ2 τh(x) = Σ2 h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 27 2) Mean representation equivalence: µTl(x) = ¯µl(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', 1 |Ul| � u∈Ul k(x, XTl)⊤(KXTlXTl + λI)−1yl,u = k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1¯yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (31) For the last within-phase index τHl = Tl, we also have the following: 1 |Ul| � u∈Ul Φ⊤ Tlyl,u = 1 |Ul| � u∈Ul tl+Tl � t=tl+1 yu,tϕ(xt) = 1 |Ul| � u∈Ul Hl � h=1 � t∈Tl(ah) yu,tϕ(xt) = 1 |Ul| � u∈Ul Hl � h=1 ϕ(ah) � t∈Tl(ah) yu,t = 1 |Ul| � u∈Ul Hl � h=1 ϕ(ah)Tl(ah)yu l (ah) = Hl � h=1 Tl(ah)yl(ah)ϕ(ah) = Φ⊤ HlWHl ¯yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (32) Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' we are ready to derive the equivalence of two representations of the mean function: 1 |Ul| � u∈Ul k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' XTl)⊤(KXTlXTl + λI)−1yl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='u = 1 |Ul| � u∈Ul ϕ(x)⊤Φ⊤ Tl(ΦTlΦ⊤ Tl + λI)−1yl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='u = 1 |Ul| � u∈Ul ϕ(x)⊤(Φ⊤ TlΦTl + λI)−1Φ⊤ Tlyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='u = ϕ(x)⊤(Φ⊤ TlΦTl + λI)−1 · 1 |Ul| � u∈Ul Φ⊤ Tlyl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='u (a) = ϕ(x)⊤(Φ⊤ HlWHlΦHl + λI)−1Φ⊤ HlWHl ¯yl = ϕ(x)⊤(Φ⊤ HlW1/2 Hl W1/2 Hl ΦHl + λI)−1Φ⊤ HlW1/2 Hl W1/2 Hl ¯yl = ϕ(x)⊤((W1/2 Hl ΦHl)⊤(W1/2 Hl ΦHl) + λI)−1(W1/2 Hl ΦHl)⊤W1/2 Hl ¯yl = ϕ(x)⊤Φ⊤ HlW1/2 Hl (W1/2 Hl ΦHlΦ⊤ HlW1/2 Hl + λI)−1W1/2 Hl ¯yl = ϕ(x)⊤Φ⊤ Hl(ΦHlΦ⊤ Hl + λW−1 Hl )−1¯yl = k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' AHl)⊤(KAHlAHl + λW−1 Hl )−1¯yl = ¯µl(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (33) where (a) is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (28) with τHl = Tl and the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 28 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Impact of Batch Schedule Strategy on Posterior Variance In our batch schedule strategy, the decision xt does not change for Tl(ah) rounds when starting choosing ah after τh−1 rounds within the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 to our setting with τ ′ = τh−1, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider any phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that τh−1 is the within-phase time index before starting choosing ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, give any set of chosen actions Ah−1 for the first h − 1 batches, for any kernel k, any x ∈ D, and any τ ∈ [τh−1 + 1, τh−1 + Tl(ah)], we have 1 ⩽ Σh−1(x) στ(x) ⩽ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 to our setting, we have 1 ⩽ σ2 τh−1(x) σ2τ(x) ⩽ 1 + Tl(ah)σ2 τh(ah).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (35) Moreover, by selecting Tl(ah) = ⌊(C2 − 1)/Σ2 h−1(ah)⌋ = ⌊(C2 − 1)/σ2 τh−1(ah)⌋ (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1) in our algorithm, we derive the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' One key step to getting the regret upper bound is to bound the confidence width, which is related to the maximal value of the posterior variance by the end of each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we provide a bound for the maximal value of the posterior variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The posterior variance after Hl batches (decisions) in the l-th phase satisfies max x∈Dl ΣHl(x) ⩽ � 2σ2C2γTl Tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (36) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that DPBE plays action ah when τ ∈ [τh−1 + 1, τh−1 + Tl(ah)] within the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' First, we have for any x ∈ Dl, any h ⩽ Hl, ΣHl(x) (a) ⩽ Σh−1(x) (b) ⩽ Σh−1(ah) = στh−1(ah), (37) where (a) holds because Σh(·) is non-increasing in h, (b) is based on our decision, and the last step is due to 29 the equivalent representation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' we have the following: max x∈Dl ΣHl(x) ⩽ 1 Tl Hl � h=1 Tl(ah)Σh−1(ah) = 1 Tl Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 Σh−1(ah) = 1 Tl Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 Σh−1(ah) στ(ah) στ(ah) (a) ⩽ 1 Tl Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 Cστ(ah) (b) = C Tl Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 στ(xtl+τ) = C Tl Tl � τ=1 στ(xtl+τ) (c) ⩽ C Tl � � � �Tl Tl � τ=1 σ2τ(xtl+τ) (d) ⩽ C Tl � Tl · 2λγTl = � 2λC2γTl Tl ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (38) where the inequality (a) is from Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2, (b) is based on our algorithm decision: xtl+τ = ah for any τ ∈ [τh−1 + 1, τh−1 + Tl(ah)], (c) is by Cauchy-Schwartz inequality, and (d) is from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Other Useful Results Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Consider any particular phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the traditional GP models, without noise in the reward observations, the difference between the ground truth and regression estimator satisfies ���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) ��� ⩽ BσTl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (39) 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Representing f(x) in the feature space, we have ���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) ��� = ���ϕ(x)⊤f − ϕ(x)⊤Φ⊤ Tl(ΦTlΦ⊤ Tl + λI)−1ΦTlf ��� = ���ϕ(x)⊤f − ϕ(x)⊤(Φ⊤ TlΦTl + λI)−1Φ⊤ TlΦTlf ��� = ���λϕ(x)⊤(Φ⊤ TlΦTl + λI)−1f ��� ⩽ ∥f∥k∥λ(Φ⊤ TlΦTl + λI)−1ϕ(x)∥k ⩽ B � λϕ(x)⊤(Φ⊤ TlΦTl + λI)−1λI(Φ⊤ TlΦTl + λI)−1ϕ(x) ⩽ B � λϕ(x)⊤(Φ⊤ TlΦTl + λI)−1(Φ⊤ TlΦTl + λI)(Φ⊤ TlΦTl + λI)−1ϕ(x) ⩽ B � λϕ(x)⊤(Φ⊤ TlΦTl + λI)−1ϕ(x) = BσTl(x), (40) where the last step is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' C Proofs of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Before proving Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we first provide the key concentration inequality under DPBE in Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any particular phase l, with probability at least 1 − 4β, the following holds |f(x) − ¯µl(x)| ⩽ wl(x), (41) where mean function ¯µl(x) and confidence width function wl(x) are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this proof, we will show the following concentration inequality holds for any x ∈ D P[|f(x) − ¯µl(x)| ⩾ wl(x)] ⩽ 4β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (42) For any x ∈ D, we let wl(x) = wl,1(x) + wl,2(x), where wl,1(x) ≜ � 2k(x, x) log(1/β) |Ul| and wl,2(x) ≜ ΣHl(x) �� 2 log(1/β) |Ul| + B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' First, for any x ∈ D, we have the following inequality: |f(x) − ¯µl(x)| ⩽ ������ f(x) − 1 |Ul| � u∈Ul fu(x) ������ + ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 31 Then, we have P [|f(x) − ¯µl(x)| ⩾ wl(x)] ⩽P � � ������ f(x) − 1 |Ul| � u∈Ul fu(x) ������ + ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,1(x) + wl,2(x) � � ⩽P � � ������ f(x) − 1 |Ul| � u∈Ul fu(x) ������ ⩾ wl,1(x) � � + P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) � � , (43) where the last inequality is from union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we try to bound the above two terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' i) Recall that each user u is associated with a local reward function fu ∼ GP(f(·), k(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, fu(x) ∼ N(f(x), k(x, x)), ∀x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (44) Note that Ul is a set of ⌈2αl⌉ independently sampled random users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we have 1 |Ul| � u∈Ul fu(x) ∼ N � f(x), k(x, x) |Ul| � , ∀x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Combining the concentration inequality for Gaussian random variables, we have P � � ������ 1 |Ul| � u∈Ul fu(x) − f(x) ������ ⩾ wl,1(x) � � ⩽ 2 exp � − |Ul|w2 l,1(x) 2k(x, x)) � = 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (45) ii) Then, we want to bound the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (43): P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) � � = � Λ P[Λ = {yl,u}u∈Ul] · P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) ������ {yl,u}u∈Ul � � , where yl,u = [yu,tl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yu,tl+Tl]⊤ denotes the realization of the local reward observations at user u in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to our assumption, the participant user u is associated with a local reward function fu sampled from Gaussian Process GP(f(·), k(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Given the points XTl = [x⊤ tl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , x⊤ tl+Tl]⊤ in D, the corresponding vector of local rewards yl,u = [yu,tl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yu,tl+Tl]⊤ has the multivariate Gaussian distribution N(f(XTl), (KXTlXTl + λI)) where f(XTl) = [f(xtl+1), · · · , f(xtl+Tl)]⊤ and KXTlXTl = [k(x, x′)]x,x′∈XTl is the kernel matrix for the Tl selected actions in the l-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Due to the properties of GPs, we have that yl,u and fu(x) are jointly Gaussian given XTl: � fu(x) yl,u � ∼ N �� f(x) f(XTl) � , � k(x, x) k(x, XTl)⊤ k(x, XTl) KXTlXTl + λI �� , (46) 32 where k(x, XTl) = [k(x, xtl+1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , k(x, xtl+Tl)]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to the basic formula for conditional distri- butions of Gaussian random vectors (see [Ras03, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2] or [Kan18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2]), we have that conditioned on yl,u (corresponding to the points XTl), the following holds: fu(x)|yl,u ∼ N(mu(x), σ2 Tl(x)), where we have mu(x) ≜ f(x) + k(x, XTl)⊤(KXTlXTl + λI)−1(yl,u − f(XTl)), (47) σ2 Tl(x) = k(x, x) − k(x, XTl)⊤(KXTlXTl + λI)−1k(x, XTl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (48) Note that we sample the participants Ul independently and that the local reward noise is also independent across participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we have the following result: � � 1 |Ul| � u∈Ul fu(x) � � ��� {yl,u}u∈Ul = 1 |Ul| � u∈Ul (fu(x) | yl,u) ∼ N � � 1 |Ul| � u∈Ul mu(x), σ2 Tl(x) |Ul| � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Combining the Gaussian concentration inequality, we have the following result P � � ������ 1 |Ul| � u∈Ul fu(x) − 1 |Ul| � u∈Ul mu(x) ������ ⩾ � 2σ2 Tl(x) log(1/β) |Ul| ������ {yl,u}u∈Ul � � ⩽ 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (49) From Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we have the following equation: 1 |Ul| � u∈Ul k(x, XTl)⊤(KXTlXTl + λI)−1yl,u = k(x, Ah)⊤(KAhAh + λW−1 h )−1¯yl = ¯µl(x), (50) which implies 1 |Ul| � u∈Ul mu(x) = ¯µl(x) + f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (51) Then, the gap between the average local function 1 |Ul| � u∈Ul fu(·) and the estimator ¯µl(·) satisfies ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩽ ������ 1 |Ul| � u∈Ul fu(x) − 1 |Ul| � u∈Ul mu(x) ������ + ���f(x) − k(x, XTl)⊤(KXTlXTl + λI)−1f(XTl) ��� (a) ⩽ ������ 1 |Ul| � u∈Ul fu(x) − 1 |Ul| � u∈Ul mu(x) ������ + BσTl(x), (52) 33 where (a) is from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Combining the result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (49), we have P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) ������ {yl,u}u∈Ul � � ⩽P � � ������ 1 |Ul| � u∈Ul fu(x) − 1 |Ul| � u∈Ul mu(x) ������ + BσTl(x0) ⩾ wl,2(x) ������ {yl,u}u∈Ul � � (a) =P � � ������ 1 |Ul| � u∈Ul fu(x) − 1 |Ul| � u∈Ul mu(x) ������ ⩾ � 2σ2 Tl(x) log(1/β) |Ul| ������ {yl,u}u∈Ul � � ⩽ 2β, (53) where (a) is from σ2 Tl(x) = Σ2 Hl(x) according to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Therefore, we derive the desired result: P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) � � = � Λ P[Λ = {yl,u}u∈Ul] · P � � ������ 1 |Ul| � u∈Ul fu(x) − ¯µl(x) ������ ⩾ wl,2(x) ������ {yl,u}u∈Ul � � ⩽ � Λ P[Λ = {yl,u}u∈Ul] · 2β = 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (54) To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we first present three main conclusions when the concentration inequality in Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 holds, then get an upper bound for the regret incurred in a particular phase l with high probability, and finally sum up the regret over all phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define a “good” event when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (41) holds in the l-th phase as: El ≜ {∀x ∈ Dl, |f(x) − ¯µl(x)| ⩽ wl(x)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We have P[El] ⩾ 1−4|D|β via the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, under event El in the l-th phase, we have the following three observations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any optimal action x∗ ∈ argmaxx∈D f(x), if x∗ ∈ Dl, then x∗ ∈ Dl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let f∗ = maxx∈D f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Supposed that x∗ ∈ Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any x ∈ Dl+1, its reward gap from the optimal reward is bounded by 4 maxx∈Dl wl(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', f∗ − f(x) ⩽ 4 max x∈Dl wl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The confidence width function satisfies max x∈Dl wl(x) ⩽ � 2κ2 log(1/β) |Ul| + � 4σ2C2γTl log(1/β) Tl|Ul| + � 2σ2B2C2γTl Tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 34 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Observation 1: Let b ∈ argmaxx∈Dl(¯µl(x) − wl(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then under event El, we have ¯µl(x∗) + wl(x∗) ⩾ f(x∗) ⩾ f(b) ⩾ ¯µl(b) − wl(b), (55) which indicates x∗ ∈ Dl+1 according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Observation 2: For any x ∈ Dl+1, we have x ∈ Dl and ¯µl(x) + wl(x) ⩾ ¯µl(b) − wl(b) ⩾ ¯µl(x∗) − wl(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (56) Then, we have the regret of choosing any action x ∈ Dl+1 satisfying f(x∗) − f(x) (a) ⩽ ¯µl(x∗) + wl(x∗) − ¯µl(x) + wl(x) (b) ⩽ 2(wl(x) + wl(x∗)) ⩽ 4 max x∈Dl wl(x), (57) where (a) holds under event El and the second inequality (b) is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, we derive Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Observation 3: Based on the result in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3, we have max x∈Dl wl(x) = max x∈Dl �� 2k(x, x) log(1/β) |Ul| + ΣHl(x) �� 2 log(1/β) |Ul| + B �� ⩽ � 2κ2 log(1/β) |Ul| + max x∈Dl ΣHl(x) �� 2 log(1/β) |Ul| + B � ⩽ � 2κ2 log(1/β) |Ul| + � 4λC2γTl log(1/β) Tl|Ul| + � 2λB2C2γTl Tl = � 2κ2 log(1/β) |Ul| + � 4σ2C2γTl log(1/β) Tl|Ul| + � 2σ2B2C2γTl Tl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (58) Then, we are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let the regret in the l-th phase be rl ≜ � t∈Tl(maxx∈D f(x)−f(xt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any l ⩾ 2, 35 we assume event El−1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' we have the following result rl = � t∈Tl (max x∈D f(x) − f(x)) ⩽ � t∈Tl 4 max x∈Dl−1 wl−1(x) ⩽ 4Tl max x∈Dl−1 wl−1(x) ⩽ 4Tl � � � 2κ2 log(1/β) |Ul−1| + � 4σ2C2γTl−1 log(1/β) Tl−1|Ul−1| + � 2σ2B2C2γTl−1 Tl−1 � � (a) ⩽ 4 · 2l−1 �� 2κ2 log(1/β) 2α(l−1) + � 4σ2C2γT log(1/β) 2(1+α)(l−1)−1 + � 2σ2B2C2γT 2l−2 � ⩽ 4 � 2κ2 log(1/β) � 2(2−α)(l−1) + 8σC � 2γT log(1/β) � 2(1−α)(l−1) + 8σBC � γT 2l−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (59) where (a) is from γTl−1 ⩽ γT and |Ul| ⩾ 2αl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define Eg as the event where the “good” event occurs in every phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', Eg ≜ �L l=1 El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It is not difficult to obtain P[Eg] ⩾ 1 − 4|D|βL by applying union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' At the same time, let Rg be the regret under event Eg, and Rb be the regret if event Eg does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the expected total regret in T is E[R(T)] = P[Eg]Rg + (1 − P[Eg])Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under event Eg, the regret in the l-th phase rl satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (59) for any l ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that r1 ⩽ 2T1Bκ ⩽ 2Bκ since T1 = 1 and for any x ∈ D, |f(x)| = |⟨f, k(x, ·)⟩k| ⩽ ∥f∥k⟨k(x, ·), k(x, ·)⟩1/2 k ⩽ Bk(x, x)1/2 ⩽ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' we have Rg = L � l=1 rl ⩽ 2Bκ + L � l=2 4 � 2κ2 log(1/β) � 2(2−α)(l−1) + L � l=2 8σC � 2γT log(1/β) � 2(1−α)(l−1) + L � l=2 8σBC � γT 2l−1 ⩽ 2Bκ + 4 � 2κ2 log(1/β) · 4 � 2(L−1)(2−α) + 8σC � 2γT log(1/β) · C1 � 2(1−α)(L−1) � C1 = √ 21−α/( √ 21−α − 1) � + 8σBC√γT · 4 √ 2L−1 ⩽ 2Bκ + 16 � 2κ2 log(1/β)T 1−α/2 + 8σC1C � 2γT log(1/β)T 1−α + 32σBC � γT T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (60) where the last step is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 l=1 Tl + 1 ⩽ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 36 On the other hand, Rb ⩽ 2BκT since | maxx∈D f(x) − f(x)| ⩽ 2Bκ for all x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Choose β = 1/(|D|T) in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Finally, we have the following results: E[R(T)] = P[Eg]Rg + (1 − P[Eg])Rb ⩽ Rg + 4|D|βL · 2BκT ⩽ 2Bκ + 16 � 2κ2 log(1/β)T 1−α/2 + 8σC1C � 2γT log(1/β)T 1−α + 32σBC � γT T + 8Bκ|D|βLT = 2Bκ + 16T 1−α/2� 2κ2 log(|D|T) + 8σC1C � 2γT T 1−α log(|D|T) + 32σBC � γT T + 8Bκ log(2T) = O(T 1−α/2� log(|D|T)) + O( � γT T 1−α log(|D)T) + O( � γT T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (61) D Proofs for Communication and Computation Results The results regarding computation complexity and communication cost highly depend on the number of batches Hl in each phase l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, we first provide the proof for Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To bound the number of batches in the l-th phase, we follow a similar line to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 in [Cal22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any 1 ⩽ h ⩽ Hl, we have Tl(ah) = � C2 − 1 Σ2 h−1(ah) � ⩾ C2 − 1 Σ2 h−1(ah) − 1 ⇒ Σ2 h−1(ah)(Tl(ah) + 1) ⩾ C2 − 1 ⇒ 2Σ2 h−1(ah)Tl(ah) ⩾ C2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (62) Recall that we use τh to denote the last within-phase time index in the h-th batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, summing the above 37 inequality across all batches up to Hl, we have Hl(C2 − 1) ⩽ Hl � h=1 2Σ2 h−1(ah)Tl(ah) ⩽ 2 Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 Σ2 h−1(ah) = 2 Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 Σ2 h−1(ah) σ2τ(ah) σ2 τ(ah) (a) ⩽ 2 Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 C2 · σ2 τ(ah) (b) = 2C2 Hl � h=1 τh−1+Tl(ah) � τ=τh−1+1 σ2 τ(xtl+τ) = 2C2 Tl � τ=1 σ2 τ(xtl+τ) (c) ⩽ 4σ2C2γTl, (63) where (a) is from Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2, (b) is based on our algorithm decision: xtl+τ = ah for any τ ∈ [τh−1 + 1, τh−1 + Tl(ah)], (c) is from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 where XTl = [x⊤ tl+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , x⊤ tl+Tl]⊤ for any phase l and λ = σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, we derive Hl ⩽ 4σ2C2 C2 − 1γTl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (64) We already analyze how to derive the computation complexity for DPBE in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8, which tells the result regarding communication cost: O(γT T α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that the communicating data in each phase between participants and the agent is the local average performance yu l (a) for each action a chosen in the corresponding batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, Nu,l ⩽ Hl for every participant u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (Here, the inequality holds when merging batches as Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Combining the bound of Hl in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6, we derive the total communication cost satisfying L � l=1 |Ul|Hl ⩽ L � l=1 4σ2C2 C2 − 1γTl · (2αl + 1) = O � σ2C2 C2 − 1 · γT T α � , (65) where the last step is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 l=1 Tl + 1 ⩽ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' E Differentially Private DPBE Extensions In this section, we extend the differentially private DPBE in Section 6 to two other celebrated DP models: the local model and the shuffle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 38 To begin with, we present the details of the DP-DPBE algorithm (see Algorithm 2) in the central DP model discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that in the central DP model, with a trusted agent, data privacy is protected by privatizing the aggregated feedback so that the output of the algorithm is indistinguishable between any two users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In a particular phase l, the aggregated feedback for each chosen action becomes ˜yl = ¯yl + (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρHl) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (15)), where ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N(0, σ2 nc) is the injected Gaussian noise for ensuring the required (ε, δ)-DP and is chosen according to the (high-probability) sensitivity of ¯yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, we set the variance of the injected Gaussian noise to the following: σnc = 2 � 2(κ2 + σ2)Hl log(2Hl/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) ε|Ul| , (66) where δ1 ∈ (0, δ) is the probability of sensitivity concentration of ¯yl (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (77) holds with probability at least 1 − δ1) and δ2 = δ − δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By accounting for privacy noise, we update the confidence width function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17) with σn = σnc � 2C2γT , where C is the rare-switching parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Differentially Private DPBE in the Local DP Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the local model, the users do not trust the agent, and thus, each is equipped with a local randomizer R to protect its own local reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let Y be the set of all possible values of the local reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Formally, a local randomizer R is (ε, δ)-local differentially private (or (ε, δ)-LDP) if for any two user inputs, the probability that R outputs two values in Y that are not different by more than a multiplicative factor of eε and an additive factor of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' To guarantee LDP, the local randomizer R at each user u injects Gaussian noise before sending the local reward observations out to the central agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, R(yu l ) = yu l + (ρu,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρu,Hl), (70) where ρu,j∼N(0, σ2 nl) is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' across both users and actions and the variance σ2 nl is chosen according to the (high-probability) sensitivity of yu l (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (80)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, the private aggregated feedback for the chosen actions in the l-th phase in the local DP model becomes ˜yl = 1 |Ul| � u∈Ul R(yu l ) = 1 |Ul| � u∈Ul (yu l + (ρu,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρu,Hl)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (71) We call the differentially private version of DPBE in the local DP model LDP-DPBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, we extend the DPBE algorithm (Algorithm 2) to LDP-DPBE by employing a local randomizer R as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (70) at each participant in the l-th phase and then using the privately aggregated feedback in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (71) to estimate the mean function ˜µl(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The injected Gaussian noise at each participant is σnl = 2√ 2(κ2+σ2)Hl log(2Hl/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2)) ε , where δ1 ∈ (0, δ) is the probability of sensitivity concentration of ¯yl (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (80) holds with probability at least 1−δ1) and δ2 = δ−δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In LDP-DPBE, we update the confidence width function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17) with σn = � 2C2σ2 nlγT |Ul| , where C is the rare-switching parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Differentially Private DPBE in the Shuffle DP Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' While local DP provides a more stringent privacy guarantee, it usually incurs larger regret cost [ZT20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The shuffle model is recently proposed to achieve a better tradeoff between regret and privacy [Che19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the shuffle model, between the users and the agent, there exists a shuffler that permutes the local feedback from the participants before they are observed by the agent so that the agent cannot distinguish between two users’ feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Thus, an additional layer of randomness is introduced via shuffling, which can often be easily implemented using Cryptographic primitives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', mixnets) due to its simple operation [Bit17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, the shuffle DP model consists of three components: a local randomizer R at each user side, a shuffler S between the users and the agent, and an analyzer A at the agent side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let UT ≜ (Ul, · · · , Ul) be the participants throughout the T rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 39 Define the (composite) mechanism Ms(UT ) ≜ ((S ◦ R)(U1), (S ◦ R)(U2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , (S ◦ R)(UL)), where (S ◦ R)(Ul) ≜ S({R(yu l )}u∈Ul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Formally, We say the DP-DPBE algorithm satisfies the shuffle differential privacy (SDP) if the composite mechanism Ms is DP, which leads to the following formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (Shuffle Differential Privacy (SDP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any ε ⩾ 0 and δ ∈ [0, 1], the DP-DPBE is (ε, δ)- shuffle differential privacy (or (ε, δ)-SDP) if for any pair UT and U ′ T that differ by one user, and for any Z ∈ Range(Ms)6: P[Ms(UT ) ∈ Z] ⩽ eεP[Ms(U ′ T ) ∈ Z] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (72) In our case, we apply a shuffle model to the feedback from participants of every particular phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' That is, the private aggregated feedback for the chosen actions in the l-th phase in the shuffle DP model becomes ˜yl = A (S ({R(yu l }u∈Ul))) , (73) where the local randomizer injects a sub-Gaussian noise with variance σ2 ns, which is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' across both users and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Thanks to our phase-then-batch strategy, the recently proposed vector summation protocol [Che21b] can be extended to our algorithm as [LZJ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We present the concrete pseudocodes of R, S, and A in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We call the differentially private version of DPBE in the shuffle model SDP-DPBE, which is extended from DP-DPBE by using the privately aggregated feedback in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (73), where R, S, and A are specified in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any δ1 ∈ (0, δ), let ∆ ≜ Bκ√Hl + � 2(κ2 + σ2)Hl log(2Hl/δ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It is not difficult to show that ∥yu l ∥2 ⩽ ∆ with probability at least 1 − δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' SDP-DPBE employs Algorithm 3 in each phase l with input {yu l }u∈Ul, ∆, and privacy parameters ε and δ2 = δ − δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to [LZJ22], the introduced error for privacy is sub-Gaussian with variance σ2 ns = O � Hl(κ2+σ2) log(Hl/δ1) ln(Hl/δ2)2 ε2|Ul|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In SDP-DPBE, we update the confidence width function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17) with σn = σns � 2C2γT , where C is the rare-switching parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Performance Guarantee For the DP-DPBE algorithm incorporated with the above local and shuffle DP models, we provide the DP guarantee and regret in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 (DP guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2, 3, and 4, for any ε > 0 and δ ∈ (0, 1), i) LDP-DPBE guarantees (ε, δ)-LDP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ii) SDP-DPBE guarantees (ε, δ)-SDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We achieve the above LDP guarantee of i) directly by employing the Gaussian mechanism given the (high-probability) sensitivity of yu l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the shuffle model, we follow the shuffle protocol for each phase in [LZJ22] and derive the corresponding SDP guarantee from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From the above results, we derive that compared to the local model the shuffle model injects much less noise (σ2 ns vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' σ2 nl) without requiring a trusted agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In the following, we present the regret performance of DP-DPBE in these two DP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 6Rang(M) denotes the range of the output of the mechanism M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 40 Table 6: Regret of DP-DPBE in Different DP Models Algorithms Regret DPBE O(T 1−α/2� log(|D|T)) CDP-DPBE O(T 1−α/2� log(|D|T)) + O � ln(1/δ)γT T 1−α√ log(|D|T) ε � LDP-DPBE O(T 1−α/2� log(|D|T)) + O � ln(1/δ)γT T 1−α/2√ log(|D|T) ε � SDP-DPBE O(T 1−α/2� log(|D|T)) + O � ln3/2(γT /δ)γT T 1−α√ log(|D|T) ε � Notes: CDP-DPBE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' LDP-DPBE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and SDP-DPBE represent the DP-DPBE algorithm in the central,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' local,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and shuffle models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' which guarantee (ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' δ)-DP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' δ)-LDP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and (ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' δ)-SDP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 (LDP-DPBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2, and 3, the LDP-DPBE algorithm with β = 1 |D|T achieves the following expected regret: E[R(T)] = O(T 1−α/2� log(|D|T)) + O � ln(1/δ)γT T 1−α/2� log(|D|T) ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (75) Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 (SDP-DPBE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Under Assumptions 1, 2, and 3, the SDP-DPBE algorithm with β = 1 |D|T achieves the following expected regret: E[R(T)] = O(T 1−α/2� log(|D|T)) + O � ln3/2(γT /δ)γT T 1−α� log(|D|T) ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (76) We omit the proofs for the above two theorems because they can be derived by directly replacing σn of the central model with σn = � 2C2σ2 nlγT |Ul| of the local model and σn = σns � 2C2γT of the shuffle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Proofs for DP Guarantees Before providing the DP guarantee of the DPBE algorithm in the three DP models, we first show the ℓ2 sensitivity of ¯yl, which is a key parameter to decide the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let UT , U′ T ⊆ U be two sets of participants in DPBE differing on a single user that is participating in the l-th phase, and let ¯yl and ¯y′ l be the corresponding average local reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any δ1 ∈ (0, 1), we have that with probability at least 1 − δ1, the maximal ℓ2 distance between ¯yl and ¯y′ l is bounded by max |¯y′ l − ¯yl| ⩽ 2 � (κ2 + σ2)Hl log(2Hl/δ1) |Ul| , (77) where Hl denotes the dimension of ¯yl and σ2 is the variance of the noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 41 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let Ul, U ′ l be the sets of participating users in l-th phase corresponding to UT and U′ T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We have |Ul| = |U ′ l| and the maximal ℓ2 distance between ¯yl, ¯y′ l is the following: max |¯y′ l − ¯yl| = max UT ,U′ T ������ 1 |Ul| � u∈U′ l yu l − 1 |Ul| � u∈Ul yu l ������ 2 = 1 |Ul| max u,u′∈U ∥yu′ l − yu l ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (78) For any chosen action a ∈ AHl, we have the following result: |yu′ l (a) − yu l (a)| = ������ 1 Tl(a) � t∈Tl(a) yu′,t − 1 Tl(a) � t∈Tl(a) yu,t ������ = ������ 1 Tl(a) � t∈Tl(a) (yu′,t − yu,t) ������ = ������ 1 Tl(a) � t∈Tl(a) (fu′(xt) + ηu′,t − fu(xt) − ηu,t) ������ ⩽ 1 Tl(a) � t∈Tl(a) ��fu′(xt) + ηu′,t − fu(xt) − ηu,t �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that fu(x) ∼ N(f(x), k(x, x)), ηu,t ∼ N(0, σ2), and the participating users are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We have (fu′(xt) + ηu′,t − fu(xt) − ηu,t) ∼ N(0, 2(k(xt, xt) + σ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' According to the concentration property of Gaussian distribution, we have with probability at least 1 − δ1, |fu′(xt) + ηu′,t − fu(xt) + ηu,t)| ⩽ 2 � (k(xt, xt) + σ2) log(2/δ1) ⩽ 2 � (κ2 + σ2) log(2/δ1), (79) which results in |yu′ l (a) − yu l (a)| ⩽ 2 � (κ2 + σ2) log(2/δ1) for any particular a ∈ AHl with probability at least 1 − δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By substituting the above result into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (78) and applying union bound, we have that with probability at least 1 − δ1, the following is satisfied: max u,u′∈U ∥yu′ l − yu l ∥2 ⩽ 2 � Hl(κ2 + σ2) log(2Hl/δ1), (80) and then with probability at least 1 − δ1, the ℓ2 distance between ¯yl and ¯y′ l is bounded by max |¯y′ l − ¯yl| ⩽ maxu,u′∈U ∥yu′ l − yu l ∥2 |Ul| ⩽ 2 � Hl(κ2 + σ2) log(2Hl/δ1) |Ul| , (81) where the last step is because Hl is the dimension of yu l and also the number of actions in AHl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For both the central model and the local model, we employ the Gaussian mechanism in the differential privacy literature, which is described in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (Gaussian Mechanism [DR14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Given any vector-valued function7 f : U∗ → Rs, define ∆2 ≜ maxU1,U′ 2⊆U ∥f(U1) − f(U2)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let σ = ∆2 � 2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ)/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The Gaussian mechanism, which adds independently drawn random noise from N(0, σ2) to each output of f(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' returning f(U) + (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρs) with ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N(0, σ2), ensures (ε, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 7We use the superscript ∗ to indicate that the length could be varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 42 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let E denote the event that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (77) holds, and thus, P[E] ⩾ 1 − δ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let ∆2 ≜ max |¯y′ l − ¯yl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' If E holds, adding independently drawn noise from N � 0, 2∆2 2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) ε � to each element of ¯yl, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', returning ¯yl + (ρ1, · · · , ρHl) with ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N � 0, 2∆2 2 ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) ε � , ensures (ε, δ2)-DP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Specifically, the following inequality holds P[M(UT ) ∈ Z|E] ⩽ eεP[M(U′ T ) ∈ Z|E] + δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (82) Then, we have P[M(UT ) ∈ Z] ⩽ P[M(UT ) ∈ Z|E]P[E] + 1 − P[E] ⩽ (eεP[M(U′ T ) ∈ Z|E] + δ2)P[E] + δ1 ⩽ eεP[M(U′ T ) ∈ Z|E]P[E] + δ2 + δ1 ⩽ eεP[M(U′ T ) ∈ Z|E]P[E] + δ2 + δ1 ⩽ eεP[M(U′ T ) ∈ Z, E] + δ2 + δ1 ⩽ eεP[M(U′ T ) ∈ Z] + δ, (83) where δ = δ1 + δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Similarly, we can derive the (ε, δ)-LDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Meanwhile, we can achieve (ε, δ)-SDP by combining the analysis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (83) and the proof for Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 in [LZJ22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3 Following a similar line to the proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we first provide the key concentration inequality under DP-DPBE in Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any particular phase l, with probability at least 1 − 6β, the following holds |f(x) − ˜µl(x)| ⩽ ˜wl(x), (84) where mean function ˜µl(x) and confidence width function ˜wl(x) are defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (16) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this proof, we will show the following concentration inequality holds for any x ∈ D P[|f(x) − ˜µl(x)| ⩾ ˜wl(x)] ⩽ 6β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (85) Let ρ ≜ (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρHl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Note that ˜µl(x) = k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1˜yl = k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1(¯yl + ρ) = ¯µl(x) + k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (86) Then, we have |f(x) − ˜µl(x)| ⩽ |f(x) − ¯µl(x)| + |k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (87) 43 For any x ∈ D, we have P [|f(x) − ˜µl(x)| ⩾ ˜wl(x)] ⩽P � |f(x) − ¯µl(x)| + ���k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ ��� ⩾ wl(x) + 2C � γT σ2n log(1/β) � ⩽P [|f(x) − ¯µl(x)| ⩾ wl(x)] + P ����k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ ��� ⩾ 2C � γT σ2n log(1/β) � ⩽4β + P ����k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ ��� ⩾ 2C � γT σ2n log(1/β) � , (88) where the first inequality is due to ˜wl(x) = wl(x) + � 2σ2n log(1/β) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17), the second inequality is from union bound, and the last one is from Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Hence, it remains to bound the second probability in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Recall that ρ = (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρHl) where ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N(0, σ2 nc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ is the sum of Hl i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Gaussian variables, and the total variance (denoted by σ2 sum) is σ2 sum = k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1(KAHlAHl + λW−1 Hl )−1k(x, AHl)σ2 nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (89) Notice that k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' AHl)⊤(KAHlAHl + λW−1 Hl )−1(KAHlAHl + λW−1 Hl )−1k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' AHl) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=ϕ(x)⊤Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl(ΦHlΦ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl + λW−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl )−1(ΦHlΦ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl + λW−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl )−1ΦHlϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=ϕ(x)⊤Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl (W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHlΦ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl + λI)−1W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl · W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl (W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHlΦ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl + λI)−1W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHlϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=ϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHl + λI)−1Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl · WHl · W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHl(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl W1/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl ΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=ϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlW2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlΦHl(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽Tlϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=Tlϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='nI)(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='− λTlϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽Tlϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='HlWHlΦHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='=Tlϕ(x)⊤(Φ⊤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='τHlΦτHl + λI)−1ϕ(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Tlσ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='τHl(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='TlΣ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='TlΣ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Hl(x) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽ 2C2γTl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (90) where (a) is from Φ⊤ HlW2 HlΦHl ⩽ Φ⊤ Hl(TlI)WHlΦHl = TlΦ⊤ HlWHlΦHl because each diagonal entry of WHl satisfies [WHl]hh = Tl(ah) ⩽ Tl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (b) is based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (28), (c) is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (23), and (d) is according to the equivalence representation in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The last step is from the result in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Substituting the above result into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (89), we have σ2 sum ⩽ 2C2γTlσ2 nc = σ2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (91) According to the tail bound of Gaussian variables, we have P ����k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1ρ ��� ⩾ � 2σ2n log(1/β) � ⩽ 2 exp � −4C2γT σ2 nc log(1/β) 2σ2sum � ⩽ 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 44 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Similar to the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1, we, to prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='3, first present three results when the concentration inequality in Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 holds, then obtain an upper bound for the regret incurred in a particular phase l > 2 with high probability, and finally sum up the regret over all phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1) Three observations when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (84) holds Define a “good” event when Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (84) holds in the l-th phase as: ˜El ≜ {∀x ∈ Dl, |f(x) − ˜µl(x)| ⩽ ˜wl(x)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We have P[ ˜El] ⩾ 1 − 6|D|β via the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, similar to the non-private case, under event ˜El in the l-th phase, we have the following three observations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any optimal action x∗ ∈ argmaxx∈D f(x), if x∗ ∈ Dl, then x∗ ∈ Dl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Let f∗ = maxx∈D f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Supposed that x∗ ∈ Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' For any x ∈ Dl+1, its reward gap from the optimal reward is bounded by 4 maxx∈Dl ˜wl(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', f∗ − f(x) ⩽ 4 max x∈Dl ˜wl(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The confidence width function in the private setting satisfies max x∈Dl ˜wl(x) ⩽ max x∈Dl wl(x) + G1γT � 2 log(1/β) |Ul| , (92) where G1 ≜ 8C2√ 2(κ2+σ2)σ2 log(1/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) ε √ C2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The first two observations can be derived similar to the non-private case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Regarding the third observation, we have the confidence width function in the private setting ˜wl(x) = wl(x) + � 2σ2n log(1/β) and � 2σ2n log(1/β) = 2C � γT σ2nc log(1/β) = 4C � 2(κ2 + σ2)HlγT log(1/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) log(1/β) ε|Ul| (a) ⩽ 8C2γT � 2(κ2 + σ2)σ2 log(1/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) log(1/β) ε|Ul| √ C2 − 1 ⩽ 8C2� 2(κ2 + σ2)σ2 log(1/δ1) ln(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='25/δ2) ε √ C2 − 1 � �� � G1 γT � 2 log(1/β) |Ul| , where (a) is from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 2) Regret in a specific phase l > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 45 Under event ˜El−1, the regret incurred in the l-th phase is � t∈Tl f∗ − f(xt) ⩽ � t∈Tl 4 max x∈Dl−1 ˜wl−1(x) ⩽4Tl max x∈Dl−1 wl−1(x) (a) ⩽4Tl max x∈Dl−1 wl−1(x) + 4Tl · G1γT � 2 log(1/β) |Ul−1| ⩽4Tl max x∈Dl−1 wl−1(x) + 4G1γT � 2 log(1/β)2(1−α)(l−1) ⩽4 � 2κ2 log(1/β) � 2(2−α)(l−1) + 8σC � 2γT log(1/β) � 2(1−α)(l−1) + 8σBC � γT 2l−1 + 4G1γT � 2 log(1/β)2(1−α)(l−1), where (a) is from Observation 3 and the last step is from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 3) Total regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Define ˜Eg as the event where the “good” event occurs in every phase in the private setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=', ˜Eg ≜ �L l=1 ˜El.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' It is not difficult to obtain P[Eg] ⩾ 1 − 6|D|βL by applying union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' At the same time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='total regret under event ˜Eg becomes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='Rg = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='l=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='t∈Tl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='(f∗ − f(xt)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽ 2Bκ + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='l=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2κ2 log(1/β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2(2−α)(l−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='l=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8σC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2γT log(1/β) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2(1−α)(l−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='l=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8σBC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='γT 2l−1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='L ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='l=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4G1γT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 log(1/β)2(1−α)(l−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽ 2Bκ + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2κ2 log(1/β) · 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2(L−1)(2−α) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ 8σC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2γT log(1/β) · C1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2(1−α)(L−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='C1 ≜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='21−α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='21−α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ 8σBC√γT · 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2L−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ 4G1γT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 log(1/β) · C22(1−α)(L−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='C2 ≜ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='21−α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='21−α − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='⩽2Bκ + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2κ2 log(1/β)T 1−α/2 + 8σC1C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2γT log(1/β)T 1−α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='+ 32σBC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='γT T + 4C2G1γT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 log(1/β)T 1−α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (93) where the last step is is due to 2L−1 ⩽ T and L ⩽ log(2T) since �L−1 l=1 Tl + 1 ⩽ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' On the other hand, Rb ⩽ 2BκT since | maxx∈D f(x) − f(x)| ⩽ 2Bκ for all x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Choose β = 1/(|D|T) in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 46 Then, the expected regret is: E[R(T)] = P[ ˜Eg]Rg + (1 − P[ ˜Eg])Rb ⩽ Rg + 6|D|βL · 2BκT ⩽ 2Bκ + 16 � 2κ2 log(1/β)T 1−α/2 + 8σC1C � 2γT log(1/β)T 1−α + 32σBC � γT T + 4C2G1γT � 2 log(1/β)T 1−α + 12Bκ|D|βLT = 2Bκ + 16T 1−α/2� 2κ2 log(|D|T) + 8σC1C � 2γT T 1−α log(|D|T) + 32σBC � γT T + 4C2G1γT � 2 log(|D|T)T 1−α + 12Bκ log(2T) = O(T 1−α/2� log(|D|T)) + O( � γT T 1−α log(|D)T) + O(G1γT T 1−α� log(|D|T)) + O( � γT T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (94) Finally, substituting G1 with δ1 = δ2 = δ/2, we have the total expected regret under the DP-DPBE with the central model is E[R(T)] = O(T 1−α/2� log(|D|T)) + O � ln(1/δ)γT T 1−α� log(kT) ε � + O( � γT T 1−α log(|D)T) + O( � γT T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (95) While the DPBE algorithm uses GP tools to define and manage the uncertainty in estimating the unknown function f, the analysis of DPBE algorithm does not rely on any Bayesian assumption about f being actually drawn from the prior GP(0, k), and it only requires f to be bounded in the kernel norm associated with the RKHS Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' F Additional Numerical Results F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 Evaluation of DP-DPBE In Section 7, we evaluated DP-DPBE on the synthetic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' In this subsection, we present additional numerical results for DP-DPBE on the standard benchmark functions and the function from real-world (light-sensor) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' By considering the same setting as for the synthetic function, we run T = 106 rounds and present how the cumulative regret at the end of T varies with privacy budget ε ∈ {5, 10, 15, 20, 25, 30} and δ = 10−6 in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Then, by choosing privacy parameters δ = 10−6 and ε = 15, we also compare the per-round regret of DP-DPBE and DPBE for the three benchmark functions and the real-world (light-sensor) data and present the results in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We perform 20 runs for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' From these results, we make similar observations to those for the synthetic function: the privacy-regret tradeoff and achieving privacy “for free”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 Comparison with State-of-the-Art In Section 8, we provide simulation results on the regret performance and running time of GP-UCB, BPE, and our algorithm DPBE with different values of α on the synthetic data generated in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='1 In this section, we add additional numerical results on three benchmark functions (Sphere, Six-hump Camel, Michalewicz) and one function from real-world data– Light sensor data [Sch].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The parameters of the 47 (a) Sphere function (b) Six-Hump Camel function (c) Michalewicz function (d) Function from light sensor data Figure 8: Performance of DP-DPBE: Final cumulative regret vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' privacy budget ε with δ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Table 7: Comparison of running time (seconds) under GP-UCB, BPE, and DPBE with different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Algorithms DPBE GP-UCB BPE α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 Sphere 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='68 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='87 Six-Hump Camel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='79 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='43 Michalewicz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='48 Light Sensor Data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='22 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='08 problem setting and the algorithms are as follows: T = 4 × 104, |D| = 100, and k = kSE with lSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (a) Sphere function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 3, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='001, λ = σ2/v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (b) Six-Hump Camel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (c) Michalewicz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (d) Functions from real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' Settings: d = 2, C = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='42, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, v2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='01, λ = σ2/v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' We plot the cumulative regret for all the algorithms in Figure 10 and present the running time in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 48 ×105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 5 10 15 20 25 30 Pivacy budget x105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 5 10 15 20 25 30 Pivacy budget x105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 5 10 15 20 25 30 Pivacy budget ×104 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 5 10 15 20 25 30 Pivacy budget (a) Sphere function (b) Six-Hump Camel function (c) Michalewicz function (d) Function from light sensor data Figure 9: Performance of DP-DPBE: Per-round regret vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' time with parameters ε = 15 and δ = 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 DP-DPBE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBE egret 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Rounds ×1061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DP-DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Rounds ×1061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DP-DPBE DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Rounds ×106DP-DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 DPBE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 Rounds ×106Algorithm 2 Differentially Private Distributed Phase-then-Batch-based Elimination (DP-DPBE) 1: Input: D ⊆ Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' α ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' β ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' rare-switching parameter C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' local noise σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' privacy parameters ε and δ 2: Initialization: l = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' D1 = D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' t1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and T1 = 1 3: while tl < T do 4: Set τ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' h = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' τ1 = 0 and Σ2 0(x) = k(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' for all x ∈ Dl 5: while τ ⩽ Tl do 6: h = h + 1 7: Choose ah ∈ argmax x∈Dl Σ2 h−1(x) (67) 8: Play action ah for Tl(ah) ≜ ⌊(C2 − 1)/Σ2 h−1(ah)⌋ times if not reaching min{T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' tl + Tl} 9: Update τ = τ + Tl(ah),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and the posterior variance Σ2 h(·) by including ah according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 10: end while 11: Let Hl = h denote the total number of batches in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 12: Randomly select ⌈2αl⌉ participants Ul # Operations at each participant 13: for each participant u ∈ Ul do 14: Collect and compute local average reward for every chosen action a ∈ AHl: yu l (a) = 1 Tl(a) � t∈Tl(a) yu,t 15: Send the local average reward for every chosen action yu l ≜ [yu l (a)]a∈AHl to the agent 16: end for 17: Aggregate local observations for each chosen action a ∈ AHl: yl(a) = 1 |Ul| � u∈Ul yu l (a) 18: Let ¯yl = [yl(a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , yl(aHl)] and ˜yl = ¯yl + (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' , ρHl), where ρj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∼ N(0, σ2 nc) and σnc is specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 19: Update ˜µl(·): ˜µl(x) ≜ k(x, AHl)⊤(KAHlAHl + λW−1 Hl )−1˜yl (68) 20: Eliminate low-rewarding actions from Dl based on the confidence width function ˜wl(·) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (17) with σn = σnc � 2C2γT : Dl+1 = � x ∈ Dl : ˜µl(x) + ˜wl(x) ⩾ max b∈Dl (˜µl(b) − ˜wl(b)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' (69) 21: Tl+1 = 2Tl, t = t + Tl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' l = l + 1 22: end while 50 Algorithm 3 M : Shuffle Protocol for a Set of Vectors with Users U [LZJ22] 1: Input: {yu}u∈U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' where each yu ∈ Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∥yu∥2 ⩽ ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' privacy parameters ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' δ2 ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 1) 2: Let � � � � � � � � � � � � � �ε = ε 18√ log(2/δ2) g ≜ max{�ε � |U|/(6 � 5 ln ((4s)/δ2)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' √s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 10} b ≜ ⌈ 180g2 ln (4s/δ2) �ε2|U| ⌉ p ≜ 90g2 ln (4s/δ2) b�ε2|U| (74) // Local Randomizer function R(yu) 3: for coordinate j ∈ [s] do 4: Shift data to enforce non-negativity: wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='j = (yu)j + ∆,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' ∀u ∈ U //randomizer for each entry 5: Set ¯wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='j ← ⌊wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='jg/(2∆)⌋ //max |(yu)j + ∆| ⩽ 2∆ 6: Sample rounding value γ1 ∼ Ber(wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='jg/(2∆) − ¯wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='j) 7: Sample privacy noise value γ2 ∼ Bin(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' p) 8: Let φu j be a multi-set of (g + b) bits associated with the j-th coordinate of user u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' where φu j consists of ¯wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='j + γ1 + γ2 copies of 1 and g + b − ( ¯wi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='j + γ1 + γ2) copies of 0 9: end for 10: Report {(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' φu j )}j∈[s] to the shuffler end function // Shuffler function S({(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' φj)}j∈[s]) //φj = (φu j )u∈U 11: for each coordinate j ∈ [s] do 12: Shuffle and output all (g + b)|U| bits in φj 13: end for end function // Analyzer function A(S({(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' φj)}j∈[s]) 14: for coordinate j ∈ [s] do 15: Compute zj ← 2∆ g|U|((�(g+b)|U| i=1 (φj)i) − b|U|p) // (φj)i denotes the i-th bit in φj 16: Re-center: oj ← zj − ∆ 17: end for 18: Output the estimator of vector average o = (oj)j∈[s] end function 51 (a) Sphere function (b) Six-Hump Camel function (c) Michalewicz function (d) Function from light sensor data Figure 10: Comparison of regret performance under DPBE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' GP-UCB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' and BPE on three benchmark functions and one function from real-world dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' The shaded area represents the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content=' 52 x103 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBE α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 中 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 BPE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 GP-UCB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0 1 2 3 4 Rounds ×104×103 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 DPBE α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 DPBE α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 4 DPBE α =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 BPE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 GP-UCB gret 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='0 0 1 2 3 4 Rounds ×104×103 5 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 中 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 BPE 4 DPBE α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 GP-UCB 2 1 0 0 1 2 3 4 Rounds ×104×103 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='4 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='8 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBE α= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='5 DPBEα=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='6 BPE DPBE α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FLT4oBgHgl3EQfXi8l/content/2301.12061v1.pdf'} +page_content='7 GP-UCB Regr 1.' 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To support relevant use cases, including sensing, edge +learning, and edge inference, all require transmission of high- +dimensional data or AI models over the air. To overcome the +bottleneck, we propose a novel framework of SEMantic DAta +Sourcing (SEMDAS) for locating semantically matched data +sources to efficiently enable edge-intelligence operations. The +comprehensive framework comprises new architecture, protocol, +semantic matching techniques, and design principles for task- +oriented wireless techniques. As the key component of SEMDAS, +we discuss a set of machine learning based semantic matching +techniques targeting different edge-intelligence use cases. More- +over, for designing task-oriented wireless techniques, we discuss +different tradeoffs in SEMDAS systems, propose the new concept +of joint semantics-and-channel matching, and point to a number +of research opportunities. The SEMDAS framework not only +overcomes the said communication bottleneck but also addresses +other networking issues including long-distance transmission, +sparse connectivity, high-speed mobility, link disruptions, and +security. In addition, experimental results using a real dataset +are presented to demonstrate the performance gain of SEMDAS. +I. INTRODUCTION +The sixth-generation (6G) mobile networks are expected to +be artificial intelligence (AI) native, featuring the ubiquitous +deployment of machine learning and AI algorithms at the +network edge [1]. On the other hand, data have replaced fuel +to become the most valuable resource in the world [2]. Mobile +data are being generated at an exponentially growing rate that +is expected to increase by three-fold to reach 324 EB/month +by 2028 [3]. The 6G edge intelligence will provide a platform +for continuous distillation of AI to support many Internet-of- +Everything (IoE) applications ranging from sensing to auto- +driving to industrial automation. Compared with “cloud AI”, +edge intelligence has the advantages of efficient processing of +mobile data, preserving user privacy, reducing network traffic, +and providing ultra-low-latency access [4]. The main challenge +in implementing edge intelligence is that the wireless transmis- +sion of high-dimensional data or AI model parameters creates +a communication bottleneck. To overcome the bottleneck, we +propose in this article a novel framework of SEMantic DAta +Sourcing (SEMDAS) for locating semantically-matched data +sources to efficiently enable edge-intelligence operations. +A. 5G Connectivity-Centric Networking +5G’s key innovation lies in providing an infrastructure +that supports heterogeneous types of services and applica- +K. Huang, Q. Lan and Z. Liu are with Dept. of EEE, The University of +Hong Kong, Hong Kong SAR. L. Yang is with Huawei Noah’s Ark Lab, +China. Contact: K. Huang (huangkb@eee.hku.hk). +tions. Specifically, to meet different requirements in rate, +reliability, and latency, three types of connectivity have been +defined including ultra-reliable-low-latency communications, +massive machine-type communication, and enhanced mobile +broadband. 5G communication techniques have been largely +designed using the rate-centric approach that is rooted in Shan- +non’s information theory. In this approach, data are essentially +treated as a sequence of bits that should be transported from a +source to a destination reliably and quickly. As suggested by +Warren Weaver in his 1953 paper [5], communications should +transcend merely solving this technical problem to address +the semantic issue, namely the accuracy in communicating +meanings of messages and the effectiveness of their use at +the destination to execute a specific task. One would argue +that many mobile applications do have semantic awareness +such as recommendations and advertising. However, the ap- +plications are add-ons to the network and interface only with +its Application Layer. In other words, they are decoupled +from the radio access layers where wireless techniques are +deployed. The traditional computation-communication sepa- +ration approach for designing wireless techniques is believed +to be sub-optimal in terms of end-to-end (E2E) performance. +As a result, the existing semantics-agnostic techniques lack +the maximum efficiencies needed to cope with the exponen- +tially growing mobile data and population of edge devices +and achieve faster-than-human (i.e., 0.1 milliseconds) latency. +Therefore, 6G researchers advocate the new computation- +communication integration approach that tightly couples com- +munication, computing, sensing, and control in designing next- +generation wireless techniques [1]. +B. 6G Semantic Communications +As a 6G paradigm, Semantic Communications (SemCom) +refers to the computation-communication integrated designs +that optimize the E2E performance metric of semantic accu- +racy or the overlapping metric of task effectiveness. In contrast +with the traditional opaque-data transmission, the semantics- +awareness can be exploited to reduce communication over- +head by avoiding transmitting information lacking relevance +and to improve the radio-resource utilization efficiency (e.g., +prioritizing packets in resource allocation based on their +data content). Two basic operations in a SemCom system +are semantic encoding, which compresses messages while +retaining their semantics or task utility, and channel encoding +for ensuring reliability in the presence of channel distortion. +In a popular approach known as joint source-channel coding, +these two operations are enabled using separate neural network +models, which are jointly trained for an E2E task such as edge +inference or sensing [6]. For implementation on a layering +network architecture, SemCom involves deep coupling of the +arXiv:2301.00403v1 [cs.IT] 1 Jan 2023 + +2 +Data-driven task +Service Requesters +SEMDAS Controller +Data Sources +Semantic query: +“Walking man” +Query +broadcast +✓ +✗ +✗ +✗ +Data uploading +Result feedback +Fig. 1. The SEMDAS network architecture. +top Application layer and bottom radio access layers [7], +[8]. To illustrate its advantage, consider the transmission of +the message “Einstein’s son, **** ****** ********, was a +professor of ********* ************ at the University of +********************.” with erroneous letters marked using +“*”. It could have been discarded as an unreliable message +in the Physical layer but can be reliably recovered in the +Application layer using a knowledge graph to be “Einstein’s +son, Hans Albert Einstein, was a professor of hydraulic +engineering at the University of California, Berkeley.” The rise +of edge intelligence provides a platform for SemCom where +powerful tools including AI algorithms, knowledge graphs, +and data analytics can automate and streamline the system +operations. Furthermore, given an AI-empowered receiver ro- +bust against perturbation, the graceful degradation of inference +performance with a decreasing number of transmitted features +of source-data samples can also allow received messages to +be useful even in the event of packet loss. +C. Proposed Semantic Data Sourcing for 6G Edge Intelligence +Aiming at efficient implementations of edge intelligence on +a SemCom platform, we propose a novel framework called +SEMantic DAta Sourcing (SEMDAS). The goal of SEMDAS +is to locate among many data sources a subset that can provide +semantically relevant data to enable communication-efficient +edge learning or inference. To this end, the SEMDAS protocol +essentially involves a service requester transmitting a query, +which characterizes a specific edge-intelligence task, to a +controller that searches for matched data sources. Then the +matched sources are connected to the requester to provide a +computing service or their data. As the key component of +SEMDAS, a set of semantic matching techniques are proposed +for three representative use cases of edge intelligence — +Internet-of-Thing (IoT) sensing, edge learning, and inference. +They build on suitably chosen existing learning algorithms +including semantic embedding, uncertainty evaluation, distri- +bution matching, and autoencoder. +The proposed SEMDAS framework has three main advan- +tages. First, SEMDAS overcomes the communication bottle- +neck of edge intelligence by avoiding unnecessary transmis- +sion of semantically irrelevant data. Second, without requiring +E2E connections, SEMDAS exploits the existence of multiple +semantically similar sources to address networking issues in- +cluding long-distance transmission, sparse connectivity, high- +speed mobility, and link disruptions. Last, data source verifica- +tion via semantics checking by SEMDAS controller provides +a mechanism to ensure security and data integrity. +We further propose new principles for designing task- +oriented wireless techniques for SEMDAS. One principle lies +in system optimization based on the tradeoffs between query +size, data overhead, privacy, and semantic matching accuracy. +The other principle is joint semantics-channel matching. We +discuss how the new tradeoffs lead to new designs of task- +oriented wireless techniques including multi-access, over-the- +air computing, radio resource management, and beamforming. +II. OVERVIEW OF SEMANTIC DATA SOURCING +Although SEMDAS applies to a broader range of applica- +tions, we focus on those of edge intelligence. The network +architecture, protocol, and advantages of SEMDAS are de- +scribed separately in the following sub-sections. +A. SEMDAS Network Architecture +The architecture, as illustrated in Fig. 1, comprises the +following key components. +• Service Requester: The node (either a device or a server) +performs a data-driven task. To this end, it sends a query +that comprises descriptors to the SEMDAS controller to +ask for semantically-matched data that helps effective +task execution. +• SEMDAS Controller: The node coordinates the SEM- +DAS process, serves as the interface between data sources +and requester, and implements the security function, and +manages mobility. +• Semantic Data Sources: Data sources, which can be +edge devices or servers, store and supply sensing/user +data, multimedia files, documents, or AI models. Se- +mantic data are the data that are categorized based +on their semantics (i.e., content and utility). Different +types of semantic information are embedded in a single +data sample, e.g., human faces and behaviors, buildings, +weather, context, and locations in the same image. For a +specific task, only selected information is useful (e.g., +human faces for surveillance). A semantic dataset is +identified by its semantics regardless of the physical loca- +tion, generation mechanism, and communication method. +The semantic similarity between two datasets can be +measured according to given descriptors such as “smiley +faces”, “German shepherd dogs”, and “hand gestures”. +Descriptors can be also in the form of multimedia objects + +D1 +F03 +Source +Selector +Inference +Model +SEMDAS +Controller +Query +Generator +Reference +image +Semantic +Matching +Semantic +Matching +Matching +scores +Channel states +Semantic +Matching +Sensor 1 +Sensor 2 +Sensor 3 +Sensor 4✓ +Device Scheduler +and Receiver +Result: “Missing person found at location N48.14° E17.13° time 12:15pm” +Send query +1 +Broadcast query +2 +Matching score feedback +3 +Signalling +4 +Image uploading +5 +Sensor 1 +Sensor 2 +Sensor 4 +Sensor 3 +Semantic +Matching +Fig. 2. SEMDAS for AI-empowered IoT sensing. +such as images, video clips, and speech signals (e.g., +the photo of a missing person in Section V). From the +perspective of task effectiveness, any two semantically +similar datasets are identical in their utility. The nodes +storing semantically similar data can all supply data to +the same requester. +• Semantic Matching: SEMDAS can be viewed as a task- +oriented semantics-based “search engine” for edge intel- +ligence. AI algorithms are applied to real-time matching +between data and a query in terms of semantic similarity. +Specific algorithms are discussed in the next section. +B. SEMDAS Protocol +A typical SEMDAS protocol comprises the following steps. +• Service Requesting: A service requester sends its query +(e.g., text or multimedia objects) to the SEMDAS con- +troller. The query is usually a low-dimensional feature +vector characterizing the desired data. The requester can +be either a user device as in the cases of IoT sensing and +edge inference or a server in the case of edge learning. +• Semantic Source Selection: The controller broadcasts +the query to all data sources in the network. Each source +generates a key of its data and compares it with the +query using a suitable semantic matching technique (see +Section III). The operation generates a matching score +that is fed back to the controller. +• Data Uploading: Using the matching scores, the server +selects the best-matched data sources to upload their data. +The data is forwarded to the requesting node for process- +ing. Alternatively, the computation can be performed at +the controller or servers with the result downloaded to +the user. +C. Advantages of SEMDAS +The proposed SEMDAS framework has the following main +advantages. First, SEMDAS represents a scalable solution of +data sourcing for edge intelligence. It overcomes the bot- +tleneck of peer-to-peer data transportation by evaluating the +semantic similarity between data sources to avoid unnecessary +transmission and exploiting the existence of multiple semanti- +cally similar sources to cope with communication issues such +as long-distance transportation, link disruption, and unreach- +able sites. Second, data source verification via checking of +its semantic content by SEMDAS controller ensures secu- +rity and data integrity. Third, SEMDAS facilitates mobility +management. Unlike 5G connectivity-centric networking, the +SEMDAS approach does not require E2E connections. A +moving user repeats sending the same query, which can be +served by different data sources with semantic similarity. +There is no need to maintain a connection to the previous +source. Thereby, SEMDAS helps to cope with unfavorable +networking conditions, e.g., sparse connectivity, high-speed +mobility, and link disruptions. +III. SEMANTIC MATCHING TECHNIQUES FOR EDGE +INTELLIGENCE +A. Semantic Matching for AI-Empowered IoT Sensing +IoT sensing is a new function of 6G that exploits cross- +network collaboration between on-device sensors to form +a large-scale sensor network for surveillance, localization, +tracking, and event detection. The feeding of multi-modal +sensing data into edge-AI models endows on the sensors the +capabilities of object/event recognition and human behavior +detection. Nevertheless, the transportation of high-dimensional +sensing data places a heavy burden on the network. The resul- +tant traffic jams can be alleviated by sensor selection based on +semantic matching. Relevant applications and techniques are +described as follows. +Consider two representative use cases. First, sensing via +crowdsourcing refers to the involvement of the sensors owned +by a group of participating users to collectively perform a +sensing task. A specific task of finding a missing person, +as illustrated in Fig. 2, is considered in the experiment in +Section V. The second use case is networked perception. The +network function involves multiple devices (e.g., vehicles and +robots) cooperating to complete a perception task such as +tracking, localization, and object recognition as coordinated by +an edge server. The use case of networked perception can be +further divided into two sub-cases — peer-assisted perception +and multi-view perception. The former overcomes the issue of +a degraded sensor at a device by using matched data collected +by a helping device. For instance, an autonomous vehicle +with faulty cameras can rely on nearby vehicles to observe +the roads and surrounding environment [9]. On the other +hand, multi-view perception uses a server to aggregate the 2D +observations of multiple camera sensors from different view + +1 +F04 +D2 +Semantic +Matching +Local +Trainer +D3 +Semantic +Matching +D2 +SEMDAS +Controller +Send query +1 +Broadcast query +2 +Matching score +feedback +3 +For FL training +4 +Source +Selector +D2 C2 +D3 C2 +Requester +Query +Generator +C2 +C1 +C2 +C1 +FL +Parameter +Sever +Forward trained model +Semantic +Matching +Local +Trainer +D1 +C2 +C1 +Local +Trainer +Fig. 3. SEMDAS for federated edge learning. +angles to improve the sensing accuracy (of, for example, object +recognition) or reconstruct a 3D object [10]. For instance, the +query in the case of peer-assisted perception can comprise a +degraded view-image captured by the user’s faulty sensor; in +the cases of multi-view perception, the query can be generated +from an image of a wild hog to locate wild intruders in the +city center. +We propose an efficient semantic matching technique that +aims to find an observation close to the query in the semantic +space. The semantic space is a feature space where samples +(e.g., phrases or images) with similar meanings are clustered +and those with different meanings are separated. In practice, +the semantic space is created using a projection neural network +model that is trained according to the sensing task of interest +and deployed to project the query and keys into the space [9]. +The matching score of a query-key pair is then obtained +with the general attention mechanism, which computes the +dot-product or cosine similarity of their projections in the +learned semantic space. For example, the Google RankBrain +system, which powers the Google search engine accessed +by billions per day, matches queries and web pages in the +semantic space instead of plain keyword-searching. It infers +the semantic intent behind the query and identifies web pages +that cover the intent, which can be interpreted as projecting +both into a semantic space and selecting based on similarity +measurements. +The optimal performance of the above semantic matching +technique requires the E2E training of two semantic encoders. +One is the query generator projecting requests into low- +dimensional queries; the other is the key generator computing +keys carrying semantic information from sensing observations. +B. Semantic Matching for Edge Learning +Edge learning refers to the edge-intelligence use case that +trains AI models using mobile data distributed in a wireless +network. It is envisioned to be a key 6G operation that distills +intelligence needed for empowering a wide range of applica- +tions. There exist two main approaches. The first is centralized +edge learning (CEEL) which directly uploads data (or their +features) from devices to a server for model training. The +other approach, federated edge learning (FEEL) is deployed +when the data ownership needs to be preserved [4]. To this +end, FEEL uses a so-called parameter server to download a +global model onto devices for updating using local datasets, +and then upload and aggregate the local models to update +the global model. Implementing the classic iterative algorithm +of stochastic gradient descent, the process is repeated until +the model converges. Both approaches are confronted with +a communication bottleneck as they require uploading of +high-dimensional data or model parameters/gradients from +potentially many devices. The bottleneck can be alleviated +using the SEMDAS approach (see Fig. 3). +Data at different devices exhibit a high level of hetero- +geneity. Indiscriminately training a model on all data may be +harmful as the inclusion of out-of-domain data can corrupt +the model. Then, the purpose of SEMDAS is to find data +that match the domain of the learning task. For instance, to +learn French-to-English machine translation, the datasets in +other languages are irrelevant. As another example, to train +a model that recognizes a famous author’s handwriting, not +all handwritten texts with the same meaning are useful except +for those matching the author’s style. Therefore, the semantic +matching in the context of edge learning refers to domain +matching between training data and the learning task. +We discuss in the sequel the domain matching techniques +for two cases where datasets are characterized by labels or +descriptors. The case with data description is relatively simple. +The data-task matching can be performed by the SEMDAS +controller that receives the data description published by data +sources and the task description from the query sent by the +data-seeking sender. Then the controller evaluates the score +of the matching from a particular dataset to the task by +projecting representative data and task descriptors into the +semantic space to evaluate their similarity, which is based +on the same technique as in Section III-A. The domain +matching in the case without dataset description is more +challenging. Recently, researchers have observed that different +implicit semantic characteristics embedded in a cluster of +samples of the same label can cause the cluster to have a +nested sub-clustered distribution [11]. For example, in the +MNIST dataset, the handwritten digits of the same label +form sub-clusters, each of which corresponds to one writing +style [11]. Therefore, domain matching can be translated to +the matching of data distribution and task. To this end, the + +2 +X35 +Expert Model K +Send query +1 +Broadcast query +2 +2 +Matching score +feedback +3 +3 +Download +4 +Query +Generator +SEMDAS +Controller +Source +Selection +Model +Downloading +… +Expert Model 1 +Expert Gate 1 +Expert Gate K +1 +4 +Fig. 4. SEMDAS for edge inference with model downloading. +query is generated to contain statistical information on the +data in the desired domain or a representative mini-batch of +such data. Then a matching score of a particular dataset is +generated by evaluating or approximating its Kullback-Leibler +distance from the desired distribution specified by the query. +The controller coordinates selected devices to upload their data +to the requesting server in the case of CEEL or to participate +in FEEL. +C. Semantic Matching for Edge Inference +Edge inference is a basic operation of 6G edge intelligence +that focuses on the efficient provisioning of AI inference +capabilities to edge devices. Depending on whether to offload +computation to edge servers, edge inference can be imple- +mented in two ways, namely on-device inference and split +inference. The former refers to on-demand downloading of AI +models to devices for local inference, depicted in Fig. 4. This +is suitable for small-to-medium model sizes and provides a +faster response speed and better protection of privacy. For the +latter, the computation load is split between users and servers. +The device sub-model extracts intermediate features from data +samples to preserve the data ownership by avoiding sharing +raw data. Then the server receives and feeds the uploaded +features into the server sub-model to perform prediction. By +offloading intensive computation to the server, split inference +provides resource-constrained devices access to large-scale +deep neural network models, for which on-device deployment +is infeasible. There exist a practically infinite number of +models stored in the network. They can be differentiated +in many dimensions, e.g., size, context, task, performance, +and topology [12]. For instance, an object recognition model +can either have a convolutional or recursive topology and +can be trained for an urban, rural, or indoor environment. +In the context of edge inference, a data source refers to +a node (device or server) that stores a particular sharable +model (i.e., its parameters and topology information). Then the +SEMDAS problem is to find data sources whose models meet +the requester’s requirements in terms of inference accuracy, +complexity, and storage. +We discuss the semantic matching techniques for edge +inference as follows. First, the technique for the case with +available model description is similar to those for sensing +and edge learning, for which matching leverages semantic +space. Next, we focus on the case where the model description +is either unavailable or insufficient for semantic matching. +The proposed techniques for this purpose require a requesting +device to generate a query containing test samples of local +data (e.g., sensor observations). Then using the query, the +SEMDAS controller performs semantic matching via either +of the following two techniques. The first is expert gateway. +Assume that the controller has access to a library of popular +AI models called expert models. Each expert has a lightweight +autoencoder pair of the encoder and the decoder apart from +the inference model itself. The controller consults each expert +by providing it with the query. In response, the expert uses +its autoencoder to extract features from the test samples and +generate reconstruction errors [13]. The controller selects the +expert with the minimum errors (i.e., the one best matching the +user’s task) to download its model. As the second technique +called server polling, if no matched model can be found among +expert models, the SEMDAS controller broadcasts the user’s +query to all available servers in the network. Each server tests +the matching level of its model by using it to perform inference +on the test samples in the query and evaluate the generated +confidence scores. The model with the highest overall score +is deemed semantically matching the query. Then the hosting +server is associated with the requesting user to provide the +inference service in the case of split inference. In the case of +model downloading, the model is retrieved and forwarded to +the user. +IV. TASK-ORIENTED COMMUNICATIONS FOR SEMDAS +Task-oriented communication techniques for SEMDAS are +designed using an E2E performance metric such as IoT sensing +accuracy, convergence speed for FEEL, and prediction error +for edge inference. In this section, we introduce this new area +by proposing new design principles and discussing several +research opportunities. +A. Tradeoffs and Optimal Designs +There exist two unique tradeoffs in wireless communica- +tions for SEMDAS. They allow optimization of the edge- +intelligence system and techniques. +The first tradeoff, called the query-data tradeoff, is due +to two opposite trends. On one hand, shortening the query +vector reduces the overhead in broadcasting to all data sources. +On the other hand, less information contained in queries +can lead to less accurate semantic matching and hence more +uploaded data that increase the uplink overhead. The opposite +trend also holds, inducing the said tradeoff between the query +and data overhead. Wireless systems for SEMDAS can be +optimized based on the tradeoff. For instance, the uplink- +downlink division of bandwidth or time can be optimized + +2 +X300 +06 +for an E2E performance metric. Furthermore, under a query- +rate constraint, the query can be designed to maximize the +overall semantic matching level at data sources or to minimize +the communication overhead for data uploading for a given +matching level. +The second tradeoff, called the privacy-accuracy tradeoff, +results from the fact that placing more details in the query +(e.g., the number of features of a missing person) reveals +more private information about the requestor (e.g., the per- +son’s identity) or his/her task but improving the semantic- +matching accuracy and hence the communication efficiency, +and vice versa. The tradeoff is of practical interest as in many +practical scenarios, the query contains private information +and its broadcasting over the network can compromise the +requester’s privacy. The consideration of the tradeoff leads +to the need for privacy-preserving communication designs. +In particular, under a privacy constraint on the requester, the +query generation can be integrated with adaptive transmission +(e.g., coding, power control, and beamforming) to maximize +the system efficiency. +B. Joint Semantic-and-Channel Matching (JSCM) +Following a rate-centric approach, selecting data sources +merely based on their channel states can result in a mis- +match between the sourced data and the computing task. +Nevertheless, the consideration of only semantics-matching +in the selection may lead to unreliable communication links. +Therefore, for supporting SEMDAS applications in wireless +systems, it is important to balance both aspects in choosing +data sources, which is called JSCM. Redesigning traditional +rate-centric wireless techniques to feature JSCM creates many +research opportunities. Several selected ones are described as +follows. +• Multi-Access: JSCM-based distributed selection of data +sources for uploading over a multi-access channel can +be implemented using a threshold on their semantic +matching scores and another on their channel gains. +The thresholds can be jointly designed for the dual +objectives of sourcing sufficient relevant data and at the +same time regulating the number of accessing devices to +avoid frequent packet collisions or insufficiency of radio +resources. +• Over-the-Air Computing (AirComp): AirComp exploits +the waveform superposition property of a multi-access +channel to realize over-the-air aggregation of views and +local models/stochastic gradients in the use cases of +sensing and FEEL, respectively [14]. The bottleneck of +suppressing AirComp error lies in those channels of trans- +mitters with unfavorable conditions. JSCM-based device +selection can be designed to balance the suppression of +AirComp error and sourcing sufficient data to optimize +the E2E system performance. +• Radio Resource Management (RRM): Traditional RRM +techniques such as sub-channel allocation, power con- +trol, and scheduling have been designed using a rate- +centric metric. To be task-oriented for SEMDAS, these +techniques should be redesigned to prioritize devices not +only based on their channel states but also their semantic +matching scores. +• Random Beamforming: The existing random beam- +forming schemes exploit multiuser channel diversity to +maximize the data rate by selecting among many users +those with beam-aligned channel vectors and large chan- +nel gains. Such schemes can be modified to simultane- +ously exploit the diversity in both channel and semantics +over multiple data sources. +V. EXPERIMENTS +A. Experiment Design +We demonstrate SEMDAS using the sensing application +of finding missing people (e.g., a child or an elderly) via a +network of wirelessly connected surveillance cameras. The +underpinning operation, technically known as re-identification +(ReID), attempts to associate camera views, which are cap- +tured in different occasions, with the same person that is +specified by the reference photo in the query sent by the +person’s families. The real-time location information shared +by the matched cameras would help locate the person. To +implement semantic matching, we adopt the LightMBN model +trained on the well-known CUHK-03 training set [15] for +semantic feature extraction and use the matching function +discussed in Section III-A. Thereby, the query photo and each +camera view are compressed into translated into 3584-by-1 +feature vectors for the purpose of semantic matching. For +communication, the query broadcast to sensors is at 32-bit +resolution per dimension; the matched sensors upload their +raw data (i.e., photos) for their views with an Acknowledge +of “person found”. We simulate a network of 20 cameras and +generate the query and camera views by drawing samples from +the CUHK-03 dataset. In each trial, among 20 camera views, +4 contain the target identity while the others are associated +with different persons. In total, the experiment contains 1229 +random trials. The channels from sensors to the SEMDAS +controller are modeled as i.i.d. Rayleigh fading and being +orthogonal with a total uplink bandwidth of 5 MHz. The +performance metric, called missing rate, is defined as the +probability of ReID failure, namely that none of the matched +cameras views actually contains the target person. The missing +rate is evaluated as a function of average uplink communica- +tion load (in Mbits) and average uplink communication latency +(in milliseconds) per request. We consider the following JSCM +and benchmarking schemes in experiments. +• (Proposed) JSCM: A given number of matched sen- +sors are selected using the criterion of maximizing the +weighted sum of semantic matching score and communi- +cation rate, where their weights are optimized numerically +as 1 and 0.09, respectively. +• Best-semantics selection (BSS) based on the criterion of +the maximum semantic matching score. +• Best-channel selection (BCS) based on the criterion of +the maximum communication rate. +• Random selection (RS) of uploading sensors. + +7 +101 +102 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +20 +40 +60 +80 +100 +Fig. 5. +The missing rate versus average communication latency for the +proposed JSCM and three benchmarking schemes — BSS, BCS and RS. +B. Performance Evaluation +The performance of JSCM is compared with that of BSS, +BCS and RS. Fig. 5 depicts the curves of missing rate versus +communication latency that grows as the number of selected +sensors increases. We can observe that BCS, a rate-centric +scheme, and RS both have unacceptable performance due to +their lack of semantic awareness. On the other hand, with +such awareness, the missing rates of the JSCM and BSS +schemes achieve much lower missing rates than the preceding +schemes, for example, more than 2-order magnitude lower at +30-ms latency. Between JSCM and BSS, the proposed design +significantly outperforms the latter. This demonstrates JSCM +being a promising solution and the need for task-oriented +wireless design. +VI. CONCLUDING REMARKS +We have proposed the SEMDAS framework to solve the +problem of communication bottleneck of 6G systems caused +by data sourcing in edge-intelligence use cases. Its basic +principle is to transport only data whose semantics match the +computing tasks so as to avoid redundant network traffic due +to transmission of irrelevant data. Based on the principle, a +comprehensive framework has been presented that comprises +the architecture, protocol, learning-based semantic matching +techniques, as well as new design principles for task-oriented +wireless techniques. At the high level, SEMDAS represents +a contribution to the ongoing development of 6G semantic +communication systems. In particular, the new framework +points to the new direction of revolutionizing the wireless net- +work architecture to enable highly efficient edge-intelligence +operations and applications. +REFERENCES +[1] Huawei, “6G: The next horizon from connected people and things to +connected intelligence,” [Online] https://www-file.huawei.com/-/media/ +corp2020/pdf/tech-insights/1/6g-white-paper-en.pdf?la=en (accessed on +Dec. 1, 2022). +[2] The +Economist, +“The +world’s +most +valuable +resource +is +no +longer +oil, +but +data +—Regulating +the +Internet +giants,” +[Online] +https://www.economist.com/leaders/2017/05/06/ +the-worlds-most-valuable-resource-is-no-longer-oil-but-data (accessed +on Dec. 1, 2022). +[3] Ericsson, “Ericsson mobility report,” [Online] https://www.ericsson.com/ +en/reports-and-papers/mobility-report/reports/november-2022 (accessed +on Dec. 1, 2022). +[4] G. Zhu, D. Liu, Y. Du, C. You, J. Zhang and K. Huang, “Toward +an intelligent edge: Wireless communication meets machine learning,” +IEEE Commun. Mag., vol. 58, no. 1, pp. 19–25, Jan. 2020. +[5] W. Weaver, “Recent contributions to the mathematical theory of com- +munication,” ETC: A Review of General Semantics, vol. 10, no. 4, pp. +261–281, 1953. +[6] M. Jankowski, D. G¨und¨uz and K. Mikolajczyk, “Wireless image retrieval +at the edge,” IEEE J. Sel. Areas Commun., vol. 39, no. 1, pp. 89–100, +Nov. 2021. +[7] Z. Qin, X. Tao, J. Lu, W. Tong and G. Y. Li, “Semantic communications: +Principles and challenges,” [Online] https://arxiv.org/pdf/2201.01389.pdf +(accessed on Dec. 1, 2022). +[8] Q. Lan, D. Wen, Z. Zhang, Q. Zeng, X. Chen, P. Popovski and K. Huang, +“What is semantic communication? A view on conveying meaning in +the era of machine intelligence,” J. Commun. Inf. Netw., vol. 6, no. 4, +pp. 336–371, Dec. 2021. +[9] Y. C. Liu, J. Tian, N. Glaser and Z. Kira, “When2com: Multi-agent +perception via communication graph grouping,” in Proc. IEEE/CVF +Conf. Comput. Vision Pattern Recogn. (CVPR), Seattle, WA, USA, Jun. +13-19, 2020. +[10] D. Wang, X. Cui, X. Chen, Z. Zou, T. Shi, S. Salcudean, Z. Wang and +R. Ward, “Multi-view 3D reconstruction with transformers,” in Proc. +IEEE/CVF Int. Conf. Comput. Vision (ICCV), Montreal, QC, Canada, +Oct. 10-17, 2021. +[11] H. Yuan, W. Morningstar, L. Ning and K. Singhal, “What do we +mean generalization in federated learning?” in Proc. Int. Conf. Learn. +Representations (ICLR), Montreal, QC, Canada, Apr. 25-29, 2022. +[12] K. Huang, H. Wu, Z. Liu and X. Qi, “In-situ model downloading to +realize versatile edge AI in 6G mobile networks,” [Online] https://arxiv. +org/pdf/2210.03555.pdf, 2022. +[13] R. Aljundi, P. Chakravarty and T. Tuytelaars, “Expert Gate: Lifelong +learning with a network of experts,” in Proc. IEEE Conf. Comput. Vision +Pattern Recogn. (CVPR), Honolulu, HI, USA, Jul. 21-26, 2017. +[14] M. M. Amiri and D. G¨und¨uz, “Machine learning at the wireless edge: +Distributed stochastic gradient descent over-the-air,” IEEE Trans. Signal +Process., vol. 68, no. 1, pp. 2155–2169, Mar. 2021. +[15] F. Herzog, X. Ji, T. Teepe, S. H¨ormann, J. Gilg and G. Rigoll, +“Lightweight multi-branch network for person re-identification,” in Proc. +IEEE Int. Conf. Image Process. (ICIP), Anchorage, AK, USA, Sept. 19- +22, 2021. +Kaibin Huang [Fellow, IEEE] is a Professor at the Department of Electrical +and Electronic Engineering, The University of Hong Kong, Hong Kong. His +research interests include mobile edge computing, edge AI, and 6G systems. +Qiao Lan received his B.Eng. degree from Southern University of Science +and Technology (SUSTech), Shenzhen, in 2019. He is now pursuing a Ph.D. +degree with The University of Hong Kong. +Zhiyan Liu received his B.Eng. degree from Tsinghua University, Beijing, in +2021. He is now pursuing a Ph.D. degree with The University of Hong Kong. +Lin Yang received the Ph.D. degree from Hong Kong University of Science +and Technology and is now a senior researcher in Huawei Noah’s Ark Lab. + diff --git a/bNAyT4oBgHgl3EQfivjL/content/tmp_files/load_file.txt b/bNAyT4oBgHgl3EQfivjL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18189f3436e1232b83a177451dcfb9e46cca525e --- /dev/null +++ b/bNAyT4oBgHgl3EQfivjL/content/tmp_files/load_file.txt @@ -0,0 +1,489 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf,len=488 +page_content='1 Semantic Data Sourcing for 6G Edge Intelligence Kaibin Huang, Qiao Lan, Zhiyan Liu, and Lin Yang Abstract—As a new function of 6G networks, edge intelligence refers to the ubiquitous deployment of machine learning and artificial intelligence (AI) algorithms at the network edge to empower many emerging applications ranging from sensing to auto-pilot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To support relevant use cases, including sensing, edge learning, and edge inference, all require transmission of high- dimensional data or AI models over the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To overcome the bottleneck, we propose a novel framework of SEMantic DAta Sourcing (SEMDAS) for locating semantically matched data sources to efficiently enable edge-intelligence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The comprehensive framework comprises new architecture, protocol, semantic matching techniques, and design principles for task- oriented wireless techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As the key component of SEMDAS, we discuss a set of machine learning based semantic matching techniques targeting different edge-intelligence use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' More- over, for designing task-oriented wireless techniques, we discuss different tradeoffs in SEMDAS systems, propose the new concept of joint semantics-and-channel matching, and point to a number of research opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The SEMDAS framework not only overcomes the said communication bottleneck but also addresses other networking issues including long-distance transmission, sparse connectivity, high-speed mobility, link disruptions, and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In addition, experimental results using a real dataset are presented to demonstrate the performance gain of SEMDAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' INTRODUCTION The sixth-generation (6G) mobile networks are expected to be artificial intelligence (AI) native, featuring the ubiquitous deployment of machine learning and AI algorithms at the network edge [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' On the other hand, data have replaced fuel to become the most valuable resource in the world [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Mobile data are being generated at an exponentially growing rate that is expected to increase by three-fold to reach 324 EB/month by 2028 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The 6G edge intelligence will provide a platform for continuous distillation of AI to support many Internet-of- Everything (IoE) applications ranging from sensing to auto- driving to industrial automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Compared with “cloud AI”, edge intelligence has the advantages of efficient processing of mobile data, preserving user privacy, reducing network traffic, and providing ultra-low-latency access [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The main challenge in implementing edge intelligence is that the wireless transmis- sion of high-dimensional data or AI model parameters creates a communication bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To overcome the bottleneck, we propose in this article a novel framework of SEMantic DAta Sourcing (SEMDAS) for locating semantically-matched data sources to efficiently enable edge-intelligence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 5G Connectivity-Centric Networking 5G’s key innovation lies in providing an infrastructure that supports heterogeneous types of services and applica- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Huang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Lan and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Liu are with Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' of EEE, The University of Hong Kong, Hong Kong SAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Yang is with Huawei Noah’s Ark Lab, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Contact: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Huang (huangkb@eee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Specifically, to meet different requirements in rate, reliability, and latency, three types of connectivity have been defined including ultra-reliable-low-latency communications, massive machine-type communication, and enhanced mobile broadband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 5G communication techniques have been largely designed using the rate-centric approach that is rooted in Shan- non’s information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In this approach, data are essentially treated as a sequence of bits that should be transported from a source to a destination reliably and quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As suggested by Warren Weaver in his 1953 paper [5], communications should transcend merely solving this technical problem to address the semantic issue, namely the accuracy in communicating meanings of messages and the effectiveness of their use at the destination to execute a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' One would argue that many mobile applications do have semantic awareness such as recommendations and advertising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' However, the ap- plications are add-ons to the network and interface only with its Application Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In other words, they are decoupled from the radio access layers where wireless techniques are deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The traditional computation-communication sepa- ration approach for designing wireless techniques is believed to be sub-optimal in terms of end-to-end (E2E) performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As a result, the existing semantics-agnostic techniques lack the maximum efficiencies needed to cope with the exponen- tially growing mobile data and population of edge devices and achieve faster-than-human (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='1 milliseconds) latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Therefore, 6G researchers advocate the new computation- communication integration approach that tightly couples com- munication, computing, sensing, and control in designing next- generation wireless techniques [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 6G Semantic Communications As a 6G paradigm, Semantic Communications (SemCom) refers to the computation-communication integrated designs that optimize the E2E performance metric of semantic accu- racy or the overlapping metric of task effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In contrast with the traditional opaque-data transmission, the semantics- awareness can be exploited to reduce communication over- head by avoiding transmitting information lacking relevance and to improve the radio-resource utilization efficiency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', prioritizing packets in resource allocation based on their data content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Two basic operations in a SemCom system are semantic encoding, which compresses messages while retaining their semantics or task utility, and channel encoding for ensuring reliability in the presence of channel distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In a popular approach known as joint source-channel coding, these two operations are enabled using separate neural network models, which are jointly trained for an E2E task such as edge inference or sensing [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For implementation on a layering network architecture, SemCom involves deep coupling of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='00403v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='IT] 1 Jan 2023 2 Data-driven task Service Requesters SEMDAS Controller Data Sources Semantic query: “Walking man” Query broadcast ✓ ✗ ✗ ✗ Data uploading Result feedback Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The SEMDAS network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' top Application layer and bottom radio access layers [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To illustrate its advantage, consider the transmission of the message “Einstein’s son, **** ****** ********, was a professor of ********* ************ at the University of ********************.” with erroneous letters marked using “*”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' It could have been discarded as an unreliable message in the Physical layer but can be reliably recovered in the Application layer using a knowledge graph to be “Einstein’s son, Hans Albert Einstein, was a professor of hydraulic engineering at the University of California, Berkeley.” The rise of edge intelligence provides a platform for SemCom where powerful tools including AI algorithms, knowledge graphs, and data analytics can automate and streamline the system operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Furthermore, given an AI-empowered receiver ro- bust against perturbation, the graceful degradation of inference performance with a decreasing number of transmitted features of source-data samples can also allow received messages to be useful even in the event of packet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Proposed Semantic Data Sourcing for 6G Edge Intelligence Aiming at efficient implementations of edge intelligence on a SemCom platform, we propose a novel framework called SEMantic DAta Sourcing (SEMDAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The goal of SEMDAS is to locate among many data sources a subset that can provide semantically relevant data to enable communication-efficient edge learning or inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To this end, the SEMDAS protocol essentially involves a service requester transmitting a query, which characterizes a specific edge-intelligence task, to a controller that searches for matched data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then the matched sources are connected to the requester to provide a computing service or their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As the key component of SEMDAS, a set of semantic matching techniques are proposed for three representative use cases of edge intelligence — Internet-of-Thing (IoT) sensing, edge learning, and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' They build on suitably chosen existing learning algorithms including semantic embedding, uncertainty evaluation, distri- bution matching, and autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The proposed SEMDAS framework has three main advan- tages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' First, SEMDAS overcomes the communication bottle- neck of edge intelligence by avoiding unnecessary transmis- sion of semantically irrelevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Second, without requiring E2E connections, SEMDAS exploits the existence of multiple semantically similar sources to address networking issues in- cluding long-distance transmission, sparse connectivity, high- speed mobility, and link disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Last, data source verifica- tion via semantics checking by SEMDAS controller provides a mechanism to ensure security and data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We further propose new principles for designing task- oriented wireless techniques for SEMDAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' One principle lies in system optimization based on the tradeoffs between query size, data overhead, privacy, and semantic matching accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The other principle is joint semantics-channel matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We discuss how the new tradeoffs lead to new designs of task- oriented wireless techniques including multi-access, over-the- air computing, radio resource management, and beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' OVERVIEW OF SEMANTIC DATA SOURCING Although SEMDAS applies to a broader range of applica- tions, we focus on those of edge intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The network architecture, protocol, and advantages of SEMDAS are de- scribed separately in the following sub-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS Network Architecture The architecture, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 1, comprises the following key components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Service Requester: The node (either a device or a server) performs a data-driven task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To this end, it sends a query that comprises descriptors to the SEMDAS controller to ask for semantically-matched data that helps effective task execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS Controller: The node coordinates the SEM- DAS process, serves as the interface between data sources and requester, and implements the security function, and manages mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Data Sources: Data sources, which can be edge devices or servers, store and supply sensing/user data, multimedia files, documents, or AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Se- mantic data are the data that are categorized based on their semantics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', content and utility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Different types of semantic information are embedded in a single data sample, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', human faces and behaviors, buildings, weather, context, and locations in the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For a specific task, only selected information is useful (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', human faces for surveillance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A semantic dataset is identified by its semantics regardless of the physical loca- tion, generation mechanism, and communication method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The semantic similarity between two datasets can be measured according to given descriptors such as “smiley faces”, “German shepherd dogs”, and “hand gestures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Descriptors can be also in the form of multimedia objects D1 F03 Source Selector Inference Model SEMDAS Controller Query Generator Reference image Semantic Matching Semantic Matching Matching scores Channel states Semantic Matching Sensor 1 Sensor 2 Sensor 3 Sensor 4✓ Device Scheduler and Receiver Result: “Missing person found at location N48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='14° E17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='13° time 12:15pm” Send query 1 Broadcast query 2 Matching score feedback 3 Signalling 4 Image uploading 5 Sensor 1 Sensor 2 Sensor 4 Sensor 3 Semantic Matching Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS for AI-empowered IoT sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' such as images, video clips, and speech signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', the photo of a missing person in Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' From the perspective of task effectiveness, any two semantically similar datasets are identical in their utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The nodes storing semantically similar data can all supply data to the same requester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Matching: SEMDAS can be viewed as a task- oriented semantics-based “search engine” for edge intel- ligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' AI algorithms are applied to real-time matching between data and a query in terms of semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Specific algorithms are discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS Protocol A typical SEMDAS protocol comprises the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Service Requesting: A service requester sends its query (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', text or multimedia objects) to the SEMDAS con- troller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The query is usually a low-dimensional feature vector characterizing the desired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The requester can be either a user device as in the cases of IoT sensing and edge inference or a server in the case of edge learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Source Selection: The controller broadcasts the query to all data sources in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Each source generates a key of its data and compares it with the query using a suitable semantic matching technique (see Section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The operation generates a matching score that is fed back to the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Data Uploading: Using the matching scores, the server selects the best-matched data sources to upload their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The data is forwarded to the requesting node for process- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Alternatively, the computation can be performed at the controller or servers with the result downloaded to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Advantages of SEMDAS The proposed SEMDAS framework has the following main advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' First, SEMDAS represents a scalable solution of data sourcing for edge intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' It overcomes the bot- tleneck of peer-to-peer data transportation by evaluating the semantic similarity between data sources to avoid unnecessary transmission and exploiting the existence of multiple semanti- cally similar sources to cope with communication issues such as long-distance transportation, link disruption, and unreach- able sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Second, data source verification via checking of its semantic content by SEMDAS controller ensures secu- rity and data integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Third, SEMDAS facilitates mobility management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Unlike 5G connectivity-centric networking, the SEMDAS approach does not require E2E connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A moving user repeats sending the same query, which can be served by different data sources with semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' There is no need to maintain a connection to the previous source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Thereby, SEMDAS helps to cope with unfavorable networking conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', sparse connectivity, high-speed mobility, and link disruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMANTIC MATCHING TECHNIQUES FOR EDGE INTELLIGENCE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Matching for AI-Empowered IoT Sensing IoT sensing is a new function of 6G that exploits cross- network collaboration between on-device sensors to form a large-scale sensor network for surveillance, localization, tracking, and event detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The feeding of multi-modal sensing data into edge-AI models endows on the sensors the capabilities of object/event recognition and human behavior detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Nevertheless, the transportation of high-dimensional sensing data places a heavy burden on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The resul- tant traffic jams can be alleviated by sensor selection based on semantic matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Relevant applications and techniques are described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Consider two representative use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' First, sensing via crowdsourcing refers to the involvement of the sensors owned by a group of participating users to collectively perform a sensing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A specific task of finding a missing person, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 2, is considered in the experiment in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The second use case is networked perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The network function involves multiple devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', vehicles and robots) cooperating to complete a perception task such as tracking, localization, and object recognition as coordinated by an edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The use case of networked perception can be further divided into two sub-cases — peer-assisted perception and multi-view perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The former overcomes the issue of a degraded sensor at a device by using matched data collected by a helping device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For instance, an autonomous vehicle with faulty cameras can rely on nearby vehicles to observe the roads and surrounding environment [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' multi-view perception uses a server to aggregate the 2D observations of multiple camera sensors from different view 1 F04 D2 Semantic Matching Local Trainer D3 Semantic Matching D2 SEMDAS Controller Send query 1 Broadcast query 2 Matching score feedback 3 For FL training 4 Source Selector D2 C2 D3 C2 Requester Query Generator C2 C1 C2 C1 FL Parameter Sever Forward trained model Semantic Matching Local Trainer D1 C2 C1 Local Trainer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS for federated edge learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' angles to improve the sensing accuracy (of, for example, object recognition) or reconstruct a 3D object [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For instance, the query in the case of peer-assisted perception can comprise a degraded view-image captured by the user’s faulty sensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' in the cases of multi-view perception, the query can be generated from an image of a wild hog to locate wild intruders in the city center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We propose an efficient semantic matching technique that aims to find an observation close to the query in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The semantic space is a feature space where samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', phrases or images) with similar meanings are clustered and those with different meanings are separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In practice, the semantic space is created using a projection neural network model that is trained according to the sensing task of interest and deployed to project the query and keys into the space [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The matching score of a query-key pair is then obtained with the general attention mechanism, which computes the dot-product or cosine similarity of their projections in the learned semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For example, the Google RankBrain system, which powers the Google search engine accessed by billions per day, matches queries and web pages in the semantic space instead of plain keyword-searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' It infers the semantic intent behind the query and identifies web pages that cover the intent, which can be interpreted as projecting both into a semantic space and selecting based on similarity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The optimal performance of the above semantic matching technique requires the E2E training of two semantic encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' One is the query generator projecting requests into low- dimensional queries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' the other is the key generator computing keys carrying semantic information from sensing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Matching for Edge Learning Edge learning refers to the edge-intelligence use case that trains AI models using mobile data distributed in a wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' It is envisioned to be a key 6G operation that distills intelligence needed for empowering a wide range of applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' There exist two main approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The first is centralized edge learning (CEEL) which directly uploads data (or their features) from devices to a server for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The other approach, federated edge learning (FEEL) is deployed when the data ownership needs to be preserved [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To this end, FEEL uses a so-called parameter server to download a global model onto devices for updating using local datasets, and then upload and aggregate the local models to update the global model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Implementing the classic iterative algorithm of stochastic gradient descent, the process is repeated until the model converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Both approaches are confronted with a communication bottleneck as they require uploading of high-dimensional data or model parameters/gradients from potentially many devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The bottleneck can be alleviated using the SEMDAS approach (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Data at different devices exhibit a high level of hetero- geneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Indiscriminately training a model on all data may be harmful as the inclusion of out-of-domain data can corrupt the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then, the purpose of SEMDAS is to find data that match the domain of the learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For instance, to learn French-to-English machine translation, the datasets in other languages are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As another example, to train a model that recognizes a famous author’s handwriting, not all handwritten texts with the same meaning are useful except for those matching the author’s style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Therefore, the semantic matching in the context of edge learning refers to domain matching between training data and the learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We discuss in the sequel the domain matching techniques for two cases where datasets are characterized by labels or descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The case with data description is relatively simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The data-task matching can be performed by the SEMDAS controller that receives the data description published by data sources and the task description from the query sent by the data-seeking sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then the controller evaluates the score of the matching from a particular dataset to the task by projecting representative data and task descriptors into the semantic space to evaluate their similarity, which is based on the same technique as in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The domain matching in the case without dataset description is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Recently, researchers have observed that different implicit semantic characteristics embedded in a cluster of samples of the same label can cause the cluster to have a nested sub-clustered distribution [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For example, in the MNIST dataset, the handwritten digits of the same label form sub-clusters, each of which corresponds to one writing style [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Therefore, domain matching can be translated to the matching of data distribution and task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To this end, the 2 X35 Expert Model K Send query 1 Broadcast query 2 2 Matching score feedback 3 3 Download 4 Query Generator SEMDAS Controller Source Selection Model Downloading … Expert Model 1 Expert Gate 1 Expert Gate K 1 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' SEMDAS for edge inference with model downloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' query is generated to contain statistical information on the data in the desired domain or a representative mini-batch of such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then a matching score of a particular dataset is generated by evaluating or approximating its Kullback-Leibler distance from the desired distribution specified by the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The controller coordinates selected devices to upload their data to the requesting server in the case of CEEL or to participate in FEEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Semantic Matching for Edge Inference Edge inference is a basic operation of 6G edge intelligence that focuses on the efficient provisioning of AI inference capabilities to edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Depending on whether to offload computation to edge servers, edge inference can be imple- mented in two ways, namely on-device inference and split inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The former refers to on-demand downloading of AI models to devices for local inference, depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' This is suitable for small-to-medium model sizes and provides a faster response speed and better protection of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For the latter, the computation load is split between users and servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The device sub-model extracts intermediate features from data samples to preserve the data ownership by avoiding sharing raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then the server receives and feeds the uploaded features into the server sub-model to perform prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' By offloading intensive computation to the server, split inference provides resource-constrained devices access to large-scale deep neural network models, for which on-device deployment is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' There exist a practically infinite number of models stored in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' They can be differentiated in many dimensions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', size, context, task, performance, and topology [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For instance, an object recognition model can either have a convolutional or recursive topology and can be trained for an urban, rural, or indoor environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In the context of edge inference, a data source refers to a node (device or server) that stores a particular sharable model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', its parameters and topology information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then the SEMDAS problem is to find data sources whose models meet the requester’s requirements in terms of inference accuracy, complexity, and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We discuss the semantic matching techniques for edge inference as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' First, the technique for the case with available model description is similar to those for sensing and edge learning, for which matching leverages semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Next, we focus on the case where the model description is either unavailable or insufficient for semantic matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The proposed techniques for this purpose require a requesting device to generate a query containing test samples of local data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', sensor observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then using the query, the SEMDAS controller performs semantic matching via either of the following two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The first is expert gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Assume that the controller has access to a library of popular AI models called expert models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Each expert has a lightweight autoencoder pair of the encoder and the decoder apart from the inference model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The controller consults each expert by providing it with the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In response, the expert uses its autoencoder to extract features from the test samples and generate reconstruction errors [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The controller selects the expert with the minimum errors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', the one best matching the user’s task) to download its model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' As the second technique called server polling, if no matched model can be found among expert models, the SEMDAS controller broadcasts the user’s query to all available servers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Each server tests the matching level of its model by using it to perform inference on the test samples in the query and evaluate the generated confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The model with the highest overall score is deemed semantically matching the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Then the hosting server is associated with the requesting user to provide the inference service in the case of split inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In the case of model downloading, the model is retrieved and forwarded to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' TASK-ORIENTED COMMUNICATIONS FOR SEMDAS Task-oriented communication techniques for SEMDAS are designed using an E2E performance metric such as IoT sensing accuracy, convergence speed for FEEL, and prediction error for edge inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In this section, we introduce this new area by proposing new design principles and discussing several research opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Tradeoffs and Optimal Designs There exist two unique tradeoffs in wireless communica- tions for SEMDAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' They allow optimization of the edge- intelligence system and techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The first tradeoff, called the query-data tradeoff, is due to two opposite trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' On one hand, shortening the query vector reduces the overhead in broadcasting to all data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' On the other hand, less information contained in queries can lead to less accurate semantic matching and hence more uploaded data that increase the uplink overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The opposite trend also holds, inducing the said tradeoff between the query and data overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Wireless systems for SEMDAS can be optimized based on the tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For instance, the uplink- downlink division of bandwidth or time can be optimized 2 X300 06 for an E2E performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Furthermore, under a query- rate constraint, the query can be designed to maximize the overall semantic matching level at data sources or to minimize the communication overhead for data uploading for a given matching level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The second tradeoff, called the privacy-accuracy tradeoff, results from the fact that placing more details in the query (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', the number of features of a missing person) reveals more private information about the requestor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', the per- son’s identity) or his/her task but improving the semantic- matching accuracy and hence the communication efficiency, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The tradeoff is of practical interest as in many practical scenarios, the query contains private information and its broadcasting over the network can compromise the requester’s privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The consideration of the tradeoff leads to the need for privacy-preserving communication designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In particular, under a privacy constraint on the requester, the query generation can be integrated with adaptive transmission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', coding, power control, and beamforming) to maximize the system efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Joint Semantic-and-Channel Matching (JSCM) Following a rate-centric approach, selecting data sources merely based on their channel states can result in a mis- match between the sourced data and the computing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Nevertheless, the consideration of only semantics-matching in the selection may lead to unreliable communication links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Therefore, for supporting SEMDAS applications in wireless systems, it is important to balance both aspects in choosing data sources, which is called JSCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Redesigning traditional rate-centric wireless techniques to feature JSCM creates many research opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Several selected ones are described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Multi-Access: JSCM-based distributed selection of data sources for uploading over a multi-access channel can be implemented using a threshold on their semantic matching scores and another on their channel gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The thresholds can be jointly designed for the dual objectives of sourcing sufficient relevant data and at the same time regulating the number of accessing devices to avoid frequent packet collisions or insufficiency of radio resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Over-the-Air Computing (AirComp): AirComp exploits the waveform superposition property of a multi-access channel to realize over-the-air aggregation of views and local models/stochastic gradients in the use cases of sensing and FEEL, respectively [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The bottleneck of suppressing AirComp error lies in those channels of trans- mitters with unfavorable conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' JSCM-based device selection can be designed to balance the suppression of AirComp error and sourcing sufficient data to optimize the E2E system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Radio Resource Management (RRM): Traditional RRM techniques such as sub-channel allocation, power con- trol, and scheduling have been designed using a rate- centric metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To be task-oriented for SEMDAS, these techniques should be redesigned to prioritize devices not only based on their channel states but also their semantic matching scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Random Beamforming: The existing random beam- forming schemes exploit multiuser channel diversity to maximize the data rate by selecting among many users those with beam-aligned channel vectors and large chan- nel gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Such schemes can be modified to simultane- ously exploit the diversity in both channel and semantics over multiple data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Experiment Design We demonstrate SEMDAS using the sensing application of finding missing people (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', a child or an elderly) via a network of wirelessly connected surveillance cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The underpinning operation, technically known as re-identification (ReID), attempts to associate camera views, which are cap- tured in different occasions, with the same person that is specified by the reference photo in the query sent by the person’s families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The real-time location information shared by the matched cameras would help locate the person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' To implement semantic matching, we adopt the LightMBN model trained on the well-known CUHK-03 training set [15] for semantic feature extraction and use the matching function discussed in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Thereby, the query photo and each camera view are compressed into translated into 3584-by-1 feature vectors for the purpose of semantic matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' For communication, the query broadcast to sensors is at 32-bit resolution per dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' the matched sensors upload their raw data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=', photos) for their views with an Acknowledge of “person found”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We simulate a network of 20 cameras and generate the query and camera views by drawing samples from the CUHK-03 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In each trial, among 20 camera views, 4 contain the target identity while the others are associated with different persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In total, the experiment contains 1229 random trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The channels from sensors to the SEMDAS controller are modeled as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Rayleigh fading and being orthogonal with a total uplink bandwidth of 5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The performance metric, called missing rate, is defined as the probability of ReID failure, namely that none of the matched cameras views actually contains the target person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The missing rate is evaluated as a function of average uplink communica- tion load (in Mbits) and average uplink communication latency (in milliseconds) per request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We consider the following JSCM and benchmarking schemes in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' (Proposed) JSCM: A given number of matched sen- sors are selected using the criterion of maximizing the weighted sum of semantic matching score and communi- cation rate, where their weights are optimized numerically as 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='09, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Best-semantics selection (BSS) based on the criterion of the maximum semantic matching score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Best-channel selection (BCS) based on the criterion of the maximum communication rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Random selection (RS) of uploading sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 7 101 102 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='8 1 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' The missing rate versus average communication latency for the proposed JSCM and three benchmarking schemes — BSS, BCS and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Performance Evaluation The performance of JSCM is compared with that of BSS, BCS and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 5 depicts the curves of missing rate versus communication latency that grows as the number of selected sensors increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' We can observe that BCS, a rate-centric scheme, and RS both have unacceptable performance due to their lack of semantic awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' On the other hand, with such awareness, the missing rates of the JSCM and BSS schemes achieve much lower missing rates than the preceding schemes, for example, more than 2-order magnitude lower at 30-ms latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Between JSCM and BSS, the proposed design significantly outperforms the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' This demonstrates JSCM being a promising solution and the need for task-oriented wireless design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' CONCLUDING REMARKS We have proposed the SEMDAS framework to solve the problem of communication bottleneck of 6G systems caused by data sourcing in edge-intelligence use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Its basic principle is to transport only data whose semantics match the computing tasks so as to avoid redundant network traffic due to transmission of irrelevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Based on the principle, a comprehensive framework has been presented that comprises the architecture, protocol, learning-based semantic matching techniques, as well as new design principles for task-oriented wireless techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' At the high level, SEMDAS represents a contribution to the ongoing development of 6G semantic communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' In particular, the new framework points to the new direction of revolutionizing the wireless net- work architecture to enable highly efficient edge-intelligence operations and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' REFERENCES [1] Huawei, “6G: The next horizon from connected people and things to connected intelligence,” [Online] https://www-file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='huawei.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Herzog, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Ji, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Teepe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' H¨ormann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Gilg and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Rigoll, “Lightweight multi-branch network for person re-identification,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' IEEE Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Image Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' (ICIP), Anchorage, AK, USA, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' 19- 22, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Kaibin Huang [Fellow, IEEE] is a Professor at the Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' His research interests include mobile edge computing, edge AI, and 6G systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Qiao Lan received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' degree from Southern University of Science and Technology (SUSTech), Shenzhen, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' He is now pursuing a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' degree with The University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Zhiyan Liu received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' degree from Tsinghua University, Beijing, in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' He is now pursuing a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' degree with The University of Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' Lin Yang received the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} +page_content=' degree from Hong Kong University of Science and Technology and is now a senior researcher in Huawei Noah’s Ark Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf'} diff --git 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M. Piva,1, ∗ J. C. Souza,2, 1, † V. Brousseau-Couture,3 Sopheak Sorn,4 K. R. Pakuszewski,2 Janas +K. John,1 C. Adriano,2 M. Cˆot´e,3 P. G. Pagliuso,2, 5 Arun Paramekanti,6, 7 and M. Nicklas1, ‡ +1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str. 40, D-01187 Dresden, Germany +2Instituto de F´ısica “Gleb Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil +3D´epartement de Physique, Universit´e de Montr´eal, C.P. 6128, +Succursale Centre-Ville, Montr´eal, Qu´ebec, Canada H3C 3J7 +4Institute for Quantum Materials and Technology, +Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany +5Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA +6Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, M5S 1A7 Canada +7S. N. Bose National Centre for Basic Sciences, +Block JD, Sector - III, Salt lake, Kolkata-700106, India +(Dated: February 1, 2023) +In the ferromagnetic (FM) Weyl semimetal CeAlSi both space-inversion and time-reversal sym- +metries are broken. Our quantum oscillation (QO) data indicate that the FM ordering modifies the +Fermi surface topology and also leads to an unusual drop in the QO amplitude. In the FM phase, +we find a pressure-induced suppression of the anomalous and the loop Hall effects. This cannot be +explained based on the electronic band structure or magnetic structure, both of which are nearly +pressure independent. Instead, we show that a simplified model describing the scattering of Weyl +fermions off FM domain walls can potentially explain the observed topological features. Our study +highlights the importance of domain walls for understanding transport in FM Weyl semimetals. +I. +INTRODUCTION +Topological phases of matter have lately received con- +siderable attention, due to the experimental realization of +exotic types of charge carriers. One example is the mass- +less Weyl fermions found in Weyl semimetals (WSMs) +[1–3], which are characterized by remarkable electronic +properties, such as surface Fermi arcs, a bulk chiral +anomaly, axial–gravitational anomaly, an extremely large +magnetoresistance (MR) and an anomalous Hall effect +(AHE) [1, 3–6]. Weyl fermions can be generated by either +breaking space-inversion (SI) or time-reversal (TR) sym- +metry of materials with a Dirac or quadratic band touch- +ing points. So far most experimentally studied WSMs +break SI symmetry [7–15]; fewer examples are known for +WSMs with broken TR symmetry, i.e. magnetic WSMs +[16–21]. +Magnetic WSMs are of fundamental interest +since they intertwine topology and strong correlations +[22, 23]. They also offer the potential to manipulate the +topological phase in a desired way, for instance using +a magnetic field to tune the position of Weyl nodes or +to control the chirality or geometry of magnetic domain +walls, which is important for next-generation spintronics +applications [24, 25]. +The family of LnAlPn (Ln = lanthanides, Pn = Ge, +Si) materials is ideal to host nontrivial topological prop- +erties due to their noncentrosymmetric crystalline struc- +∗ Mario.Piva@cpfs.mpg.de +† Present address: +Department of Condensed Matter Physics, +Weizmann Institute of Science, Rehovot, Israel. +‡ Michael.Nicklas@cpfs.mpg.de +ture (I41md), which is the same as in the TaAs family +of WSMs [7, 9, 26–29]. Multiple Weyl nodes and a large +spin Hall effect were predicted to exist in LaAlGe and +LaAlSi [30]. Weyl cones were experimentally observed +for LaAlGe [31] and a π Berry phase was recently found +in LaAlSi [32]. Remarkably, magnetic members of the +family host rare-earth moments which can order and ad- +ditionally break TR symmetry - many of them are pre- +dicted to feature Weyl nodes near the Fermi level [33–36]. +Experiments have discovered an anomalous Hall effect +(AHE) in PrAlGe1−xSix [33], chiral surface Fermi arcs in +PrAlGe [37, 38], and a topological magnetic phase and +singular angular MR in the semimetal CeAlGe [39–41]. +In addition, Weyl fermions have been found to mediate +magnetism in NdAlSi [42] and a π Berry phase was re- +ported for quantum oscillations (QO) in SmAlSi [34]. +In this Article, we focus on the ferromagnetic Weyl +semimetal CeAlSi. CeAlSi, which hosts an in-plane non- +collinear ferromagnetic (FM) order below the Curie tem- +perature TC ≈ 8 K with a large anisotropy, the c-axis +being the magnetically hard axis [36]. Ce3+ spins in ad- +jacent FM planes display an angle of ≈ 70◦ [36]. Recent +angle resolved photo emission spectroscopy experiments +in the paramagnetic phase of CeAlSi above TC revealed +Fermi arcs and several Weyl nodes lying close to the +Fermi energy which stem from the non-centrosymmetric +structure [43]. +Going below TC, into the FM state, +a magnetic field applied parallel to the [100] direction +reveals an AHE, while a [001] field leads to an unex- +plained hysteretic loop Hall effect (LHE) [36]. +In ad- +dition, CeAlSi may exhibit nontrivial magnetic domain +walls [44]; indeed, chiral domain walls were recently de- +tected in this system [45]. Furthermore, magnetoelastic +arXiv:2301.13707v1 [cond-mat.str-el] 31 Jan 2023 + +2 +couplings give rise to picometer displacements in the unit +cell due to the internal FM field, which can lead to dif- +ferent domain wall spin textures [46]. +The presence of this magnetoelastic effect suggests +that external pressure may lead to a strong tuning of +magnetism and to associated large changes in the AHE +and LHE [46]. Hydrostatic pressure has previously been +shown to be an effective tool in tuning the electronic +structure without introducing any additional disorder +and was successfully used to tune Weyl points closer to +the Fermi energy in certain topological materials [47–50]. +Furthermore, application of pressure is known to system- +atically modify the magnetic properties in Ce-based ma- +terials [51]. Here, we use hydrostatic pressure as a tool +to investigate the origin of the features characteristic of +the nontrivial topological behavior in CeAlSi, focusing +on longitudinal and Hall transport experiments and on +quantum oscillation measurements. +We combine these +with ab initio density functional theory (DFT) calcula- +tions and phenomenological models for scattering of Weyl +fermions off magnetic domain walls to shed light on our +unusual observations. +II. +METHODS +Single crystals of CeAlSi and LaAlSi were grown by +the Al-flux technique similar to [52]. +High purity el- +ements with starting composition Ce [La] (99.99%) : +Al (99.999+%) : Si (99.999+%), 1 : 20 : 1, were place into +an alumina crucible and sealed in an evacuated quartz +tube. The samples were heated to 1200◦C, kept at this +temperature for 15 hours and cooled down to 720◦C at +2◦C/h. The excess of Al was removed by spinning the +tube upside down in a centrifuge. The crystal structure +was confirmed by x-ray powder diffraction. Energy dis- +persive x-ray spectroscopy shows, within the experimen- +tal uncertainty, a Ce:Al:Si proportion of 1 : 1 : 1. +Electrical transport experiments were carried out by a +four-probe configuration using a low-frequency AC resis- +tance bridge. Temperatures down to 1.8 K and magnetic +fields up to 9 T were achieved in a physical property +measurement system (PPMS, Quantum Design) and in a +liquid helium cryostat (Janis). Magnetization measure- +ments were conducted in a magnetic property measure- +ment system (MPMS, Quantum Design). Pressures up to +2.7 GPa (electrical transport) and 1 GPa (magnetization) +were generated using self-contained piston-cylinder-type +pressure cells using silicon oil as pressure transmitting +medium. A piece of lead (tin) served as manometer. +Density functional theory (DFT) calculations were per- +formed with the local density approximation functional +(LDA) and projector-augmented wave (PAW) method as +implemented in the Abinit software package [53], using +Jollet-Torrent-Holzwarth (JTH) pseudopotentials [54]. +Spin-orbit coupling (SOC) and non-collinear magnetism +are taken into account. An on-site Coulomb interaction +with U = 6 eV was added for the Ce f electrons within +the LDA+U scheme. We use a 16 × 16 × 16 Monkhorst- +Pack k-point grid and a plane-wave energy cutoff of +25 hartree. The lattice parameters and relevant inter- +nal atomic coordinates were optimized at respectively +0 GPa and 3 GPa until all forces on the atoms were +below 10−6 hartree/bohr3. At 0 GPa (3 GPa), we ob- +tain a = 7.926 bohr (a = 7.832 bohr) and c = 27.397 bohr +(c = 27.192 bohr). +III. +RESULTS +Temperature – pressure phase diagram +At ambient pressure, the FM ordering transition in +CeAlSi is marked by a singular magnetization M(T) and +sharp drop in electrical resistivity as a function of tem- +perature ρ(T) at TC ≈ 8 K. Figure 1 shows the effect of +external pressure on the magnetic phase (see Appendix +A for additional data). Application of pressure linearly +enhances TC(p) with a slope of 0.62(2) K/GPa, driving +TC from 7.8 K at ambient pressure to 9.4 K at 2.7 GPa +(values taken from the resistivity data). More important +2 +4 +6 +8 +10 +12 +14 +0 +1 +2 +3 +0 +1 +2 +3 +7.6 +8.0 +8.4 +8.8 +9.2 +9.6 +-6 -4 -2 +0 +2 +4 +6 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 + p (GPa) + 0 + 0.53 + 0.98 +M (103emu/mol) +T (K) +CeAlSi +(a) +H || [100] +40 +50 +60 +70 +80 + p (GPa) + 0.1 + 1.7 + 0.6 + 2.2 + 1.1 + 2.7 +ρ (µΩcm) +TC (K) +p (GPa) + χ[100] +(b) + ρ s1 + ρ s1 - 2nd run + ρ s2 + p (GPa) + 0 + 0.38 + 0.53 + 0.81 + 0.98 +M (µB/f.u.) +� 0H (T) +T = 2 K +(c) +H || [100] +FIG. 1. +(a) Magnetization (M) (left axis), obtained in an +applied field of 50 mT along the [100] crystal axis, and elec- +trical resistivity (right axis) as a function of temperature for +selected pressures. (b) Temperature–pressure phase diagram. +The solid lines are linear fits. +(c) Magnetization measure- +ments for several pressures. + +3 +Γ +Σ N Σ1 +Z +Γ +X +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +E-EF (eV) +0 +10 +20 +DOS (states/eV) +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +0GPa +3GPa +0GPa +3GPa +Ce-f +FIG. 2. Electronic bands and DOS at ambient pressure (blue) and 3 GPa (red). The hatched region of right panel corresponds +to the partial DOS associated with Ce f states. +is our finding that, for different pressures, the in-plane +magnetization curves M(H) at 2 K as a function of the +applied magnetic field along the [100] direction lie on +top of each other. This result indicates a negligible pres- +sure effect on the non-collinear planar magnetic structure +found at ambient pressure [36]. +Electronic band structure +Our DFT calculations at ambient pressure and 3 GPa, +which incorporate spin-orbit coupling and non-collinear +magnetic order, reveal only a negligible effect of pres- +sure on the electronic band structure and the electronic +density of states (DOS) at the Fermi energy [see Fig 2] +as well as on the ordered moments and their orienta- +tion. The bands contributing to the hole pockets at the +Fermi surface (FS) barely display any variation of their +intercepts of the Fermi energy in k-space, suggesting a +negligible variation of the FS area (see Appendix D for +further information). The only noticeable modification +of the electronic structure is a small shift of the bands +associated with Ce f-electrons to higher energies with re- +spect to the Fermi energy. As these bands lie about 2 eV +below the Fermi level, they most likely do not directly +contribute to the transport properties. +Quantum oscillations +Next we turn to the results of our QO measurements. +Longitudinal conductivity data σxx well above TC at +T = 15 K and in the FM state at T = 2 K at several +pressures are presented in the inset of Fig. 3(a), where +we have subtracted a smooth background yielding ∆σxx. +Within the investigated field range, the ∆σxx data as +well as its fast Fourier transform (FFT) analysis reveals +Ds +xx + (kS/m) +1/(m +0 +H) (T +-1 +) +2 K +15 K +FFT Amplitude (arb. units) +T (K) + p (GPa) + 0 + 2.2 + 1.2 + 2.6 +CeAlSi +H || [001] +(a) + p (GPa) + 0 + 2.2 +N +1/(m +0 +H) (T +-1 +) +(b) +H || [001] +T = 15 K + p (GPa) + 0 + 2.2 +N +1/(m +0 +H) (T +-1 +) +(c) +H || [001] +T = 2 K +FIG. 3. +(a) Fast Fourier transformation (FFT) amplitude +as a function of temperature at different applied pressures. +The solid lines are simulations considering the best fits using +the Lifshitz-Kosevich formula. The inset shows longitudinal +conductivity for H ∥ [001] after subtraction of a third order +polynomial background ∆σxx as a function of 1/(µ0H) at +15 K (top) and 2 K (bottom) for selected pressures. +The +curves at 2 K were shifted by −10 kS/m for clarity. (b) and (c) +Landau fan diagrams for CeAlSi at 15 and 2 K, respectively. +a single QO frequency f ≈ 20(5) T, which is found to be +independent of pressure and temperature (see Appendix +B). We notice two main features: i. the amplitude of +the oscillations at 15 K is larger than that at 2 K and +ii. the amplitude of the oscillations is suppressed by in- +creasing pressure [see Fig. 3(a)]. Generally, the thermal +damping of the QO amplitude can be described by the + +a8.0 +( +0 +0 +-a18- +0 +S +4 +eSO +2s +r.0 +-JO +2- +0 +2 +JOS.0 +0'330 +40 +0S.0 +2S.00 +r.0 +0' +81c. +0 +S +40 +r +3 +Q4 +Lifshitz-Kosevich (LK) formula [55]. However, our FFT +signal follows the LK prediction only in the paramagnetic +(PM) region above TC [solid lines in Fig. 3(a)]. In the +FM state we observe an unusual reduction of the QO am- +plitude upon cooling. This remarkable response of the +oscillation amplitude as a function of temperature has +not been observed in any other members of the LnAlPn +family [32, 34, 36, 42, 56], and was previously reported +in just a few materials [57, 58]. In SmSb, for instance, +a sudden decrease of the Shubnikov-de Haas oscillations +takes place once the material becomes antiferromagnetic, +which was conjectured to be due to the presence of a non- +trivial Berry phase [58]. +To further analyze the QO, Landau fan diagrams are +shown in Figs. 3(b) and 3(c). Our analysis indicates a +change in the nature of the topological properties be- +tween the PM and FM phase. At 15 K in the PM phase +the intercept is around −5/8, which suggests the pres- +ence of topologically trivial charge carriers [59]. In con- +trast to that, at 2 K the intercept is −9/8, which for 3D +magnetic WSMs can be associated with linear dispersive +charge carriers and a nontrivial Berry phase [59]. +Our QO data suggest that the momentum space sep- +aration between nearby Weyl nodes with opposite topo- +logical charges gets enhanced in the FM state leading +to a change in FS topology - from one which encloses +both Weyl nodes above TC to a split FS enclosing iso- +lated well-separated Weyl nodes below TC. An enhanced +separation of the Weyl nodes in the FM phase has been +also previously found in band-structure calculations (SI +of Ref. [36]). This can explain the change in the inter- +cept in our Landau fan plot. Such a change in the FS +topology could nonetheless preserve the area of certain +extremal orbits, so that the observed QO frequency can +remain nearly unchanged (see Appendix E for details). If +the topological Fermi pockets are only weakly split below +TC, the large density of states due to proximity to the +Lifshitz transition [60] can lead to an increased scattering +rate for states on the extremal orbits, thus enhancing the +Dingle temperature and suppressing the QO amplitude +for T < TC. We note that our results cannot rule out +other possible scenarios for the suppression of the ampli- +tude of the quantum oscillations upon cooling. However, +a Lifshitz transition in the FS of CeAlSi can explain both, +the suppression of the QO and the phase shift upon en- +tering the FM phase revealed by our measurements. +Hall effect +For magnetic field along the [100] direction and cur- +rent along [010] [see sketch in Fig. 4(a)] we find a large +AHE in CeAlSi in its ferromagnetic state. +The AHE +signal has been extracted by fitting the Hall resistiv- +ity to the form ρyz(H) = R0H + ρAHE, where R0 is +the ordinary Hall effect coefficient and ρAHE = RsMx +with Rs being the anomalous Hall coefficient and Mx +being the magnetization along [100] (see Appendix C). +-1.2 +-0.9 +-0.6 +-0.3 +0.0 +0.3 +0.6 +0.9 +1.2 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0 +1 +2 +3 +0.4 +0.8 +1.2 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +-1.2 +-0.8 +-0.4 +0.0 +0.4 +0.8 +1.2 +1.6 +H || [001] + p (GPa) + 0.1 + 2.2 + 1.1 + 2.7 +ρLHE (µΩcm) +µ0H (T) +CeAlSi - s1 +T = 2 K +LaAlSi (0 GPa) +(b) +0.0 +0.2 +0.4 +0.6 +0.8 +0 +1 +2 +ρxy (µΩcm) +Diff. +∆ρAHE (µΩcm) +p (GPa) + p (GPa) + 0.1 + 1.1 + 0.6 + 2.7 +ρAHE (µΩcm) +µ0H (T) +T = 2 K +CeAlSi - s2 - H || [100] +LaAlSi (0 GPa) +(a) +FIG. 4. +(a) Anomalous Hall effect (AHE) at 2 K as a func- +tion of magnetic fields H ∥ [001] for different applied pres- +sures. The top inset shows the AHE jump as a function of +pressure and the bottom inset displays a scheme of the cir- +cuit used in this measurement. s1 and s2 denote samples 1 +and 2, respectively. (b) Loop Hall effect (LHE) at 2 K as a +function of magnetic field for H ∥ [001] for selected applied +pressures. The left inset shows the Hall resistivity measured +upon increasing (red) and decreasing (blue) magnetic field +and the difference of both curves (green) at 0.1 GPa. The +right inset displays a schematic drawing of the measurement +circuit. Data of the nonmagnetic reference material LaSiAl +at ambient pressure is shown as gray line in both panels +We confirm that an AHE is absent in the non-magnetic +analog LaAlSi [Fig. 4(a)]. +We have fitted the longi- +tudinal and ordinary Hall conductivities to a simple +two-band model to obtain information on the density +of the electron- and hole-like charge carriers and their +mobilities (see Appendix C). At low temperatures and +ambient pressure we find 5.9(1) × 1019 holes/cm3 and +2.5(1) × 1019 electrons/cm3. The application of exter- +nal pressure suppresses the extracted hole density only +slightly, which reaches 4.6(1)×1019 holes/cm3 at 2 K and +2.6 GPa, whereas the electrons density remains nearly +unchanged. +Moreover, the corresponding mobilities at +2 K and ambient pressure are about 1.4(1)×103 cm2/Vs +(3.2(1)×103 cm2/Vs) for holes (electrons). These values +are nearly unaffected by application of external pressure +and are on the same order of magnitude compared with +other Weyl semimetals [61, 62]. Application of external +pressure suppresses the jump of the AHE (defined as dif- + +As +B +ca +p ++ +c +B5 +ference in ρAHE between positive and negative magnetic +fields) up to 1.5 GPa [top inset of Fig. 4(a)]. +Above +1.5 GPa the anomalous Hall jump saturates to around +0.5(1) µΩcm. As we have shown above, the M(H) curves +taken at different pressures fall on top of each other [see +Fig. 1(b)], suggesting the absence of changes in the mag- +netic structure as a source for the suppression of the +AHE. Moreover, the electronic bands close to the Fermi +level are only slightly affected by pressure [see Fig. 1(c)], +making it unlikely that this significant decrease results +from a pressure-induced change in the position of the +Weyl nodes. We find that while Rs scales nearly linearly +with ρxx for pressure ≲ 1 GPa, the scaling deviates sig- +nificantly from linear behavior at higher pressures (see +Appendix C). A linear relation between Rs and ρxx sug- +gests that the observed AHE at ambient pressure has +a significant extrinsic skew-scattering contribution [63], +which gets suppressed at high pressures. Given the ro- +bustness of the electronic structure and magnetic order +against pressure, the most plausible explanation for this +is a pressure-dependent change in the nature or distri- +bution of domain walls. Previous work has shown that +Weyl fermions can undergo skew scattering from mag- +netic domain walls which contain the axis of the average +magnetization, leading to an extrinsic contribution to the +AHE qualitatively consistent with our data [64]. +An even more unusual Hall response is observed for +field applied along [001] [see sketch in Fig. 4(b)]. +We +note that in this geometry the magnetic field is applied +perpendicular to the ferromagnetically ordered moments +in the ⟨001⟩ plane and this Hall response thus cannot +arise from the bulk in-plane magnetization. +This so- +called loop Hall effect (LHE) is displayed in Fig. 4(b). +It is only observed in the ferromagnetic regime, displays +hysteresis even in the absence of any observable Mz mag- +netization hysteresis, and is absent in the non-magnetic +analog LaAlSi. +ρLHE(H) is obtained by recording the +Hall resistivity ρxy for H ∥ [001] upon increasing and +decreasing magnetic field and taking the difference be- +tween both curves, as shown in the left inset of Fig. 4(b) +for 0.1 GPa as an example. Similar to the AHE, the ap- +plication of external pressure leads to a decrease in the +LHE [see Fig. 4(b)]. While the existence of the LHE in +CeAlSi has been argued to be tied to the presence of the +Weyl nodes near the Fermi energy [36], no clear physical +mechanism has been provided for its origin. +IV. +DISCUSSION +In the following we present a simplified model for Weyl +fermions in a non-centrosymmetric FM, and show that a +domain wall scattering mechanism, similiar to that previ- +ously explored to understand the AHE [64], can also lead +to the LHE for domain walls which are perpendicular to +the average magnetization. Our key idea here is that the +bulk magnetic domains in CeAlSi host an in-plane mag- +netization with hard axis along [001], so that the out-of- +(b) +x +y +z +Domain wall +MDW(x) = M sin q(x) +(a) +Z +(c) +MDW <0 +Z +MDW >0 +Z +FIG. 5. +(a) Domain wall between two bulk magnetic +domains showing twisted magnetization configuration with +M DW +z +(x) = M sin θ(x). (b) Skew scattering of Weyl fermions +off hysteretic domain wall loops, with red regions having +M DW +z +> 0 and blue having M DW +z +< 0, provides a mech- +anism for the loop Hall effect (LHE). The black arrows are +classical representations of the Weyl fermions trajectories. (c) +Calculated LHE shown as a ratio of Landauer conductances, +versus the maximal out of plane tilt angle θ of the domain +wall magnetization. +plane bulk contribution to the field-induced magnetiza- +tion M bulk +z +is not expected to be hysteretic as we tune the +magnetic field Hz. We instead argue that the hysteretic +LHE must be attributed to the hysteretic domain wall +magnetization M DW +z +as we tune Hz, as schematically de- +picted in Fig. 4(a) and 4(b). Our calculations show that +the intra-node skew scattering of Weyl fermions as they +cross a domain wall with nonzero M DW +z +can explain the +LHE. +To illustrate this physics, we study a model with 4 +pairs of Weyl nodes in the kz = 0 plane (see Appendix E +for details). These could be viewed as a caricature of the +W ′ +3 Weyl nodes found to lie ∼ 46 meV above the Fermi +level close to the kz = 0 plane in CeAlSi [36, 43]. For a +single Weyl node, with chirality +1, we consider a simple +linearized Hamiltonian: +H+ = vF σiG(+) +ij qj + σiMi +(1) +where vF is the nodal velocity, σ is the spin Pauli matrix, +q denotes the momentum relative to the Weyl node po- +sition, and the tensor Gij is chosen to yield an elliptical +Fermi surface at fixed kz with its major axis rotated away +from the x, y axes. The Hamiltonian for the other Weyl +nodes can be reconstructed using symmetries. The Weiss +field M is nonzero in the magnetically ordered phase and +tunes the momentum of the Weyl node; we assume this +leads to topologically nontrivial FS pockets enclosing sin- +gle Weyl nodes. For a magnetic field Hz applied along +the z-axis, the system will support domains of M, with +magnetization aligned along different in-plane directions, +which are separated by domain walls. As shown in [45], +the domain wall magnetization in such cases supports an +out-of-plane component M DW +z +. Fig. 5(c) shows the com- +puted Hall conductance scaled by the longitudinal con- + +E +2 +(10-3) +1 +xx6/ +0 +-1 +9 +-2 +3 +0.50 +0.25 +0.00 +0.25 +0.50 +u/e6 +ductance showing that it is an odd function of M DW +z +(see +Appendix F for details). Crudely, this small Landauer +conductance [65, 66] ratio is expected to be related to the +ratio of loop Hall to longitudinal resistivity - our exper- +iments show that ρLHE +xy +/ρxx ∼ 10−3, in reasonable agree- +ment with the theoretical estimate in Fig. 5. We thus +propose that the mechanism for the aptly named LHE +is the skew scattering of Weyl fermions off hysteretic do- +main wall loops or surfaces. Since we expect the domain +wall magnetization M DW +z +≪M bulk +z +, the hysteretic behav- +ior of M DW +z +cannot be resolved in bulk magnetization +measurements. Our model might also help to understand +the observation of a similar LHE reported previously in +other compounds, in which Weyl fermions were predict +to exist [67–69]. +V. +CONCLUSIONS +In summary, our study emphasizes, through a key tun- +ing parameter (hydrostatic pressure), the importance of +ferromagnetism for the low temperature topological fea- +tures in CeAlSi. Our QO data show a difference in the +Berry phase above and below TC, indicating that FM or- +dering shifts oppositely charged Weyl nodes away from +each other in momentum space, leading to a change in +the Fermi-surface topology. +We have argued that this +also leads to an increase in the scattering rate below TC, +and thus to a drop in the amplitude of Shubnikov – de +Haas oscillations in contrast to the conventional LK for- +mula. This result calls for angular dependent Shubnikov +– de Haas and de Haas – van Alphen experiments in +an extended field range. We have also discovered pres- +sure dependent changes in the AHE and LHE below TC. +Since our DFT calculations indicate that the electronic +band structure is robust against pressure, we argue that +these changes in the AHE and LHE must arise from dif- +ferences in domain wall defects and we have shown how +Weyl fermions scattering off hysteretic domain walls can +lead to the LHE. +DATA AVAILABILITY +Data that underpin the findings of this study are avail- +able at Edmond – the open research data repository of +the Max Planck Society at [70]. +ACKNOWLEDGMENTS +We +acknowledge +fruitful +discussions +with +A. +P. +Mackenzie. +We also thank U. Burkhardt for carry- +ing out energy dispersive x-ray analysis on the sam- +ples. +This project has received funding from the +European Union’s Horizon 2020 research and inno- +vation programme under the Marie Sk�lodowska-Curie +grant +agreement +No +101019024. +This +work +was +also +supported +by +the +S˜ao +Paulo +Research +Foun- +dation (FAPESP) grants 2017/10581-1, +2018/11364- +7, +2020/12283-0, +CNPq +grants +# +304496/2017-0, +310373/2019-0 and CAPES, Brazil. This research was +financially supported by the Natural Sciences and En- +gineering Research Council of Canada (NSERC), under +the Discovery Grants program grant No. RGPIN-2016- +06666. Computations were made on the supercomput- +ers Beluga and Narval managed by Calcul Qu´ebec and +the Digital Research Alliance of Canada. The operation +of these supercomputers is funded by the Canada Foun- +dation for Innovation, the Minist`ere de la Science, de +l’´Economie et de l’Innovation du Qu´ebec, and the Fonds +de recherche du Qu´ebec – Nature et technologies. V.B.- +C. and M.C are members of the Regroupement qu´eb´ecois +sur les mat´eriaux de pointe (RQMP). +APPENDICES +APPENDIX A: ELECTRICAL RESISTIVITY +Figures 6(a) and 6(b) present the electrical resistivity +(ρ) as a function of temperature at several pressures for +two different samples of CeAlSi. At high temperatures +ρ(T) exhibits a metallic behavior for both samples at all +studied pressures. Moreover, a clear kink is observed at +low-temperatures characterizing the ferromagnetic tran- +sition, in good agreement with previous reports at am- +bient pressure [36, 45, 46]. A broad shoulder is observed +at around 80 K. It shows up as a maximum in the tem- +perature derivative of the electrical resistivity [see insets +of Figs. 6(a) and 6(b)]. The shape and the position of +the maximum is nearly unaffected by the application of +external pressure, suggesting that the gap between the +ground state and the first excited crystalline electrical +field state does not change with increasing pressure. +APPENDIX B: QUANTUM OSCILLATIONS +The left panels of Fig. 7 present the longitudinal con- +ductivity measured with H ∥ [001] after subtraction of a +third order polynomial (fit between 5 and 9 T) ∆σxx as +a function of 1/(µ0H). We note that quantum oscilla- +tions are clearly seen up to 40 K at all studied pressures. +Furthermore, the unusual behavior of the quantum oscil- +lations amplitudes can be seen by the naked eye. The os- +cillations in the paramagnetic state at 15 K are more pro- +nounced than the oscillations in the ferromagnetic state +at 2 K. The panels on the right side of Fig. 7 present +the Fast Fourier transformation (FFT) of ∆σxx, using a +Hamming window from 0.2 to 0.11 T−1, as a function of +frequency (f) at several temperatures for selected pres- +sures. +Only one oscillation frequency f ≈ 20(5) T is +present. It is unaffected by changes in pressure and/or +temperature. + +7 +0 +50 +100 +150 +200 +250 +300 +40 +60 +80 +100 +120 +0 +50 +100 +150 +200 +250 +300 +25 +50 +75 +100 +125 +50 +100 +150 +200 +-1 +0 +1 +2 +3 +50 +100 +150 +200 +-1 +0 +1 +2 +3 + p (GPa) + 0 + 1.7 + 0.7 + 2.2 + 1.2 + 2.6 + 1.4 + + +ρ (µΩcm) +T (K) +(a) CeAlSi - s1 +ρ (µΩcm) +T (K) + p (GPa) + 0.1 + 2.2 + 0.6 + 2.7 + 1.1 + 1.7 +(b) CeAlSi - s2 +dρ/dT + p (GPa) + 0 + 1.4 + 2.6 +dρ/dT + p (GPa) + 0.1 + 1.7 + 2.7 +FIG. 6. (a) and (b) Electrical resistivity (ρ) as a function of temperature at several pressures for two different samples of +CeAlSi. The insets displays a magnified view of the low-temperature range. The insets show the temperature derivative of ρ +as a function of temperature at several pressures for two different samples of CeAlSi. +The effective mass (m∗) was estimated in the param- +agnetic state of CeAlSi by fitting the FFT amplitude as +a function of temperature by the Lifshitz-Kosevich (LK) +formula [55]: +RT = +αTm∗ +B sinh(αTm∗/B), +(2) +in which α = 2π2kB/eℏ ≈ 14.69 T/K, T is the tempera- +ture, B is the magnetic field and m∗ the effective mass. +As shown in Fig. 8, the application of external pressure +leads to a decrease in the value of m∗. +APPENDIX C: HALL EFFECT +Anomalous Hall Effect +Figure 9 presents the Hall resistivity (ρyz) as a function +of magnetic field at 2 K for several pressures. A linear +background was determined by performing a linear fit in +the range 0.2 T ⩽ H ⩽ 0.6 T. We obtained the anomalous +Hall (ρAHE) effect by subtracting the linear background +using ρyz = R0H + ρAHE. +As we can see in Fig. 10, the linear dependence between +the anomalous Hall coefficient (RS) and the longitudi- +nal resistivity (ρxx) characterizes the presence of a skew +scattering contribution to the AHE in CeAlSi at ambi- +ent pressure [63]. The observation of this contribution in +a good metal regime can be attributed to domain wall +scattering of Weyl fermions (see Sec. IV), as this contri- +bution should be the dominant one in highly conducting +samples (σxx ⩾ 0.5 × 106 (Ωcm)−1) [63]. Furthermore, +the application of external pressure suppresses the lin- +ear relation between Rs and ρxx, which is better seen in +the inset of Fig. 10, where the exponents obtained with +allometric fits (RS = a + bρn +xx) are shown as a function +of pressure. One can clearly see the increase of the ex- +ponent n as a function of increasing pressure, reaching +1.21(1) at 1.1 GPa, indicating that the skew scattering +contribution of the AHE from the domain walls is be- +ing suppressed by application of pressure. The domains +themselves (bulk) also contribute to the AHE. It is pos- +sible to differentiate both contributions, as analyzed in +great detail in Ref. [64], by considering that the domain +wall scattering contribution to the AHE is limited by the +electron mean free path, whereas the bulk contribution +is not. The total Hall resistivity is therefore an average +between the bulk and domain wall contributions. Our +results suggests that at low pressures (p ⩽ 1.5 GPa) the +AHE is dominated by the skew scattering contribution +coming from the domain walls, while in the high-pressure +range (p ⩾ 1.5 GPa), where the AHE is not skew scat- +tering type, it is dominated by the contribution of the +domains. +Two-band model fits +To accurately estimate the carrier densities and mobil- +ities of CeAlSi, we have simultaneously fit the longitudi- +nal (σxx) and the Hall (σxy) conductivities considering a +two-band model described by: +σxx = e +� +neµe +1 + µ2e (µ0H)2 + +nhµh +1 + µ2 +h (µ0H)2 +� +σxy = e (µ0H) +� +neµ2 +e +1 + µ2e (µ0H)2 − +nhµ2 +h +1 + µ2 +h (µ0H)2 +� +, +where n denotes the electron (e) and hole (h) carrier den- +sities, and µe and µh are the electron and hole mobilities, +respectively. Figure 11(a) presents a representative plot +of the fits at 2 K and 1.2 GPa, in which a good agreement +between the experimental data and the fits is observed. +Figure 11(b) displays the carrier densities as a function of + +8 +0 +40 +80 +120 +160 +200 +0 +1 +2 +3 +4 +0 +50 +100 +150 +200 +0 +1 +2 +3 +4 +5 +6 +7 +8 +0.12 +0.14 +0.16 +0.18 +0.20 +-20 +-15 +-10 +-5 +0 +5 +10 +15 +0 +40 +80 +120 +160 +200 +0 +1 +2 +3 +0.12 +0.14 +0.16 +0.18 +0.20 +-6 +-4 +-2 +0 +2 +4 +0.12 +0.14 +0.16 +0.18 +0.20 +-8 +-6 +-4 +-2 +0 +2 +4 +0.12 +0.14 +0.16 +0.18 +0.20 +-12 +-8 +-4 +0 +4 +8 +0 +40 +80 +120 +160 +200 +0 +1 +2 +3 +4 +5 +6 +FFT Amplitude ∆σxx +f (T) + T (K) + 2 + 12.5 + 3.5 + 15 + 5 + 20 + 7.5 + 30 + 10 + 40 +CeAlSi - s1 +H || [001] +p - 2.2 GPa +FFT Amplitude ∆σxx +f (T) + T (K) + 2 + 15 + 3.5 + 20 + 5 + 30 + 7.5 + 40 + 10 +CeAlSi - s2 H || [001] p - 0 GPa +∆σxx (kS/m) +1/(µ0H) (T-1) + T (K) + 2 + 7.5 + 20 + 3.5 + 10 + 30 + 5 + 15 + 40 +CeAlSi - s2 +H || [001] +p - 0 GPa +FFT Amplitude ∆σxx +f (T) + T (K) + 2 + 12.5 + 3.5 + 15 + 5 + 20 + 7.5 + 30 + 10 + 40 +CeAlSi - s1 H || [001] +p - 2.6 GPa +∆σxx (kS/m) +1/(µ0H) (T-1) + T (K) + 2 + 7.5 + 15 + 40 + 3.5 + 10 + 20 + 5 + 12.5 + 30 +CeAlSi - s1 H || [001] p - 2.6 GPa +∆σxx (kS/m) +1/µ0H (T-1) + T (K) + 2 + 7.5 + 15 + 40 + 3.5 + 10 + 20 + 5 + 12.5 + 30 +CeAlSi - s1 H || [001] p - 2.2 GPa +∆σxx (kS/m) +1/(µ0H) (T-1) + T (K) + 2 + 7.5 + 20 + 3.5 + 10 + 30 + 5 + 15 + 40 +CeAlSi - s1 H || [001] +p - 1.2 GPa +FFT Amplitude ∆σxx +f (T) + T (K) + 2 + 15 + 3.5 + 20 + 5 + 30 + 7.5 + 40 + 10 +CeAlSi - s1 H || [001] +p - 1.2 GPa +FIG. 7. (Left panels) Longitudinal conductivity after subtraction of a third order polynomial ∆σxx as a function of 1/(µ0H) +at several temperatures for selected pressures. (Right panels) Fast Fourier transformation of ∆σxx shown in the left. + +9 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.052 +0.056 +0.060 +0.064 +0.068 +m* (m +e +) +p (GPa) +CeAlSi +H || [001] +FIG. 8. Effective mass m∗ as a function of pressure for mag- +netic fields parallel to [001]. +FIG. 9. Hall resistivity (ρyz) as a function of magnetic field +applied parallel to [100] at 2 K for several pressures. +The +solid orange line is an extrapolation of a linear fit performed +in the range 0.2 T ⩽ H ⩽ 0.6 T, which yields the ordinary +background of ρyz. +temperature at several pressures. Figure 11(c) shows the +mobilities as a function of temperature at several pres- +sures. +APPENDIX D: BANDSTRUCTURE +CALCULATIONS +Figure 12 shows the electronic bands and DOS zoomed +in the vicinity of the Fermi level, to emphasize the negli- +gible effect of pressure on the bands contributing to the +AHE and LHE. Note, also that no crossing feature nor +electron pocket was found in our ambient pressure calcu- +lation along the Γ − X high symmetry path, in contrary +to Fig. 3(a) of [36]. This discrepancy could be attributed +to the different exchange-correlation functional or to our +use of theoretically relaxed lattice parameters, while [36] +used experimental values which, in the case of the PBE- +40 +50 +60 +70 +80 +90 +100 +110 +120 +130 +-250 +-200 +-150 +-100 +-50 +0 +50 + p (GPa) + 0.1 + 1.7 + 0.6 + 2.2 + 1.1 + 2.7 +R +S + (mWcm/T) +r +xx + (mWcm) +H || [100] +T = 2 K +0 +1 +2 +3 +1.0 +1.5 +2.0 +2.5 +n +p (GPa) +R +S + = a + br +n +xx +FIG. 10. Anomalous Hall coefficient (RS) as a function of the +longitudinal resistivity (ρxx) at several pressures. +GGA functional used in their paper, will be smaller than +the theoretical one. Nevertheless, from the pressure de- +pendence of the electronic bands relative to the Fermi +level, an electron pocket could likely appear along this +path upon further increasing the pressure. +We further refine the analysis of the electronic struc- +ture by calculating the orbital decomposition of the +electronic wavefunction inside the atom-centered PAW +spheres for Ce 5d states (left panels, red), as well as for +Al (middle panels, green) and Si (right panels, blue) 2p +states, in the same energy range as Fig. 12. The relative +weights of the different orbitals at ambient pressure (top +panels) and 3 GPa (bottom panels) are essentially iden- +tical, thus confirming the negligible effect of pressure on +the electronic bands. +The calculated magnetic structure does not display +any significant differences between 0 and 3 GPa either, +in agreement with the experimental observations (see +Fig. 1(b) of the main text). +For 0 GPa (3 GPa), we +find a magnetic moment of 0.880 µB (0.878 µB) inside +the PAW spheres of the 2 inequivalent Ce atoms in the +unit cell. Considering that one moment points mostly in +the ˆx direction and the other in the ˆy direction with an +angle of 87.3◦ (89.8◦) between them, we find a total net +magnetic moment of 1.311 µB (1.284 µB) oriented along +[110] for the whole unit cell. Note that the net size of the +magnetic moment depends strongly on the choice of U. +APPENDIX E: SIMPLIFIED MODEL FOR WEYL +NODES +We choose a simple model for CeAlSi with 4 pairs of +Weyl nodes as shown in Fig. 14 (top left panel). We have +chosen the Weyl nodes and Fermi pockets for T > TC +to be consistent with the C4v and mirror Mx, My crys- +tal symmetries of CeAlSi, as well as time-reversal sym- +metry. These nodes crudely mimic the W ′ +3 nodes found +slightly above the Fermi level in previous ab initio elec- + +6 +CeAISi - s2 - H Il [100] +T= 2 K +4 +(woun) +2 +0 +Pyz +p (GPa) +0.1 +-2 +0.6 +b +1.1 +-4 +1.7 +B +2.2 +Vy +-6 +2.7 +-1.0 -0.8 -0.6 -0.4 -0.2 +¥0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +μoH (T)10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +0 +500 +1000 +1500 +2000 +2500 +3000 +0 +50 +100 +150 +200 +250 +300 +0 +4 +8 +12 +16 +0 +50 +100 +150 +200 +250 +300 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +σ (kS/m) +µ0H (T) + σxx + σxy + Two-band fits +CeAlSi - s1 p = 1.2 GPa +T = 2 K, H || [001] +(a) +n (1019 cm-3) +T (K) + p (GPa) + 0.1 + 1.7 + 0.7 + 2.2 + 1.2 + 2.6 + 1.4 +H || [001] +CeAlSi - s1 +open - holes +solid - electrons +(b) +µ (103 cm2/Vs) +T (K) + p (GPa) + 0.1 + 0.7 + 1.2 + 1.4 + 1.7 + 2.2 + 2.6 +CeAlSi - s1 +H || [001] +open - holes +solid - electrons +(c) +FIG. 11. (a) Longitudinal (σxx) and Hall (σxy) conductivities at 2 K and 1.2 GPa. Carrier densities (b) and mobilities (c) +obtained from the two-band fits as a function of temperature at several pressures. +tronic structure calculations. +With the onset of mag- +netism, the Weyl nodes get displaced with opposite chi- +rality nodes being displaced in opposite directions. As +shown in Fig. 14(top right panel), this can lead to a topo- +logical transition of the Fermi surface where each pocket +now encloses a single Weyl node. +At the same time, +for the type of Fermi surface sketched above, certain +extremal orbits can remain unchanged in area (dashed +lines), so that the QO frequency will be unaffected as +observed. +If the Weyl nodes are not widely separated +even after the topological Fermi surface phase transition, +proximity to a Lifshitz transition may lead to an enhance- +ment of the electron scattering rate (in the presence of +weak disorder) due to a large density of states, which can +enhance the Dingle temperature and explain the strong +observed deviation from the Lifshitz-Kosevich formula. +Within the symmetry broken for T < TC, it is reason- +able to consider the physics of isolated Weyl nodes. We +model a single Weyl node as having an elliptical pocket +with velocity tensor and a simple coupling to the Weiss +field from the magnetization. +H+ = σiG(+) +ij ˜qj + σiMi, +(3) +G(+) +ij += +� +� +� +� +|a| +1 +|a| +1 +� +� +� +� +� +� +� +� +cos β +sin β 0 +− sin β cos β 0 +0 +0 +1 +� +� +� +� , +(4) +Here ˜qi = vF qi where q denotes the momentum mea- +sured from the Weyl node location, and vF is a velocity +scale. The real matrix, Gij, defined this way results in +an ellipsoidal Fermi surface, whose xy-plane cross section +has an elliptical shape with the major and minor axis, |a| +and 1/|a| respectively for |a| > 1, and the major axis is +rotated from the qy-axis by the angle β. For |a| < 1, the +major axis is instead rotated from the qx axis by β. +Position of Weyl points: When Mi = 0, the Weyl +point resides at q = 0. When Mi ̸= 0, the Weyl point +shifts to the point satisfying the following equation +˜q∗ +i = − +� +G(+)�−1 +ij Mj +(5) + +11 +Γ +Σ N Σ1 +Z +Γ +X +-1 +0 +1 +E-EF (eV) +0 +2 +4 +6 +8 +DOS (states/eV) +-1 +0 +1 +0GPa +3GPa +0GPa +3GPa +FIG. 12. Electronic bands and DOS at ambient pressure (blue) and 3 GPa (red), zoomed in the vicinity of the Fermi level. +N +1 +Z +X +1.0 +0.5 +0.0 +0.5 +1.0 +E-EF (eV) +Ce-5d, 0GPa +N +1 +Z +X +Al-2p, 0GPa +N +1 +Z +X +Si-2p, 0GPa +(a) +N +1 +Z +X +1.0 +0.5 +0.0 +0.5 +1.0 +E-EF (eV) +Ce-5d, 3GPa +N +1 +Z +X +Al-2p, 3GPa +N +1 +Z +X +Si-2p, 3GPa +(b) +FIG. 13. Orbital decomposition of the electronic wavefunction at (a) ambient pressure and (b) 3 GPa, zoomed in the vicinity +of the Fermi level. + +12 +Eigenspectrum: The eigenvalues of H+ are given by +E = ± +� +˜qi[G(+)]T +ij[G(+)]jl˜ql + 2Mi[G(+)]ij ˜qj + MiMi +(6) +Written in the form which is useful for numerics: +0 = ˜q2 +x +� +|a|2c2 + +1 +|a|2 s2 +� ++˜qx +� +2cs +� +|a|2 − +1 +|a|2 +� +˜qy + 2 +� +|a|cMx − 1 +|a|sMy +�� ++ +� +˜q2 +z + 2Mz ˜qz + +� +|a|2s2 + +1 +|a|2 c2 +� +˜q2 +y ++ 2 +� +|a|sMx + 1 +|a|cMy +� +˜qy + M 2 − E2 +� +, +(7) +where s ≡ sin β and c ≡ cos β. The quadratic equation +allows us to determine the mover modes given the Fermi +energy and the Weiss field. If the ˜qx solutions are real- +valued, we obtain travelling waves; the complex-valued +solutions correspond to evanescent waves. +T > Tc +T < Tc +kx +ky +kz +kz +ky +kx +kx +ky +T < Tc +FIG. 14. +Top: +Illustrative example of 4 topologically +trivial Fermi surface pockets for T +> Tc, each enclos- +ing a pair of Weyl points (WP) with opposite topological +charge, located at momenta (K0, ±K1, 0), (−K0, ±K1, 0), +(±K1, K0, 0), (±K1, −K0, 0). For T < Tc, the in-plane mag- +netization leads to a momentum space displacement of the +Weyl points, leading to transition into 8 topologically non- +trivial Fermi pockets. +The area of certain maximal orbits +(dashed lines) can remain unchanged across this transition. +Bottom: Projected view of the elliptical cross sections of the +topologically nontrivial Fermi surfaces for T 0, +1 +√ +2d(d+d3) +� +id2 − d1 +d3 + d +� +, for E = −d < 0. +(8) +The group velocity for a mover are given by +vi = +� +G(+)�T +ij +�� +G(+)� +jl ˜ql + Mj +� +E +(9) +Negative-chirality node: With M = 0, we can use +C4v, time-reversal, and mirror symmetries Mx, My to +write out the Hamiltonian for all 8 Weyl nodes. For in- +stance a negative chirality node is obtained under a mir- +ror operation, where we can relate the g-tensor part of +the Hamiltonian H(+) to the g-tensor part of H(−). For +the Weyl point related to the original one by a mirror +My, we have +H(−) = σiG(−) +ij ˜qj + σiMi, +(10) +G(−) = − +� +� +� +� +|a| +1 +|a| +1 +� +� +� +� +� +� +� +� +cos β − sin β 0 +sin β +cos β +0 +0 +0 +1 +� +� +� +� . +(11) +The distinctions from G(+) are (i) the prefactor -1 which +leads to the negative determinant and (ii) the sin β which +used to be − sin β in G(+). +The latter amounts to a +rotation of the Fermi surface about qz-axis by −β instead +of β in H(+). All the formulae derived in earlier in this +section can be straightforwardly generalized for H(−). +Choice of parameters: As an illustrative example, we +choose |a| = 0.5 and β = π/4. This results in ellipti- +cal cross-sections (at any given kz) for the Fermi sur- +faces near a Weyl point with major : minor axis ratio of +4 : 1. The major axis of the ellipse is rotated by π/4, +so that it points along the 45◦ direction in the (kx, ky) +plane. We also choose other parameters to be reason- +able values in line with the ab initio calculations, namely +vF = 500 meV˚Aand chemical potential µ = −30 meV +(below the Weyl node). This leads to Fermi pockets with +a typical size kF ∼ 0.06 ˚A−1. We fix the Weiss field to +have a magnitude |M| = |EF |/4. We note that our re- +sults do not change qualitatively if we choose somewhat +different parameters - however, it is important that the +elliptical Fermi pockets are not aligned along the tetrag- +onal x or y axes (see Fig. 14). +APPENDIX F: MODELING THE DOMAIN WALL +We assume a domain wall width N × w = 40 nm (cor- +responding to N = 40.) We consider a domain wall be- + ++13 +x +y +z +Mz > 0 +Domain wall +I +II +III +FIG. 15. +Evolution of the magnetization across a do- +main wall. +Region I and region III indicate bulk do- +mains, and region-II is the domain wall region. +Going +across the domain wall, the magnetization vector twists, with +the perpendicular domain wall magnetization M DW +z +> 0 +for the depicted configuration. +We will denote M(x) = +M(cos θ(x) cos γ(x), cos θ(x) sin γ(x), sin θ(x)). In region-I, we +choose θ = 0, γ = π/4, while we set θ = 0, γ = −π/4 in region +III. In region II, we assume a twisting magnetization profile, +with the maximum out of plane component determined by +θmax which is achieved at the center of region II. +kx +ky +kx +ky +REGION-I +REGION-III +FIG. 16. Illustration of the Fermi surfaces and the domain +wall induced intra-node scattering (dashed arrow). For sim- +plicity, we have not shown the displacements of the Fermi +pockets relative to each other in the two domains or their dif- +ference in spin textures, but this is taken into account in our +calculations as given below. +tween a left and a right region (see illustration in Fig. 15) +with the following Weiss fields, respectively, +MI = M (cos γ, sin γ, 0)T , +(12) +MIII = M (cos γ, − sin γ, 0)T . +(13) +The domain wall region, region II, has a width Nw, which +is partitioned into N intervals, each with width w. The +Weiss field in j-th interval is given by +Mj = M (cos θj cos γj, cos θj sin γj, sin θj)T , +(14) +γj = γ − 2γ j − 1/2 +N +, +(15) +θj = +� +1 − 2|j − 1 +2 − N +2 | +N +� +θ, +(16) +where θ is the angle at the center of region II. θj mono- +tonically decreases away from the center of region II. A +large N models a smooth variation of the Weiss field in +region II. +A. +Transmission and reflection coefficients +The domain wall will lead to a scattering between Weyl +Fermi surfaces. For simplicity, we assume a smooth do- +main wall and only take into account the intra-node scat- +tering as shown in Fig. 16. We now sketch the compu- +tation of the transmission coefficient (TC) and reflection +coefficient (RC) at the domain wall, defined earlier. For +concreteness, we show the calculation for H(+). The step- +by-step summary is given below +• For a given Fermi energy E and the parallel mo- +menta (qy, qz), we compute the x-momenta for all +the regions. +• Compute the eigenfunctions +• Wave function in each region is a superposition of +a left mover and a right mover, except in Region +III, where the wave function consists of only a right +mover +ΨI = χReiqR·x + rχLeiqL·x +(17) +ΨII,j = c(j) +1 η(j) +1 eip(j) +1 +·x + c(j) +2 η(j) +2 eip(j) +2 +·x, +(18) +ΨIII = tξReikR·x, +(19) +where t and r are the transmission and reflection +amplitude respectively. +• We then match the wave function at each boundary +at xj = jw for j = 0, 1, · · · , N to determine r, t +and c(j) +1,2. This can be formulated in transfer matrix +form. This can be seen below. + +14 +χR + rχL = c(1) +1 η(1) +1 ++ c(1) +2 η(1) +2 , +(20) +c(1) +1 η(1) +1 eip(1) +1x w + c(1) +2 η(1) +2 eip(1) +2x w = c(2) +1 η(2) +1 eip(2) +1x w + c(2) +2 η(2) +2 eip(2) +2x w +(21) +c(2) +1 η(2) +1 eip(2) +1x 2w + c(2) +2 η(2) +2 eip(2) +2x 2w = c(3) +1 η(3) +1 eip(3) +1x 2w + c(3) +2 η(3) +2 eip(3) +2x 2w +(22) +c(j) +1 η(j) +1 eip(j) +1x jw + c(j) +2 η(j) +2 eip(j) +2x jw = c(j+1) +1 +η(j+1) +1 +eip(j+1) +1x +jw + c(j+1) +2 +η(j+1) +2 +eip(j+1) +2x +jw +(23) +c(N) +1 +η(N) +1 +eip(N) +1x Nw + c(N) +2 +η(N) +2 +eip(N) +2x Nw = tξReikRxNw. +(24) +Rewriting in matrix form +� +χR χL +� +� +�1 +r +� +� = +� +η(1) +1 +η(1) +2 +� +� +�c(1) +1 +c(1) +2 +� +� , +(25) +� +η(j) +1 eip(j) +1x jw η(j) +2 eip(j) +2x jw +� +� +�c(j) +1 +c(j) +2 +� +� = +� +η(j+1) +1 +eip(j+1) +1x +jw η(j+1) +2 +eip(j+1) +2x +jw +� +� +�c(j+1) +1 +c(j+1) +2 +� +� +(26) +� +η(N) +1 +eip(N) +1x Nw η(N) +2 +eip(N) +2x Nw +� +� +�c(N) +1 +c(N) +2 +� +� = tξReikRxNw. +(27) +From above, we can solve for r and t by the transfer matrix T (j): +� +�1 +r +� +� = teikRxNw � +χR χL +�−1 +T (1) · · · T (N)ξR ≡ t +� +�u1 +u2 +� +� +(28) +T (j) = +� +η(j) +1 eip(j) +1x (j−1)w η(j) +2 eip(j) +2x (j−1)w +� � +η(j) +1 eip(j) +1x jw η(j) +2 eip(j) +2x jw +�−1 +, +(29) +where, in the definition of the transfer matrix, the two +matrices differ from each other, apart from the inverse +operation, by the phase factors: one involves (j − 1)w, +whereas the other involves jw. We finally obtain +t = 1/u1 +(30) +r = u2/u1. +(31) +TC and RC are given by +TC = |vx,trans| +|vx,inc| |t|2, +(32) +RC = |vx,refl| +|vx,inc| |r|2. +(33) +The longitudinal conductance gxx and the transverse +conductance gyx are then computed using TC and RC +[64] (see also Refs. 71 and 72.) +B. +Results: case |a| = 0.5, β = π/4 +1. +Parameters +The xy-plane cross section of the Fermi surface near a +Weyl point is an ellipse whose major axis is rotated by +π/4. +The results corresponds to the following parameters: +• Fermi velocity vF = 500 meV.˚A +• Fermi wave vector kF ∼ 0.06 ˚A−1 +• Fermi energy EF = −30 meV +• Weiss field |M| = 0.015 a.u., which corresponds to +|EF |/4. +• Domain wall width N × w = 40 nm for N = 40. +2. +Results +We will show results of the anomalous Hall contribu- +tion obtained by antisymmetrizing the off-diagonal con- +ductance: gA +yx(θ) = +1 +2(gyx(θ) − gyx(−θ)), namely anti- +symmetrizing w.r.t merely reversing the Mz component +of the Weiss field. Figure 17 shows gA +yx for the 4 pairs +of WPs: (i) Fermi surfaces in each pair are related by +either Mx or My mirror operation at zero Weiss field +(see Fig. 14), and (ii) different pairs are related by a C4v +rotation at zero Weiss field (see Fig. 14.) +A few main results are summarized below: +(1) AHE contributions from the two Fermi surfaces in +each pair has opposite signs, yet they do not cancel +each other out, so AHE is still non vanishing. + +15 +(2) AHE from the four WPs related by C4z rotations +has the same sign (see Fig. 17.) +(3) Interchanging region I and region III leads to the +same θ-dependence of gA +yx (see Fig. +18.) +This +suggests that as long as the total z-component of +the Weiss field does not vanish, AHE contribution +from the domain wall is non-zero. +(4) Ratio gA +yx/gxx is of the order 10−3 at small angle, +see Fig. 5(c). +0.50 +0.25 +0.00 +0.25 +0.50 +/ +15 +10 +5 +0 +5 +10 +15 +gA +yx [a.u.] +WP Pair 1 +WP Pair 2 +WP Pair 3 +WP Pair 4 +FIG. 17. 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Lett. 113, 172405 +(2018). + diff --git a/bdFST4oBgHgl3EQfCjh5/content/tmp_files/load_file.txt b/bdFST4oBgHgl3EQfCjh5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fbf586ca4e8bbcafc3f5867252e563d7da41384 --- /dev/null +++ b/bdFST4oBgHgl3EQfCjh5/content/tmp_files/load_file.txt @@ -0,0 +1,1784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf,len=1783 +page_content='Topological features in the ferromagnetic Weyl semimetal CeAlSi: Role of domain walls M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Piva,1, ∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Souza,2, 1, † V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Brousseau-Couture,3 Sopheak Sorn,4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Pakuszewski,2 Janas K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' John,1 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Adriano,2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Cˆot´e,3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Pagliuso,2, 5 Arun Paramekanti,6, 7 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Nicklas1, ‡ 1Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 40, D-01187 Dresden, Germany 2Instituto de F´ısica “Gleb Wataghin”, UNICAMP, 13083-859, Campinas, SP, Brazil 3D´epartement de Physique, Universit´e de Montr´eal, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 6128, Succursale Centre-Ville, Montr´eal, Qu´ebec, Canada H3C 3J7 4Institute for Quantum Materials and Technology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany 5Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA 6Department of Physics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' George Street, Toronto, ON, M5S 1A7 Canada 7S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Bose National Centre for Basic Sciences, Block JD, Sector - III, Salt lake, Kolkata-700106, India (Dated: February 1, 2023) In the ferromagnetic (FM) Weyl semimetal CeAlSi both space-inversion and time-reversal sym- metries are broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our quantum oscillation (QO) data indicate that the FM ordering modifies the Fermi surface topology and also leads to an unusual drop in the QO amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In the FM phase, we find a pressure-induced suppression of the anomalous and the loop Hall effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This cannot be explained based on the electronic band structure or magnetic structure, both of which are nearly pressure independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Instead, we show that a simplified model describing the scattering of Weyl fermions off FM domain walls can potentially explain the observed topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our study highlights the importance of domain walls for understanding transport in FM Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' INTRODUCTION Topological phases of matter have lately received con- siderable attention, due to the experimental realization of exotic types of charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' One example is the mass- less Weyl fermions found in Weyl semimetals (WSMs) [1–3], which are characterized by remarkable electronic properties, such as surface Fermi arcs, a bulk chiral anomaly, axial–gravitational anomaly, an extremely large magnetoresistance (MR) and an anomalous Hall effect (AHE) [1, 3–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Weyl fermions can be generated by either breaking space-inversion (SI) or time-reversal (TR) sym- metry of materials with a Dirac or quadratic band touch- ing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' So far most experimentally studied WSMs break SI symmetry [7–15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' fewer examples are known for WSMs with broken TR symmetry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' magnetic WSMs [16–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Magnetic WSMs are of fundamental interest since they intertwine topology and strong correlations [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' They also offer the potential to manipulate the topological phase in a desired way, for instance using a magnetic field to tune the position of Weyl nodes or to control the chirality or geometry of magnetic domain walls, which is important for next-generation spintronics applications [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The family of LnAlPn (Ln = lanthanides, Pn = Ge, Si) materials is ideal to host nontrivial topological prop- erties due to their noncentrosymmetric crystalline struc- ∗ Mario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='Piva@cpfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='de † Present address: Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' ‡ Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='Nicklas@cpfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='de ture (I41md), which is the same as in the TaAs family of WSMs [7, 9, 26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Multiple Weyl nodes and a large spin Hall effect were predicted to exist in LaAlGe and LaAlSi [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Weyl cones were experimentally observed for LaAlGe [31] and a π Berry phase was recently found in LaAlSi [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Remarkably, magnetic members of the family host rare-earth moments which can order and ad- ditionally break TR symmetry - many of them are pre- dicted to feature Weyl nodes near the Fermi level [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Experiments have discovered an anomalous Hall effect (AHE) in PrAlGe1−xSix [33], chiral surface Fermi arcs in PrAlGe [37, 38], and a topological magnetic phase and singular angular MR in the semimetal CeAlGe [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In addition, Weyl fermions have been found to mediate magnetism in NdAlSi [42] and a π Berry phase was re- ported for quantum oscillations (QO) in SmAlSi [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In this Article, we focus on the ferromagnetic Weyl semimetal CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' CeAlSi, which hosts an in-plane non- collinear ferromagnetic (FM) order below the Curie tem- perature TC ≈ 8 K with a large anisotropy, the c-axis being the magnetically hard axis [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Ce3+ spins in ad- jacent FM planes display an angle of ≈ 70◦ [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Recent angle resolved photo emission spectroscopy experiments in the paramagnetic phase of CeAlSi above TC revealed Fermi arcs and several Weyl nodes lying close to the Fermi energy which stem from the non-centrosymmetric structure [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Going below TC, into the FM state, a magnetic field applied parallel to the [100] direction reveals an AHE, while a [001] field leads to an unex- plained hysteretic loop Hall effect (LHE) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In ad- dition, CeAlSi may exhibit nontrivial magnetic domain walls [44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' indeed, chiral domain walls were recently de- tected in this system [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Furthermore, magnetoelastic arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='13707v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='str-el] 31 Jan 2023 2 couplings give rise to picometer displacements in the unit cell due to the internal FM field, which can lead to dif- ferent domain wall spin textures [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The presence of this magnetoelastic effect suggests that external pressure may lead to a strong tuning of magnetism and to associated large changes in the AHE and LHE [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Hydrostatic pressure has previously been shown to be an effective tool in tuning the electronic structure without introducing any additional disorder and was successfully used to tune Weyl points closer to the Fermi energy in certain topological materials [47–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Furthermore, application of pressure is known to system- atically modify the magnetic properties in Ce-based ma- terials [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Here, we use hydrostatic pressure as a tool to investigate the origin of the features characteristic of the nontrivial topological behavior in CeAlSi, focusing on longitudinal and Hall transport experiments and on quantum oscillation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We combine these with ab initio density functional theory (DFT) calcula- tions and phenomenological models for scattering of Weyl fermions off magnetic domain walls to shed light on our unusual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' METHODS Single crystals of CeAlSi and LaAlSi were grown by the Al-flux technique similar to [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' High purity el- ements with starting composition Ce [La] (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='99%) : Al (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='999+%) : Si (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='999+%), 1 : 20 : 1, were place into an alumina crucible and sealed in an evacuated quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The samples were heated to 1200◦C, kept at this temperature for 15 hours and cooled down to 720◦C at 2◦C/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The excess of Al was removed by spinning the tube upside down in a centrifuge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The crystal structure was confirmed by x-ray powder diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Energy dis- persive x-ray spectroscopy shows, within the experimen- tal uncertainty, a Ce:Al:Si proportion of 1 : 1 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Electrical transport experiments were carried out by a four-probe configuration using a low-frequency AC resis- tance bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Temperatures down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 K and magnetic fields up to 9 T were achieved in a physical property measurement system (PPMS, Quantum Design) and in a liquid helium cryostat (Janis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Magnetization measure- ments were conducted in a magnetic property measure- ment system (MPMS, Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Pressures up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 GPa (electrical transport) and 1 GPa (magnetization) were generated using self-contained piston-cylinder-type pressure cells using silicon oil as pressure transmitting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' A piece of lead (tin) served as manometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Density functional theory (DFT) calculations were per- formed with the local density approximation functional (LDA) and projector-augmented wave (PAW) method as implemented in the Abinit software package [53], using Jollet-Torrent-Holzwarth (JTH) pseudopotentials [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Spin-orbit coupling (SOC) and non-collinear magnetism are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' An on-site Coulomb interaction with U = 6 eV was added for the Ce f electrons within the LDA+U scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We use a 16 × 16 × 16 Monkhorst- Pack k-point grid and a plane-wave energy cutoff of 25 hartree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The lattice parameters and relevant inter- nal atomic coordinates were optimized at respectively 0 GPa and 3 GPa until all forces on the atoms were below 10−6 hartree/bohr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' At 0 GPa (3 GPa), we ob- tain a = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='926 bohr (a = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='832 bohr) and c = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='397 bohr (c = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='192 bohr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' RESULTS Temperature – pressure phase diagram At ambient pressure, the FM ordering transition in CeAlSi is marked by a singular magnetization M(T) and sharp drop in electrical resistivity as a function of tem- perature ρ(T) at TC ≈ 8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Figure 1 shows the effect of external pressure on the magnetic phase (see Appendix A for additional data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Application of pressure linearly enhances TC(p) with a slope of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='62(2) K/GPa, driving TC from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 K at ambient pressure to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 K at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 GPa (values taken from the resistivity data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' More important 2 4 6 8 10 12 14 0 1 2 3 0 1 2 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 6 -4 -2 0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 p (GPa) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='98 M (103emu/mol) T (K) CeAlSi (a) H || [100] 40 50 60 70 80 p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 ρ (µΩcm) TC (K) p (GPa) χ[100] (b) ρ s1 ρ s1 - 2nd run ρ s2 p (GPa) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='98 M (µB/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=') � 0H (T) T = 2 K (c) H || [100] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) Magnetization (M) (left axis), obtained in an applied field of 50 mT along the [100] crystal axis, and elec- trical resistivity (right axis) as a function of temperature for selected pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (b) Temperature–pressure phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The solid lines are linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (c) Magnetization measure- ments for several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3 Γ Σ N Σ1 Z Γ X 4 3 2 1 0 1 2 3 4 E-EF (eV) 0 10 20 DOS (states/eV) 4 3 2 1 0 1 2 3 4 0GPa 3GPa 0GPa 3GPa Ce-f FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Electronic bands and DOS at ambient pressure (blue) and 3 GPa (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The hatched region of right panel corresponds to the partial DOS associated with Ce f states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' is our finding that, for different pressures, the in-plane magnetization curves M(H) at 2 K as a function of the applied magnetic field along the [100] direction lie on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This result indicates a negligible pres- sure effect on the non-collinear planar magnetic structure found at ambient pressure [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Electronic band structure Our DFT calculations at ambient pressure and 3 GPa, which incorporate spin-orbit coupling and non-collinear magnetic order, reveal only a negligible effect of pres- sure on the electronic band structure and the electronic density of states (DOS) at the Fermi energy [see Fig 2] as well as on the ordered moments and their orienta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The bands contributing to the hole pockets at the Fermi surface (FS) barely display any variation of their intercepts of the Fermi energy in k-space, suggesting a negligible variation of the FS area (see Appendix D for further information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The only noticeable modification of the electronic structure is a small shift of the bands associated with Ce f-electrons to higher energies with re- spect to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As these bands lie about 2 eV below the Fermi level, they most likely do not directly contribute to the transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Quantum oscillations Next we turn to the results of our QO measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Longitudinal conductivity data σxx well above TC at T = 15 K and in the FM state at T = 2 K at several pressures are presented in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3(a), where we have subtracted a smooth background yielding ∆σxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Within the investigated field range, the ∆σxx data as well as its fast Fourier transform (FFT) analysis reveals Ds xx (kS/m) 1/(m 0 H) (T 1 ) 2 K 15 K FFT Amplitude (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' units) T (K) p (GPa) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 CeAlSi H || [001] (a) p (GPa) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 N 1/(m 0 H) (T 1 ) (b) H || [001] T = 15 K p (GPa) 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 N 1/(m 0 H) (T 1 ) (c) H || [001] T = 2 K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) Fast Fourier transformation (FFT) amplitude as a function of temperature at different applied pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The solid lines are simulations considering the best fits using the Lifshitz-Kosevich formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The inset shows longitudinal conductivity for H ∥ [001] after subtraction of a third order polynomial background ∆σxx as a function of 1/(µ0H) at 15 K (top) and 2 K (bottom) for selected pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The curves at 2 K were shifted by −10 kS/m for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (b) and (c) Landau fan diagrams for CeAlSi at 15 and 2 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' a single QO frequency f ≈ 20(5) T, which is found to be independent of pressure and temperature (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We notice two main features: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' the amplitude of the oscillations at 15 K is larger than that at 2 K and ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' the amplitude of the oscillations is suppressed by in- creasing pressure [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Generally, the thermal damping of the QO amplitude can be described by the a8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 ( 0 0 a18- 0 S 4 eSO 2s r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 JO 2- 0 2 JOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content="0 0'330 40 0S." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='00 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content="0 0' 81c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 0 S 40 r 3 Q4 Lifshitz-Kosevich (LK) formula [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' However, our FFT signal follows the LK prediction only in the paramagnetic (PM) region above TC [solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In the FM state we observe an unusual reduction of the QO am- plitude upon cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This remarkable response of the oscillation amplitude as a function of temperature has not been observed in any other members of the LnAlPn family [32, 34, 36, 42, 56], and was previously reported in just a few materials [57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In SmSb, for instance, a sudden decrease of the Shubnikov-de Haas oscillations takes place once the material becomes antiferromagnetic, which was conjectured to be due to the presence of a non- trivial Berry phase [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' To further analyze the QO, Landau fan diagrams are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3(b) and 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our analysis indicates a change in the nature of the topological properties be- tween the PM and FM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' At 15 K in the PM phase the intercept is around −5/8, which suggests the pres- ence of topologically trivial charge carriers [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' In con- trast to that, at 2 K the intercept is −9/8, which for 3D magnetic WSMs can be associated with linear dispersive charge carriers and a nontrivial Berry phase [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our QO data suggest that the momentum space sep- aration between nearby Weyl nodes with opposite topo- logical charges gets enhanced in the FM state leading to a change in FS topology - from one which encloses both Weyl nodes above TC to a split FS enclosing iso- lated well-separated Weyl nodes below TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' An enhanced separation of the Weyl nodes in the FM phase has been also previously found in band-structure calculations (SI of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This can explain the change in the inter- cept in our Landau fan plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Such a change in the FS topology could nonetheless preserve the area of certain extremal orbits, so that the observed QO frequency can remain nearly unchanged (see Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' If the topological Fermi pockets are only weakly split below TC, the large density of states due to proximity to the Lifshitz transition [60] can lead to an increased scattering rate for states on the extremal orbits, thus enhancing the Dingle temperature and suppressing the QO amplitude for T < TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We note that our results cannot rule out other possible scenarios for the suppression of the ampli- tude of the quantum oscillations upon cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' However, a Lifshitz transition in the FS of CeAlSi can explain both, the suppression of the QO and the phase shift upon en- tering the FM phase revealed by our measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Hall effect For magnetic field along the [100] direction and cur- rent along [010] [see sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(a)] we find a large AHE in CeAlSi in its ferromagnetic state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The AHE signal has been extracted by fitting the Hall resistiv- ity to the form ρyz(H) = R0H + ρAHE, where R0 is the ordinary Hall effect coefficient and ρAHE = RsMx with Rs being the anomalous Hall coefficient and Mx being the magnetization along [100] (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='3 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 H || [001] p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 ρLHE (µΩcm) µ0H (T) CeAlSi - s1 T = 2 K LaAlSi (0 GPa) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 0 1 2 ρxy (µΩcm) Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' ∆ρAHE (µΩcm) p (GPa) p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 ρAHE (µΩcm) µ0H (T) T = 2 K CeAlSi - s2 - H || [100] LaAlSi (0 GPa) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) Anomalous Hall effect (AHE) at 2 K as a func- tion of magnetic fields H ∥ [001] for different applied pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The top inset shows the AHE jump as a function of pressure and the bottom inset displays a scheme of the cir- cuit used in this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' s1 and s2 denote samples 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (b) Loop Hall effect (LHE) at 2 K as a function of magnetic field for H ∥ [001] for selected applied pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The left inset shows the Hall resistivity measured upon increasing (red) and decreasing (blue) magnetic field and the difference of both curves (green) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The right inset displays a schematic drawing of the measurement circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Data of the nonmagnetic reference material LaSiAl at ambient pressure is shown as gray line in both panels We confirm that an AHE is absent in the non-magnetic analog LaAlSi [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We have fitted the longi- tudinal and ordinary Hall conductivities to a simple two-band model to obtain information on the density of the electron- and hole-like charge carriers and their mobilities (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' At low temperatures and ambient pressure we find 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='9(1) × 1019 holes/cm3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5(1) × 1019 electrons/cm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The application of exter- nal pressure suppresses the extracted hole density only slightly, which reaches 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6(1)×1019 holes/cm3 at 2 K and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 GPa, whereas the electrons density remains nearly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Moreover, the corresponding mobilities at 2 K and ambient pressure are about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4(1)×103 cm2/Vs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2(1)×103 cm2/Vs) for holes (electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' These values are nearly unaffected by application of external pressure and are on the same order of magnitude compared with other Weyl semimetals [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Application of external pressure suppresses the jump of the AHE (defined as dif- As B ca p + c B5 ference in ρAHE between positive and negative magnetic fields) up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 GPa [top inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Above 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 GPa the anomalous Hall jump saturates to around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5(1) µΩcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As we have shown above, the M(H) curves taken at different pressures fall on top of each other [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 1(b)], suggesting the absence of changes in the mag- netic structure as a source for the suppression of the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Moreover, the electronic bands close to the Fermi level are only slightly affected by pressure [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 1(c)], making it unlikely that this significant decrease results from a pressure-induced change in the position of the Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We find that while Rs scales nearly linearly with ρxx for pressure ≲ 1 GPa, the scaling deviates sig- nificantly from linear behavior at higher pressures (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' A linear relation between Rs and ρxx sug- gests that the observed AHE at ambient pressure has a significant extrinsic skew-scattering contribution [63], which gets suppressed at high pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Given the ro- bustness of the electronic structure and magnetic order against pressure, the most plausible explanation for this is a pressure-dependent change in the nature or distri- bution of domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Previous work has shown that Weyl fermions can undergo skew scattering from mag- netic domain walls which contain the axis of the average magnetization, leading to an extrinsic contribution to the AHE qualitatively consistent with our data [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' An even more unusual Hall response is observed for field applied along [001] [see sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We note that in this geometry the magnetic field is applied perpendicular to the ferromagnetically ordered moments in the ⟨001⟩ plane and this Hall response thus cannot arise from the bulk in-plane magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This so- called loop Hall effect (LHE) is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' It is only observed in the ferromagnetic regime, displays hysteresis even in the absence of any observable Mz mag- netization hysteresis, and is absent in the non-magnetic analog LaAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' ρLHE(H) is obtained by recording the Hall resistivity ρxy for H ∥ [001] upon increasing and decreasing magnetic field and taking the difference be- tween both curves, as shown in the left inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(b) for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 GPa as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Similar to the AHE, the ap- plication of external pressure leads to a decrease in the LHE [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' While the existence of the LHE in CeAlSi has been argued to be tied to the presence of the Weyl nodes near the Fermi energy [36], no clear physical mechanism has been provided for its origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' DISCUSSION In the following we present a simplified model for Weyl fermions in a non-centrosymmetric FM, and show that a domain wall scattering mechanism, similiar to that previ- ously explored to understand the AHE [64], can also lead to the LHE for domain walls which are perpendicular to the average magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our key idea here is that the bulk magnetic domains in CeAlSi host an in-plane mag- netization with hard axis along [001], so that the out-of- (b) x y z Domain wall MDW(x) = M sin q(x) (a) Z (c) MDW <0 Z MDW >0 Z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) Domain wall between two bulk magnetic domains showing twisted magnetization configuration with M DW z (x) = M sin θ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (b) Skew scattering of Weyl fermions off hysteretic domain wall loops, with red regions having M DW z > 0 and blue having M DW z < 0, provides a mech- anism for the loop Hall effect (LHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The black arrows are classical representations of the Weyl fermions trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (c) Calculated LHE shown as a ratio of Landauer conductances, versus the maximal out of plane tilt angle θ of the domain wall magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' plane bulk contribution to the field-induced magnetiza- tion M bulk z is not expected to be hysteretic as we tune the magnetic field Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We instead argue that the hysteretic LHE must be attributed to the hysteretic domain wall magnetization M DW z as we tune Hz, as schematically de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 4(a) and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our calculations show that the intra-node skew scattering of Weyl fermions as they cross a domain wall with nonzero M DW z can explain the LHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' To illustrate this physics, we study a model with 4 pairs of Weyl nodes in the kz = 0 plane (see Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' These could be viewed as a caricature of the W ′ 3 Weyl nodes found to lie ∼ 46 meV above the Fermi level close to the kz = 0 plane in CeAlSi [36, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' For a single Weyl node, with chirality +1, we consider a simple linearized Hamiltonian: H+ = vF σiG(+) ij qj + σiMi (1) where vF is the nodal velocity, σ is the spin Pauli matrix, q denotes the momentum relative to the Weyl node po- sition, and the tensor Gij is chosen to yield an elliptical Fermi surface at fixed kz with its major axis rotated away from the x, y axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The Hamiltonian for the other Weyl nodes can be reconstructed using symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The Weiss field M is nonzero in the magnetically ordered phase and tunes the momentum of the Weyl node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' we assume this leads to topologically nontrivial FS pockets enclosing sin- gle Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' For a magnetic field Hz applied along the z-axis, the system will support domains of M, with magnetization aligned along different in-plane directions, which are separated by domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As shown in [45], the domain wall magnetization in such cases supports an out-of-plane component M DW z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 5(c) shows the com- puted Hall conductance scaled by the longitudinal con- E 2 (10-3) 1 xx6/ 0 1 9 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='50 u/e6 ductance showing that it is an odd function of M DW z (see Appendix F for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Crudely, this small Landauer conductance [65, 66] ratio is expected to be related to the ratio of loop Hall to longitudinal resistivity - our exper- iments show that ρLHE xy /ρxx ∼ 10−3, in reasonable agree- ment with the theoretical estimate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We thus propose that the mechanism for the aptly named LHE is the skew scattering of Weyl fermions off hysteretic do- main wall loops or surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Since we expect the domain wall magnetization M DW z ≪M bulk z , the hysteretic behav- ior of M DW z cannot be resolved in bulk magnetization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our model might also help to understand the observation of a similar LHE reported previously in other compounds, in which Weyl fermions were predict to exist [67–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' CONCLUSIONS In summary, our study emphasizes, through a key tun- ing parameter (hydrostatic pressure), the importance of ferromagnetism for the low temperature topological fea- tures in CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our QO data show a difference in the Berry phase above and below TC, indicating that FM or- dering shifts oppositely charged Weyl nodes away from each other in momentum space, leading to a change in the Fermi-surface topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We have argued that this also leads to an increase in the scattering rate below TC, and thus to a drop in the amplitude of Shubnikov – de Haas oscillations in contrast to the conventional LK for- mula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This result calls for angular dependent Shubnikov – de Haas and de Haas – van Alphen experiments in an extended field range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We have also discovered pres- sure dependent changes in the AHE and LHE below TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Since our DFT calculations indicate that the electronic band structure is robust against pressure, we argue that these changes in the AHE and LHE must arise from dif- ferences in domain wall defects and we have shown how Weyl fermions scattering off hysteretic domain walls can lead to the LHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' DATA AVAILABILITY Data that underpin the findings of this study are avail- able at Edmond – the open research data repository of the Max Planck Society at [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' ACKNOWLEDGMENTS We acknowledge fruitful discussions with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We also thank U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Burkhardt for carry- ing out energy dispersive x-ray analysis on the sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and inno- vation programme under the Marie Sk�lodowska-Curie grant agreement No 101019024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This work was also supported by the S˜ao Paulo Research Foun- dation (FAPESP) grants 2017/10581-1, 2018/11364- 7, 2020/12283-0, CNPq grants # 304496/2017-0, 310373/2019-0 and CAPES, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This research was financially supported by the Natural Sciences and En- gineering Research Council of Canada (NSERC), under the Discovery Grants program grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' RGPIN-2016- 06666.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Computations were made on the supercomput- ers Beluga and Narval managed by Calcul Qu´ebec and the Digital Research Alliance of Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The operation of these supercomputers is funded by the Canada Foun- dation for Innovation, the Minist`ere de la Science, de l’´Economie et de l’Innovation du Qu´ebec, and the Fonds de recherche du Qu´ebec – Nature et technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='- C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='C are members of the Regroupement qu´eb´ecois sur les mat´eriaux de pointe (RQMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' APPENDICES APPENDIX A: ELECTRICAL RESISTIVITY Figures 6(a) and 6(b) present the electrical resistivity (ρ) as a function of temperature at several pressures for two different samples of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' At high temperatures ρ(T) exhibits a metallic behavior for both samples at all studied pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Moreover, a clear kink is observed at low-temperatures characterizing the ferromagnetic tran- sition, in good agreement with previous reports at am- bient pressure [36, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' A broad shoulder is observed at around 80 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' It shows up as a maximum in the tem- perature derivative of the electrical resistivity [see insets of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 6(a) and 6(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The shape and the position of the maximum is nearly unaffected by the application of external pressure, suggesting that the gap between the ground state and the first excited crystalline electrical field state does not change with increasing pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' APPENDIX B: QUANTUM OSCILLATIONS The left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 7 present the longitudinal con- ductivity measured with H ∥ [001] after subtraction of a third order polynomial (fit between 5 and 9 T) ∆σxx as a function of 1/(µ0H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We note that quantum oscilla- tions are clearly seen up to 40 K at all studied pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Furthermore, the unusual behavior of the quantum oscil- lations amplitudes can be seen by the naked eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The os- cillations in the paramagnetic state at 15 K are more pro- nounced than the oscillations in the ferromagnetic state at 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The panels on the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 7 present the Fast Fourier transformation (FFT) of ∆σxx, using a Hamming window from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='11 T−1, as a function of frequency (f) at several temperatures for selected pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Only one oscillation frequency f ≈ 20(5) T is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' It is unaffected by changes in pressure and/or temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 7 0 50 100 150 200 250 300 40 60 80 100 120 0 50 100 150 200 250 300 25 50 75 100 125 50 100 150 200 1 0 1 2 3 50 100 150 200 1 0 1 2 3 p (GPa) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 ρ (µΩcm) T (K) (a) CeAlSi - s1 ρ (µΩcm) T (K) p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 (b) CeAlSi - s2 dρ/dT p (GPa) 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 dρ/dT p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) and (b) Electrical resistivity (ρ) as a function of temperature at several pressures for two different samples of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The insets displays a magnified view of the low-temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The insets show the temperature derivative of ρ as a function of temperature at several pressures for two different samples of CeAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The effective mass (m∗) was estimated in the param- agnetic state of CeAlSi by fitting the FFT amplitude as a function of temperature by the Lifshitz-Kosevich (LK) formula [55]: RT = αTm∗ B sinh(αTm∗/B), (2) in which α = 2π2kB/eℏ ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='69 T/K, T is the tempera- ture, B is the magnetic field and m∗ the effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 8, the application of external pressure leads to a decrease in the value of m∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' APPENDIX C: HALL EFFECT Anomalous Hall Effect Figure 9 presents the Hall resistivity (ρyz) as a function of magnetic field at 2 K for several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' A linear background was determined by performing a linear fit in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 T ⩽ H ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We obtained the anomalous Hall (ρAHE) effect by subtracting the linear background using ρyz = R0H + ρAHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 10, the linear dependence between the anomalous Hall coefficient (RS) and the longitudi- nal resistivity (ρxx) characterizes the presence of a skew scattering contribution to the AHE in CeAlSi at ambi- ent pressure [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The observation of this contribution in a good metal regime can be attributed to domain wall scattering of Weyl fermions (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' IV), as this contri- bution should be the dominant one in highly conducting samples (σxx ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 × 106 (Ωcm)−1) [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Furthermore, the application of external pressure suppresses the lin- ear relation between Rs and ρxx, which is better seen in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 10, where the exponents obtained with allometric fits (RS = a + bρn xx) are shown as a function of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' One can clearly see the increase of the ex- ponent n as a function of increasing pressure, reaching 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='21(1) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 GPa, indicating that the skew scattering contribution of the AHE from the domain walls is be- ing suppressed by application of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The domains themselves (bulk) also contribute to the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' It is pos- sible to differentiate both contributions, as analyzed in great detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' [64], by considering that the domain wall scattering contribution to the AHE is limited by the electron mean free path, whereas the bulk contribution is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The total Hall resistivity is therefore an average between the bulk and domain wall contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Our results suggests that at low pressures (p ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 GPa) the AHE is dominated by the skew scattering contribution coming from the domain walls, while in the high-pressure range (p ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 GPa), where the AHE is not skew scat- tering type, it is dominated by the contribution of the domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Two-band model fits To accurately estimate the carrier densities and mobil- ities of CeAlSi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' we have simultaneously fit the longitudi- nal (σxx) and the Hall (σxy) conductivities considering a two-band model described by: σxx = e � neµe 1 + µ2e (µ0H)2 + nhµh 1 + µ2 h (µ0H)2 � σxy = e (µ0H) � neµ2 e 1 + µ2e (µ0H)2 − nhµ2 h 1 + µ2 h (µ0H)2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' where n denotes the electron (e) and hole (h) carrier den- sities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' and µe and µh are the electron and hole mobilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Figure 11(a) presents a representative plot of the fits at 2 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa, in which a good agreement between the experimental data and the fits is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Figure 11(b) displays the carrier densities as a function of 8 0 40 80 120 160 200 0 1 2 3 4 0 50 100 150 200 0 1 2 3 4 5 6 7 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='20 20 15 10 5 0 5 10 15 0 40 80 120 160 200 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='20 6 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='20 8 6 4 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='20 12 8 4 0 4 8 0 40 80 120 160 200 0 1 2 3 4 5 6 FFT Amplitude ∆σxx f (T) T (K) 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 15 5 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 30 10 40 CeAlSi - s1 H || [001] p - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa FFT Amplitude ∆σxx f (T) T (K) 2 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 20 5 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 40 10 CeAlSi - s2 H || [001] p - 0 GPa ∆σxx (kS/m) 1/(µ0H) (T-1) T (K) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 10 30 5 15 40 CeAlSi - s2 H || [001] p - 0 GPa FFT Amplitude ∆σxx f (T) T (K) 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 15 5 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 30 10 40 CeAlSi - s1 H || [001] p - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 GPa ∆σxx (kS/m) 1/(µ0H) (T-1) T (K) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 15 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 10 20 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 30 CeAlSi - s1 H || [001] p - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 GPa ∆σxx (kS/m) 1/µ0H (T-1) T (K) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 15 40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 10 20 5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 30 CeAlSi - s1 H || [001] p - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa ∆σxx (kS/m) 1/(µ0H) (T-1) T (K) 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 10 30 5 15 40 CeAlSi - s1 H || [001] p - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa FFT Amplitude ∆σxx f (T) T (K) 2 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 20 5 30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 40 10 CeAlSi - s1 H || [001] p - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (Left panels) Longitudinal conductivity after subtraction of a third order polynomial ∆σxx as a function of 1/(µ0H) at several temperatures for selected pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (Right panels) Fast Fourier transformation of ∆σxx shown in the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='068 m* (m e ) p (GPa) CeAlSi H || [001] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Effective mass m∗ as a function of pressure for mag- netic fields parallel to [001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Hall resistivity (ρyz) as a function of magnetic field applied parallel to [100] at 2 K for several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The solid orange line is an extrapolation of a linear fit performed in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 T ⩽ H ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 T, which yields the ordinary background of ρyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' temperature at several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Figure 11(c) shows the mobilities as a function of temperature at several pres- sures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' APPENDIX D: BANDSTRUCTURE CALCULATIONS Figure 12 shows the electronic bands and DOS zoomed in the vicinity of the Fermi level, to emphasize the negli- gible effect of pressure on the bands contributing to the AHE and LHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Note, also that no crossing feature nor electron pocket was found in our ambient pressure calcu- lation along the Γ − X high symmetry path, in contrary to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 3(a) of [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' This discrepancy could be attributed to the different exchange-correlation functional or to our use of theoretically relaxed lattice parameters, while [36] used experimental values which, in the case of the PBE- 40 50 60 70 80 90 100 110 120 130 250 200 150 100 50 0 50 p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 R S (mWcm/T) r xx (mWcm) H || [100] T = 2 K 0 1 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 n p (GPa) R S = a + br n xx FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Anomalous Hall coefficient (RS) as a function of the longitudinal resistivity (ρxx) at several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' GGA functional used in their paper, will be smaller than the theoretical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Nevertheless, from the pressure de- pendence of the electronic bands relative to the Fermi level, an electron pocket could likely appear along this path upon further increasing the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We further refine the analysis of the electronic struc- ture by calculating the orbital decomposition of the electronic wavefunction inside the atom-centered PAW spheres for Ce 5d states (left panels, red), as well as for Al (middle panels, green) and Si (right panels, blue) 2p states, in the same energy range as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The relative weights of the different orbitals at ambient pressure (top panels) and 3 GPa (bottom panels) are essentially iden- tical, thus confirming the negligible effect of pressure on the electronic bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The calculated magnetic structure does not display any significant differences between 0 and 3 GPa either, in agreement with the experimental observations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 1(b) of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' For 0 GPa (3 GPa), we find a magnetic moment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='880 µB (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='878 µB) inside the PAW spheres of the 2 inequivalent Ce atoms in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Considering that one moment points mostly in the ˆx direction and the other in the ˆy direction with an angle of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='3◦ (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8◦) between them, we find a total net magnetic moment of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='311 µB (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='284 µB) oriented along [110] for the whole unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Note that the net size of the magnetic moment depends strongly on the choice of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' APPENDIX E: SIMPLIFIED MODEL FOR WEYL NODES We choose a simple model for CeAlSi with 4 pairs of Weyl nodes as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 14 (top left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We have chosen the Weyl nodes and Fermi pockets for T > TC to be consistent with the C4v and mirror Mx, My crys- tal symmetries of CeAlSi, as well as time-reversal sym- metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' These nodes crudely mimic the W ′ 3 nodes found slightly above the Fermi level in previous ab initio elec- 6 CeAISi - s2 - H Il [100] T= 2 K 4 (woun) 2 0 Pyz p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 Vy 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 ¥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 μoH (T)10 0 1 2 3 4 5 6 7 8 9 0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 300 0 4 8 12 16 0 50 100 150 200 250 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 σ (kS/m) µ0H (T) σxx σxy Two-band fits CeAlSi - s1 p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa T = 2 K, H || [001] (a) n (1019 cm-3) T (K) p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 H || [001] CeAlSi - s1 open - holes solid - electrons (b) µ (103 cm2/Vs) T (K) p (GPa) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='6 CeAlSi - s1 H || [001] open - holes solid - electrons (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' (a) Longitudinal (σxx) and Hall (σxy) conductivities at 2 K and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='2 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Carrier densities (b) and mobilities (c) obtained from the two-band fits as a function of temperature at several pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' tronic structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' With the onset of mag- netism, the Weyl nodes get displaced with opposite chi- rality nodes being displaced in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 14(top right panel), this can lead to a topo- logical transition of the Fermi surface where each pocket now encloses a single Weyl node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' At the same time, for the type of Fermi surface sketched above, certain extremal orbits can remain unchanged in area (dashed lines), so that the QO frequency will be unaffected as observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' If the Weyl nodes are not widely separated even after the topological Fermi surface phase transition, proximity to a Lifshitz transition may lead to an enhance- ment of the electron scattering rate (in the presence of weak disorder) due to a large density of states, which can enhance the Dingle temperature and explain the strong observed deviation from the Lifshitz-Kosevich formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Within the symmetry broken for T < TC, it is reason- able to consider the physics of isolated Weyl nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' We model a single Weyl node as having an elliptical pocket with velocity tensor and a simple coupling to the Weiss field from the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' H+ = σiG(+) ij ˜qj + σiMi, (3) G(+) ij = � � � � |a| 1 |a| 1 � � � � � � � � cos β sin β 0 − sin β cos β 0 0 0 1 � � � � , (4) Here ˜qi = vF qi where q denotes the momentum mea- sured from the Weyl node location, and vF is a velocity scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The real matrix, Gij, defined this way results in an ellipsoidal Fermi surface, whose xy-plane cross section has an elliptical shape with the major and minor axis, |a| and 1/|a| respectively for |a| > 1, and the major axis is rotated from the qy-axis by the angle β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' For |a| < 1, the major axis is instead rotated from the qx axis by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Position of Weyl points: When Mi = 0, the Weyl point resides at q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' When Mi ̸= 0, the Weyl point shifts to the point satisfying the following equation ˜q∗ i = − � G(+)�−1 ij Mj (5) 11 Γ Σ N Σ1 Z Γ X 1 0 1 E-EF (eV) 0 2 4 6 8 DOS (states/eV) 1 0 1 0GPa 3GPa 0GPa 3GPa FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Electronic bands and DOS at ambient pressure (blue) and 3 GPa (red), zoomed in the vicinity of the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' N 1 Z X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 E-EF (eV) Ce-5d, 0GPa N 1 Z X Al-2p, 0GPa N 1 Z X Si-2p, 0GPa (a) N 1 Z X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content='0 E-EF (eV) Ce-5d, 3GPa N 1 Z X Al-2p, 3GPa N 1 Z X Si-2p, 3GPa (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Orbital decomposition of the electronic wavefunction at (a) ambient pressure and (b) 3 GPa, zoomed in the vicinity of the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 12 Eigenspectrum: The eigenvalues of H+ are given by E = ± � ˜qi[G(+)]T ij[G(+)]jl˜ql + 2Mi[G(+)]ij ˜qj + MiMi (6) Written in the form which is useful for numerics: 0 = ˜q2 x � |a|2c2 + 1 |a|2 s2 � +˜qx � 2cs � |a|2 − 1 |a|2 � ˜qy + 2 � |a|cMx − 1 |a|sMy �� + � ˜q2 z + 2Mz ˜qz + � |a|2s2 + 1 |a|2 c2 � ˜q2 y + 2 � |a|sMx + 1 |a|cMy � ˜qy + M 2 − E2 � , (7) where s ≡ sin β and c ≡ cos β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The quadratic equation allows us to determine the mover modes given the Fermi energy and the Weiss field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' If the ˜qx solutions are real- valued, we obtain travelling waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' the complex-valued solutions correspond to evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' T > Tc T < Tc kx ky kz kz ky kx kx ky T < Tc FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Top: Illustrative example of 4 topologically trivial Fermi surface pockets for T > Tc, each enclos- ing a pair of Weyl points (WP) with opposite topological charge, located at momenta (K0, ±K1, 0), (−K0, ±K1, 0), (±K1, K0, 0), (±K1, −K0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' For T < Tc, the in-plane mag- netization leads to a momentum space displacement of the Weyl points, leading to transition into 8 topologically non- trivial Fermi pockets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' The area of certain maximal orbits (dashed lines) can remain unchanged across this transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf'} +page_content=' Bottom: Projected view of the elliptical cross sections of the topologically nontrivial Fermi surfaces for T 2, +the positivity of ρ adds some constraints on K, which is +strictly contained in the set {( +1 +√ +N , v) : ∥v∥ ≤ +� +1 − 1 +N }; +the vectors corresponding to pure states constitute a proper +subset of ∂K (more precisely, they correspond to the extreme +points of K). + +In coherence representation, equation (1) becomes +˙vρ = +� +MH + ML +� +vρ, MH and ML being respectively the rep- +resentations of the operators −i[H, ·] and L(·). It is worth +notice (see [3], [15] for more details) that, due to the fact that +the evolution induced by (1) is trace-preserving, the matrices +MH and ML always have the block forms +MH = +� +0 0 +0 � +H +� +ML = +� +0 +0 +v0 � +D +� +, +(2) +where �H ∈ so(N 2 − 1), v0 ∈ RN 2−1 and �D is a square +matrix of dimension N 2 − 1. +The coherence representation of equation (1) subject to +coherent control is +˙vρ = +� +MH0 + +� +j +ujMHj + ML +� +vρ. +(3) +The controllability properties of equation (3) have been +studied in [3], where, in particular, it is stated that the system +is never STLC. We recall moreover the following result. +Theorem 1: Let q0 ∈ ˚ +K and q1 ∈ ∂K. There is no +essentially bounded control function that steers q0 to q1 in +finite (positive) time. +C. The system +In this paper, we focus on coherently controlled composite +open quantum systems, composed by two interacting qubits +(called A and B); we recall that a qubit is a quantum +system living in a two-dimensional Hilbert space. According +to quantum mechanics ([6], [12], [20]), the total system +A + B evolves on the tensor product of two Hilbert spaces, +H = HA ⊗ HB, with HA = HB = C2. Also, we recall +that the state of each subsystem (denoted respectively with +ρA or ρB) can be extracted from the state ρ of the total +system, by means of the partial trace: indeed, we stress +that, if ρ is a density matrix (i.e. Hermitian positive semi- +definite and of trace one) on H, then ρA = TrBρ and ρB = +TrAρ are density matrices on HA and HB, respectively. +Physically, taking the partial trace on A can be interpreted as +“averaging” on the information about A, so that the reduced +state ρB describes the state of the subsystem B. Analogous +considerations hold for ρA. +The system A + B evolves according to equation (1). We +remark that we can always assume that the matrices H and +Lk are traceless, as adding them multiples of the identity +leaves invariant the right-hand side of (1). Then, H can be +uniquely written as H = HA + HI + HB, where HA = +hA ⊗ I2, HB = InA ⊗ hB, with hA, hB ∈ her0(2) and +HI ∈ her0(2)⊗her0(2). Actually, HI denotes the interaction +between the subsystems A and B, while HA and HB the +unitary free evolution of the two subsystems. +We make these further assumptions on the dynamics: +· the subsystem B does not interact directly with the +environment; in particular, this implies that the jump +operators can be taken of the form Lk = ℓk ⊗I2, where +the ℓk’s are traceless matrices on HA. +· the control directly affects only the subsystem A; in +particular, the Hamiltonian H can be written as +H = +� +hA0 + +3 +� +j=1 +ujhAj +� +⊗ I2 + HI + I2 ⊗ hB, +where the matrices hAj, j ≥ 0, HI and hB are constant +and uj are functions of time. +It is clear that, if HI = 0, then the two subsystems are +completely independent, and, in particular, the evolution of +B is not influenced by the control. +As already said, in [3] it is proved that the coherently +controlled Lindblad equation is never STLC and Theorem 1 +states that some transitions cannot be realized in finite time. +Nevertheless, it is still interesting to understand what can be +said about the controllability of the state of the subsystem +B only. In particular, the main questions that one may ask +include +1) is it possible to “protect” the subsystem B from dis- +sipation, that is, to implement a unitary dynamics on +B, at least on some submanifolds (for instance, the +submanifold Trρ2 +B = 1)? +2) is it possible to “control” the state ρB, regardless of ρ? +Question 2) needs to be further clarified, as several notions +of “partial controllability” are possible (see for instance [9]); +in any of its declination, tackling the issue is a very hard task, +even in the simplest case of two qubits. +Question 1) seems to be more affordable. In this paper, we +provide a first step towards the answer: for three particular +choices of the interaction HI (the well-known dispersive +and resonant couplings, very common in experimental set- +ups), and a generic choice of the dissipative term, we show +that the trajectories keeping ρB pure are trivial, or follow +a free evolution which is not affected by the controls. In +our opinion, more general interaction would lead to similar +results. +III. ON THE STRUCTURE OF K +In the case of two interacting qubits, we choose the +following orthonormal basis of her0(4) +Λi = 1 +2σi ⊗ I2, +i = 1, 2, 3 +Λ3i+j = 1 +2σi ⊗ σj, +i = 1, 2, 3, j = 1, 2, 3, +Λ12+j = 1 +2I2 ⊗ σj, +j = 1, 2, 3, +where σi, i = 1, 2, 3, denote the Pauli matrices +σ1 = +� +0 +1 +1 +0 +� +, σ2 = +� +0 +−i +i +0 +� +, σ3 = +� +1 +0 +0 +−1 +� +. +For such system, in the following we will adopt also the +more intuitive notation +vρ = +�1 +2, vA, vAB, vB� +, +with vA, vB ∈ R3, and vAB ∈ R9. +Thanks to the peculiarity of the Pauli matrices (in partic- +ular, the fact that σiσj + σjσi = 2δijI2), the matrices �H + +and �D and the vector v0 in equation (2) have the following +block structure +�H = + + + +ˆhA +�HIt +0 +�HIl +ˆhA ⊗ I3 + I3 ⊗ ˆhB +�HIr +0 +�HIb +ˆhB + + + +(4) +�D = + + +ˆd +0 +0 +0 +ˆd ⊗ I3 +v0 ⊗ I3 +0 +0 +0 + + +v0 = (v01, v02, v03, 0, . . . , 0) +(more details on the objects appearing here above are pro- +vided in Appendix). +In order to answer to question 1, we look for a character- +ization of states ρ such that TrAρ is pure. +Proposition 2: Consider ρ ∈ P such that Trρ2 +B = 1, +where ρB = TrAρ. Then Φ(ρ) ∈ ∂K and there exist +vA, vB ∈ R3 with ∥vA∥2 ≤ 1 +4 and ∥vB∥2 = 1 +4 such that +Φ(ρ) = +�1 +2, vA, 2vA ⊗ vB, vB� +. +(5) +Proof: The first claim is an easy consequence of [17, +Proposition 2.1]. +Let us now prove the second part. The bound on ∥vA∥ +and the value of ∥vB∥ yield from the fact that Tr(TrBρ)2 = +1 +2 + 2 �3 +j=1(vA +j )2 and Tr(TrAρ)2 = 1 +2 + 2 �3 +j=1(vB +j )2. +Moreover +1 +4 +d +dtTrρ2 +B = ⟨vB,�hBvB⟩ + ⟨vB, �HIbvAB⟩ +(6) += +3 +� +r,s=1 +λrs⟨vB, Ts(vAB +(r−1)+1, vAB +(r−1)+2, vAB +(r−1)+3)⟩, +where we used equation (8) and the fact that �hB is antisym- +metric. +By +hypothesis, +Trρ2 +B +cannot +increase +with +time. +On +the +other hand, as +{T1, T2, T3} +constitute a +ba- +sis +of +so(3), +and +equation +(6) +holds +for +any +HI, +⟨vB, A(vAB +(r−1)+1, vAB +(r−1)+2, vAB +(r−1)+3)⟩ must be zero for ev- +ery antisymmetric matrix A and for r = 1, 2, 3. Thus, the +vectors (vAB +(r−1)+1, vAB +(r−1)+2, vAB +(r−1)+3) must be collinear to +vB, and ρ can be written as +ρ = 1 +4I4 + σA ⊗ I2 +√ +2 ++ 1 +2 +� +γσ1 + βσ2 + θσ3 + I2 +� +⊗ +� +vB +1 σ1 + vB +2 σ2 + vB +3 σ3), +for some σA = +1 +√ +2 +� +vA +1 σ1+vA +2 σ2+vA +3 σ3 +� +. Let {ψ1, ψ2} be +the (normalized) eigenvectors of vB +1 σ1 +vB +2 σ2 +vB +3 σ3, rel- +ative to the eigenvalues µ1 = − 1 +2 and µ2 = 1 +2, respectively, +and let {φ1, φ2} be any (possibly different) orthonormal basis +of C2; let λ be an eigenvalue of ρ, and ϕ a corresponding +eigenvector, which can be written as ϕ = �2 +i,j=1 aijφi ⊗ψj. +Set Aj = σA +√ +2 + µj +2 (γσ1+βσ2+θσ3+I2)+ 1 +4I2, for j = 1, 2. +We remark that the spectrum of ρ is given by the union of +the spectra of A1 and A2. +In particular, the eigenvalues of A1 are given by +ν± = ± +� +�vA +1 +2 − γ +4 +�2 + +�vA +2 +2 − β +4 +�2 + +�vA +3 +2 − θ +4 +�2. +As they belong also to the spectrum of ρ, which is positive +semidefinite, then it must be vA +1 = γ +2 , vA +2 = β +2 , and vA +3 = θ +2, +and the proposition is proved. +Remark 1: Proposition 2 can be easily generalized to the +case in which the subsystem A has (complex) dimension +nA ≥ 2. +IV. FIRST ANSWERS TO QUESTION 1) +Proposition 2 imposes a constraint on the structure of +states ρ whose reduction to B is pure, and can be thus +exploited to study the controllability of the reduced states. +First of all, together with Theorem 1, it yields the following +fact. +Corollary 1: Let ρ0, ρT ∈ P such that Φ(ρ0) belongs to +the interior of K and Tr(TrAρT )2 = 1. Then, there is no +essentially bounded control function that can send ρ0 to ρT +in finite time. +In other to give partial answers to Question 1, we study +the following problem. +Problem 1: Let ρ0 such that Tr +� +TrAρ0)2 = 1, and call +ρ(t) the solution at time t of (1), under the assumption in +Section II-C, with ρ(0) = ρ0. Is it possible to find ǫ > 0 and +a control function defined on [0, ǫ] such that Tr +� +TrAρ(t))2 = +1 for t ∈ [0, ǫ]? +Assume that it is true, for some piecewise-C∞ con- +trol +function +ˆu +: +[0, ǫ] +→ +R3, +Set +Φ(ρ(t)) += +� 1 +2, vA(t), vAB(t), vB(t) +� +. By Proposition 2, if Trρ2 +B(t) = +1 for t ∈ [0, ǫ], then vAB(t) = 2vA(t) ⊗ vB(t). In +particular, for every k ≥ 1, we have +dk +dtk vAB(t) = 2 +k +� +j=0 +�k +j +� djvA +dtj +⊗ dk−jvB +dtk−j . +(7) +In the following, we will use equations (7) to find out the +controls satisfying the claim (if any), for different expres- +sions of the interaction HI. In order to do it, we define the +vector +w(t) = ˙vAB(t) − 2 +� +j=0,1 +� k +j +� djvA +dtj +⊗ dk−jvB +dtk−j . +Without loss of generality, we can choose bases on HA +and HB such that +HA0 = ωaσ3 +HB0 = ωbσ3 +Eventually performing a linear transformation in the control +space, we also assume that HAi = σi for i = 1, 2, 3. +“Dispersive” coupling: HI = gσ3 ⊗ σ3. Computing the +vector w, we notice that +w3 = gvA +2 (1 − 4(vB +3 )2) +w8 = gvB +2 (1 − 4(vA +3 )2), + +so that equation (7) is satisfied only if (vB +3 )2 is identically +equal to 1 +4 (which implies vB +1 ≡ vB +2 ≡ 0) or if (vA +3 )2 ≡ 1 +4 +(which implies vA +1 ≡ vA +2 ≡ 0). +We remark that the first scenario corresponds to freezing +ρB to the state 1 +2 +� +I2+σ3 +� +or to the state 1 +2 +� +I2−σ3 +� +, that is, +ρB is constant. In particular, by computation we notice that, +for this choice of HI, equation (3) has the block-triangular +form + + +˙z1 +˙z2 +˙vB +3 + + = + + +C11 +C12 +C13 +0 +C22 +0 +0 +0 +0 + + + + +z1 +z2 +vB +3 + + +with z2 = (vAB +1 +, vAB +2 +, vAB +4 +, vAB +5 +, vAB +7 +, vAB +8 +, vB +1 , vB +2 ). If +vB +3 = | 1 +2| and ρB is pure at t = 0, then z2(0) = 0, so that +z2(t) is zero for all t ≥ 0, and every value of the control. In +particular, the submanifolds of P +{ρA ⊗ 1 +2 +� +I2 + σ3 +� +: ρA = ρ† +A, ρA ≥ 0, TrρA = 1} +{ρA ⊗ 1 +2 +� +I2 − σ3 +� +: ρA = ρ† +A, ρA ≥ 0, TrρA = 1}, +are invariant for equation (1), for every choice of the control +and for any dissipation. +Let us now consider the second case; first of all, by (3), +we notice that vA +3 can be constantly equal to | 1 +2| only if +v03 +2 +vA +3 ˆd33 = 0, which restrict the class of dissipative terms +L that allow such behavior. Some examples of such operators +are the so-called amplitude damping channels ([20]), that is, +associated respectively to the jump operators σ+ ⊗ I2 or +σ− ⊗ I2, where σ± = σ1 ± iσ2. +On the other hand, taking into account the fact that the +state is factorized, the equation for vB becomes + + +˙vB +1 +˙vB +2 +˙vB +3 + += + + +0 +−ωb − 2gvA +3 +0 +ωb + 2gvA +3 +0 +0 +0 +0 +0 + + + + +vB +1 +vB +2 +vB +3 + +, +which is unaffected by the control. +“Resonant” coupling: HI = g(σ+⊗σ−+σ−⊗σ+). First of +all, we remark that we can write HI = g +2(σ1 ⊗σ1+σ2⊗σ2). +As we did above, we try to find conditions that guarantee +that w is null. As +w = g + + + + + + + + + + + + + + +−4(vA +3 vB +1 vB +2 + vA +1 vA +2 vB +3 ) +−(vA +3 (−1 + 4(vB +2 )2) + vB +3 (1 − 4(vA +1 )2) +(4vA +1 (vA +2 vB +1 − vA +1 vB +2 ) + vB +2 (1 − 4vA +3 vB +3 )) +(vA +3 (−1 + 4(vB +1 )2) + vB +3 (1 − 4(vA +2 )2)) +4(vA +3 vB +1 vB +2 + vA +1 vA +2 vB +3 ) +(4vA +2 (vA +2 vB +1 − vA +1 vB +2 ) + vB +1 (−1 + 4vA +3 vB +3 )) +(4vB +1 (vA +2 vB +1 − vA +1 vB +2 ) + vA +2 (1 − 4vA +3 vB +3 )) +(−4vB +2 (vA +2 vB +1 − vA +1 vB +2 ) + vA +1 (−1 + 4vA +3 vB +3 )) +4(vA +2 vB +1 − vA +1 vB +2 )(vA +3 − vB +3 ), + + + + + + + + + + + + + + +we deduce that, if w is null, then vA +2 vB +1 = vA +1 vB +2 and/or +vA +3 = vB +3 . +In the first case, plugging the equality into the expression +of w and setting it to zero, we obtain + + + + + + + + + +vA +1 (1 − 4vA +3 vB +3 ) = 0 +vA +2 (1 − 4vA +3 vB +3 ) = 0 +vB +1 (1 − 4vA +3 vB +3 ) = 0 +vB +2 (1 − 4vA +3 vB +3 ) = 0. +These four equations are satisfied if vA +1 += vA +2 = vB +1 += +vB +2 = 0, or if vA +3 vB +3 = 1/4; in both cases, due to the fact +that ρB is pure and to the constraints on the length of vA +and vB, we obtain that |vA +3 | = |vB +3 | = 1/2 and that also +ρA is a pure state (and, as a consequence of the “factorized +structure”, the whole state ρ is pure). +If instead vA +3 = vB +3 , setting to zero the second and the +fourth components of w, we obtain that |vA +1 | = |vB +2 | and +|vA +2 | = |vB +1 |, which again implies that ρA is a pure state. +Summing up, in presence of resonant coupling, it is not +possible to keep the partial state ρB pure if the whole state +ρ itself is not kept pure by the evolution. On the other hand, +as +d +dtTrρ2 = 2Tr(ρL(ρ)), it follows that the state ρB can +be kept pure only if ρ evolves in the set Tr(ρL(ρ)) = 0; +for “factorized states” (that is, of the form (5)), this happens +only for states such that v0 +2 + ˆdvA = 0; depending on the +particular choice of L, this equation may not have solutions +vA of norm 1/2. +HI = gσ3 ⊗ σ1. We finally discuss a further case in which +the interaction does not commute with the free Hamiltonian +HB. +Setting w ≡ 0, we find the following constraints: either +vB +2 ≡ vB +3 ≡ 0 and (vB +1 )2 ≡ 1/4, or vA +1 ≡ vA +2 ≡ 0 and +(vA +3 )2 ≡ 1/4. +The first case does not correspond to an admissible so- +lution of the control system: indeed, by computations, it is +possible to see that no control can keep ρ in a state of the +form ρA ⊗ 1 +2 +� +I2 ± σ1 +� +on a nonzero time interval. +Let us now look for admissible trajectories along which +vA +1 ≡ vA +2 ≡ 0 and (vA +3 )2 ≡ 1/4. First of all, as ˙vA +3 = +v03 +2 +vA +3 ˆd33+vA +1 ( ˆd31−u2)+vA +2 ( ˆd32+u1), such trajectories +are admissible only if vA +3 ˆd33 + v03 +2 += 0, that is true for +some particular dissipation terms only (as we already saw, +the amplitude damping channels satisfy such a constraint). +Inspecting the differential equations for vA +1 and vA +2 , we +see that they stay constant only for the choice of the control +u1 = v02 + 2vA +3 ˆd23 +2vA +3 +u2 = −v01 + 2vA +3 ˆd13 +2vA +3 +. +Moreover, as this guarantees w ≡ 0, that is, vρ has the form +(5), we can substitute the values of vAB into the differential +equations for vB, getting + + +˙vB +1 +˙vB +2 +˙vB +3 + + = + + +0 +−ωb +0 +ωb +0 +−2gvA +3 +0 +2gvA +3 +0 + + + + +vB +2 +vB +2 +vB +3 + + , +that is, the dynamics of ρB are protected from dissipation, +but the control does not affect them. +V. CONCLUSIONS +In this paper, we presented a first analysis on the indirect +controllability properties of a 2-qubit system, in the case in +which the ancilla is subject to dissipation. +First of all, we observed that states ρ such that their +reduction TrAρ is pure are not reachable (in finite time) from +the interior of the space P, thus obtaining a first obstruction + +to the indirect controllability of the system; it would be +interesting to investigate if such states are reachable from +any other point of the boundary of P. +We then focus on the possibility of preserve the subsystem +B from dissipation, that is, to find admissible trajectories +ρ(t) such that TrAρ(t) is pure. We investigated three par- +ticular cases of interaction (among them, the well known +dispersive and resonant couplings), and we found that the +only admissible trajectories are either trivial (i.e. their re- +duction TrAρ(t) is constant) or are unaffected by the action +of the control. In our opinion, similar results hold also for +other interaction Hamiltonians HI. +APPENDIX +First of all we recall that the matrices +T1 = +� 0 0 +0 +0 0 −1 +0 1 +0 +� +T2 = +� 0 +0 1 +0 +0 0 +−1 0 0 +� +T3 = +� 0 −1 0 +1 +0 +0 +0 +0 +0 +� +are the representations on R3 of the operators −i[ σj +2 , ·], j = +1, 2, 3. Then, if hA = �3 +j=1 +αj +2 σj and hB = �3 +j=1 +βj +2 σj +(we recall that we assumed them traceless), the matrices +ˆhA and ˆhB in (4) are simply ˆhA = �3 +j=1 αjTj and ˆhB = +�3 +j=1 βjTj. +Writing HI = 1 +2 +�3 +i,j=1 λijσi ⊗ σj, long but easy com- +putations give +�HIt = +3 +� +j=1 +Tj ⊗ (λj1, λj2, λj3) +�HIb = +3 +� +j=1 +(λ1j, λ2j, λ3j) ⊗ Tj +(8) +and the blocks �HIl and �HIr can be recovered by antisym- +metry. +REFERENCES +[1] C. D. Aiello and P. Cappellaro. Time-optimal control by a quantum +actuator. Phys Rev A, 91(3):042340, 2015. +[2] F. Albertini and D. D’Alessandro. +Notions of controllability for +bilinear multilevel quantum systems. IEEE Transactions on Automatic +Control, 48(8):1399–1403, 2003. +[3] C. Altafini. Controllability properties for finite dimensional quantum +Markovian master equations. J. Math. Phys., 44(6):2357–2372, 2003. +[4] C. Altafini and F. Ticozzi. +Modeling and control of quantum sys- +tems: An introduction. +IEEE Transactions on Automatic Control, +57(8):1898–1917, 2012. +[5] J. Avron and O. Kenneth. An elementary introduction to the geometry +of quantum states with pictures. Reviews in Mathematical Physics, +32(02):2030001, 2020. +[6] H.-P. Breuer and F. Petruccione. +The Theory of Open Quantum +Systems. Oxford University Press, 2002. +[7] D. Burgarth, K. Maruyama, M. Murphy, S. Montangero, T. Calarco, +F. Nori, and M. B. Plenio. Scalable quantum computation via local +control of only two qubits. Physical Review A, 81(4):040303, 2010. +[8] D’Alessandro D. and R. Romano. Indirect controllability of quantum +systems; a study of two interacting quantum bits. IEEE Transactions +on Automatic Control, 57:2009–2020, 2012. +[9] D. D’Alessandro, F. Albertini, and R. Romano. +Exact algebraic +conditions for indirect controllability of quantum systems. SIAM J. +Control Optim., 53(3):150–1542, 2015. +[10] A. Gorini, A. Kossakowski, and E.C.G. Sudarshan. +Completely +positive dynamical semigroups of N-level systems. +J. Math. Phys, +17:821, 1976. +[11] D. Grimmer, D. Layden, R. B. Mann, and E. Mart´ın-Mart´ınez. Open +dynamics under rapid repeated interaction. Phys. Rev. A, 94:032126, +Sep 2016. +[12] S. Haroche and J.M. Raimond. +Exploring the Quantum: Atoms, +Cavities, and Photons. Oxford Graduate Texts. OUP Oxford, 2006. +[13] G. Kimura. The Bloch vector for N-level systems. Journal of the +Physical Society of Japan, 72:185–188, 2003. +[14] K. Kraus. +States, Effects, and Operations : Fundamental Notions +of Quantum Theory. +Lecture Notes in Physics. Springer Berlin +Heidelberg, Berlin, Heidelberg, 1983. +[15] I. Kurniawan, G. Dirr, and U. Helmke. +Controllability aspects of +quantum dynamics: a unified approach for closed and open systems. +IEEE Transactions on Automatic Control, 57(8):1984–1996, 2012. +[16] D. Layden, E. Mart´ın-Mart´ınez, and A. Kempf. Universal scheme for +indirect quantum control. Phys. Rev. A, 93:040301, Apr 2016. +[17] M. Lin. +A treatment of a determinant inequality of Fiedler and +Markham. Czechoslovak Mathematical Journal, 66(3):737–742, 2016. +[18] G. Lindblad. On the generators of quantum dynamical semigroups. +Comm Math Phys, 48(2):119–130, 1976. +[19] M. Mirrahimi, Z. Leghtas, V.V. Albert, S. Touzard, R.J. Schoelkopf, +L. Jiang, and Devoret M.H. +Dynamically protected cat-qubits: a +new paradigm for universal quantum computation. +New J. Phys, +16(4):045014, 2014. +[20] M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum +Information. Cambridge University Press, 2000. + diff --git a/cNE0T4oBgHgl3EQf4wL5/content/tmp_files/load_file.txt b/cNE0T4oBgHgl3EQf4wL5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b519b2abc3dbe88867cdfe5dacf3d47ddee62a6 --- /dev/null +++ b/cNE0T4oBgHgl3EQf4wL5/content/tmp_files/load_file.txt @@ -0,0 +1,441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf,len=440 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='02744v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='OC] 6 Jan 2023 Indirect controllability of two interacting qubits in presence of dissipation: a first analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Verzhanska1 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Chittaro1 Abstract— We consider a bipartite open quantum system constituted by two interacting qubits A and B, assuming that the former is coupled to the environment and is directly affected by coherent control, while the latter does not interact directly with the environment and the control fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We are interested in the controllability properties of the subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In this paper, we give a first analysis of the problem and provide some negative answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' INTRODUCTION Quantum control deals with the manipulation of dynamical systems at the molecular and atomic scale, where the dy- namics are governed by quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As in classical control, the notion of controllability refers to the possibility to steer a given initial state to any desired target state, by applying appropriate external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For finite-dimensional closed quantum systems, the Lie Algebraic Rank Condition (LARC) is a necessary and suffi- cient conditions for controllability of the bilinear Schr¨odinger equation ([2], [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For open quantum systems, the problem is more delicate: in [3], the author proved that an open quantum system is never small-time locally controllable (STLC) by means of coherent control, and that some configurations are not reachable in finite time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in [15], it has been pointed out that, as the underlying group is not compact, then LARC is a sufficient condition for accessibility, but does not imply controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Yet, in many experimental set-ups, there is no need to control the whole system: for instance, in typical situations the system of interest (B) is well isolated from the envi- ronment, and interacts with an (eventually open) accessory system (or ancilla), which can be directly controlled ([1], [7], [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In such situations, a natural question is to analyze the controllability properties of the subsystem B only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' this notion is called indirect controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For closed quantum systems, a detailed analysis of indirect controllability has been carried out in the papers [8], [9], in terms of Lie groups theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For open quantum systems, the problem has been investigated in specific situations (see for instance [11], [16]) but, to our knowledge, a general theoretical analysis is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In this paper, we focus on a composite quantum system made by two interacting qubits A and B, such that the control acts directly only on the subsystem A, which is subject to the interaction with the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' This work was supported by QUACO, PRC ANR-17-CE40-0007-01 and by CARTT-IUT de Toulon 1LIS, UMR CNRS 7020, Universit´e de Toulon, Aix Marseille Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=', France kateryna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='verzhanska@lis-lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='fr francesca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='chittaro@univ-tln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='fr the system of interest B does not interact directly with the controls and the environment (only through its interaction with A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As a first step towards the characterization of indirect controllability for such systems, we restrict our attention to a particular class of target states: the states (of the whole system) such that their reduction to B is “pure” ([20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, we ask ourselves the following question: is it possible to “purify” the state of the subsystem B and/or to keep it pure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Even if the existence of purifying dynamics is well known ([3]), they achieve complete purification only asymptotically in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Based on this fact, we remark in Corollary 1 that also “partial purification” is only asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We then investigate the possibility of protecting the sub- system B from dissipation, that is, to keep its state pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We analyze three possible interactions between the qubits, and show that the only dynamics that conserve partial purity are trivial or not affected by the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' These partial results provide a first negative answer to the question of indirect controllability in presence of dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The structure of the paper is the following: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' II we introduce the minimal relevant notions on bipartite quantum systems, and we exhibit the class of systems we are interested into;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' III, we discuss the structure of the set of admissible (physical) states, with a particular interest on states that correspond to pure reduced states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' finally, in Section IV we derived some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' STATEMENT OF THE PROBLEM A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Basic facts on open quantum systems In this section, we just provide a minimal description of the formalism of open quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For more details, we refer to the monographs [6], [12], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let us first start with some notations: we denote with her(n) (respectively, her0(n)) the set of Hermitian (respec- tively, traceless) n dimensional matrices, with su(n) the set of anti-Hermitian traceless n dimensional matrices and with so(n) the set of anti-symmetric real n dimensional matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Consider the Hilbert space CnA ⊗ CnB, for nA, nB ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The partial trace over CnA is the unique linear operator TrA : her(nAnB) → her(nB) such that for every MA ∈ her(nA) and every MB ∈ her(nB) it holds TrA(MA ⊗ MB) = MB(TrMA), where Tr denotes the usual trace operation on her(nA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The partial trace over CnB is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In the standard formulation of quantum mechanics, the state of a finite-dimensional quantum system is represented by a positive semi-definite Hermitian operator with trace one, acting on a finite-dimensional complex Hilbert space H (usually identified with CN, for some N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' such operator is called density matrix (or density operator) and is usually denoted by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In this paper, we denote with P the set of density matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=', positive semi-definite and Hermitian with trace one) on H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' P is a compact and convex subset of her(N), and its extreme points coincide with rank one projection operators on CN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' these states, called pure states in the language of quantum mechanics, are characterized by the the property Trρ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The other elements of P, characterized by Trρ2 < 1, are called statistical mixture or mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The quantity Trρ2 is thus called the purity of the state ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Pure states are particularly important in quantum mechan- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Indeed, the description of quantum systems in terms of pure density matrices is completely equivalent to the one provided by the state vector (or wavefunction) |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Pure states are, indeed, projectors on one dimensional subspaces of H, thus they uniquely determine a state vector, up to a physically irrelevant global phase ([6], [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In the language of quantum mechanics, we say that a quantum system is closed if it is isolated from other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The evolution of an closed quantum system is described by the Liouville-von Neumann equation ˙ρ = −i[H, ρ], where H is a Hermitian matrix, called the Hamiltonian of the system, representing the internal energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' By adding ex- ternal control fields (such as tunable electromagnetic fields), the perturbed system is governed by a new Hamiltonian which, in most relevant physical situations, can be written as H(u) = H0+� i uiHi, where H0 still represents the internal energy of the unperturbed system and Hi are associated with the external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In the literature, such controls are usually called coherent controls ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' With a little abuse of notation, in the field of quantum control we say that a system is closed if its dynamics are described by the Liouville equation, even in presence of interactions with external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' If the dependence of H on time is regular enough, −iH(t) is the generator of a unitary evolution group Ut,0 and, for every ρ0 ∈ P, the solution of the Liouville equation with initial condition ρ(0) = ρ0 can be written as ρ(t) = Ut,0ρ0U † t,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We remark that the Liouville equation conserves the spectrum of ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' as a consequence, Tr(ρ2(t)) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In other words, coherent control conserves the purity of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' When a quantum system interacts with a surrounding environment, its evolution is no longer unitary and reversible, and the general formalism of open quantum systems is required ([6]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' under some hypothesis on the environment (such as, Markovianity) the evolution of the quantum system can be described by the Gorini-Kossakowski-Sudarshan- Lindblad master equation (see [10], [18]) ˙ρ = −i[H, ρ] + LD(ρ), (1) where the operator LD can be written as LD(ρ) = N 2−1 � k=1 LkρL† k − 1 2L† kLkρ − 1 2ρL† kLk, the Lk being square matrices called jump (or noise) oper- ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In general, the number and the choice of the jump operators describing the same operator LD is not unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' nevertheless, there is always a choice of at most N 2 − 1 operators representing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' When L ̸= 0, the evolution governed by (1) preserves the trace and the positivity of the density matrix, but it is no more unitary and isospectral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For this reason, usually the term L is called dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Thanks to Choi-Kraus’ Theorem ([14]), for every t ≥ 0 the solution of (1) with initial condition ρ0 can be written as ρ(t) = � i Mi,tρ0M † i,t, where {Mi,t}i is a family of matrices on CN such that � i M † i,tMi,t = IN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' as for the jump operators, in general, this family is not uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Coherence vector representation Coherence vector representation is a well known tool in quantum control, that permits to write equation (1) as a linear differential equation on some real linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' First of all, we endow the space of N dimensional complex square matrices with the Frobenius scalar product ⟨⟨A, B⟩⟩ = Tr � A†B � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' noticing that every density matrix can be written as 1 N IN + ˆρ, with ˆρ ∈ her0(N), we choose an orthonormal basis {Λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' , ΛN 2−1} of her0(N) and we define the map Φ : P → RN 2 as Φ(ρ) = � 1 √ N , Tr(ρΛ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' , Tr(ρΛN 2−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The map Φ is called coherence representation of ρ ([3], [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Setting vρ = Φ(ρ) and vρ = ( 1 √ N , vρ), we call vρ ∈ RN 2−1 the vector of coherences or Bloch vector of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' It is easy to prove that Tr(ρ2) = ∥vρ∥2, which in particular implies that ∥vρ∥2 ≤ 1 − 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let us call K the image of P via the coherence represen- tation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' it is easy to see that Φ is an isomorphism between P and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The properties of K are well-known in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' we are resuming them in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Proposition 1 ([3], [5], [15], [13]): Let q0 = ( 1 √ N , 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' , 0), and consider the affine hyperplane W0 = {( 1 √ N , v) : v ∈ RN 2−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' K is a compact convex neighborhood of q0 in W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' ∂K is the set of all vρ corresponding to singular density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' If N = 2, then K coincides with the set {( 1 √ 2, v) : ∥v∥ ≤ 1 √ 2} and can be identified with the three-dimensional ball of radius 1 √ 2 (called the Bloch ball);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' all vectors in the surface of the Bloch ball correspond to pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For N > 2, the positivity of ρ adds some constraints on K, which is strictly contained in the set {( 1 √ N , v) : ∥v∥ ≤ � 1 − 1 N };' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' the vectors corresponding to pure states constitute a proper subset of ∂K (more precisely, they correspond to the extreme points of K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In coherence representation, equation (1) becomes ˙vρ = � MH + ML � vρ, MH and ML being respectively the rep- resentations of the operators −i[H, ·] and L(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' It is worth notice (see [3], [15] for more details) that, due to the fact that the evolution induced by (1) is trace-preserving, the matrices MH and ML always have the block forms MH = � 0 0 0 � H � ML = � 0 0 v0 � D � , (2) where �H ∈ so(N 2 − 1), v0 ∈ RN 2−1 and �D is a square matrix of dimension N 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The coherence representation of equation (1) subject to coherent control is ˙vρ = � MH0 + � j ujMHj + ML � vρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' (3) The controllability properties of equation (3) have been studied in [3], where, in particular, it is stated that the system is never STLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We recall moreover the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Theorem 1: Let q0 ∈ ˚ K and q1 ∈ ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' There is no essentially bounded control function that steers q0 to q1 in finite (positive) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The system In this paper, we focus on coherently controlled composite open quantum systems, composed by two interacting qubits (called A and B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' we recall that a qubit is a quantum system living in a two-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' According to quantum mechanics ([6], [12], [20]), the total system A + B evolves on the tensor product of two Hilbert spaces, H = HA ⊗ HB, with HA = HB = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Also, we recall that the state of each subsystem (denoted respectively with ρA or ρB) can be extracted from the state ρ of the total system, by means of the partial trace: indeed, we stress that, if ρ is a density matrix (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Hermitian positive semi- definite and of trace one) on H, then ρA = TrBρ and ρB = TrAρ are density matrices on HA and HB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Physically, taking the partial trace on A can be interpreted as “averaging” on the information about A, so that the reduced state ρB describes the state of the subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Analogous considerations hold for ρA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The system A + B evolves according to equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We remark that we can always assume that the matrices H and Lk are traceless, as adding them multiples of the identity leaves invariant the right-hand side of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Then, H can be uniquely written as H = HA + HI + HB, where HA = hA ⊗ I2, HB = InA ⊗ hB, with hA, hB ∈ her0(2) and HI ∈ her0(2)⊗her0(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Actually, HI denotes the interaction between the subsystems A and B, while HA and HB the unitary free evolution of the two subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We make these further assumptions on the dynamics: the subsystem B does not interact directly with the environment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in particular, this implies that the jump operators can be taken of the form Lk = ℓk ⊗I2, where the ℓk’s are traceless matrices on HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' the control directly affects only the subsystem A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in particular, the Hamiltonian H can be written as H = � hA0 + 3 � j=1 ujhAj � ⊗ I2 + HI + I2 ⊗ hB, where the matrices hAj, j ≥ 0, HI and hB are constant and uj are functions of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' It is clear that, if HI = 0, then the two subsystems are completely independent, and, in particular, the evolution of B is not influenced by the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As already said, in [3] it is proved that the coherently controlled Lindblad equation is never STLC and Theorem 1 states that some transitions cannot be realized in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Nevertheless, it is still interesting to understand what can be said about the controllability of the state of the subsystem B only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, the main questions that one may ask include 1) is it possible to “protect” the subsystem B from dis- sipation, that is, to implement a unitary dynamics on B, at least on some submanifolds (for instance, the submanifold Trρ2 B = 1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' 2) is it possible to “control” the state ρB, regardless of ρ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Question 2) needs to be further clarified, as several notions of “partial controllability” are possible (see for instance [9]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in any of its declination, tackling the issue is a very hard task, even in the simplest case of two qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Question 1) seems to be more affordable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In this paper, we provide a first step towards the answer: for three particular choices of the interaction HI (the well-known dispersive and resonant couplings, very common in experimental set- ups), and a generic choice of the dissipative term, we show that the trajectories keeping ρB pure are trivial, or follow a free evolution which is not affected by the controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In our opinion, more general interaction would lead to similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' ON THE STRUCTURE OF K In the case of two interacting qubits, we choose the following orthonormal basis of her0(4) Λi = 1 2σi ⊗ I2, i = 1, 2, 3 Λ3i+j = 1 2σi ⊗ σj, i = 1, 2, 3, j = 1, 2, 3, Λ12+j = 1 2I2 ⊗ σj, j = 1, 2, 3, where σi, i = 1, 2, 3, denote the Pauli matrices σ1 = � 0 1 1 0 � , σ2 = � 0 −i i 0 � , σ3 = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' For such system, in the following we will adopt also the more intuitive notation vρ = �1 2, vA, vAB, vB� , with vA, vB ∈ R3, and vAB ∈ R9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Thanks to the peculiarity of the Pauli matrices (in partic- ular, the fact that σiσj + σjσi = 2δijI2), the matrices �H and �D and the vector v0 in equation (2) have the following block structure �H = \uf8eb \uf8ec \uf8ed ˆhA �HIt 0 �HIl ˆhA ⊗ I3 + I3 ⊗ ˆhB �HIr 0 �HIb ˆhB \uf8f6 \uf8f7 \uf8f8 (4) �D = \uf8eb \uf8ed ˆd 0 0 0 ˆd ⊗ I3 v0 ⊗ I3 0 0 0 \uf8f6 \uf8f8 v0 = (v01, v02, v03, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' , 0) (more details on the objects appearing here above are pro- vided in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In order to answer to question 1, we look for a character- ization of states ρ such that TrAρ is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Proposition 2: Consider ρ ∈ P such that Trρ2 B = 1, where ρB = TrAρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Then Φ(ρ) ∈ ∂K and there exist vA, vB ∈ R3 with ∥vA∥2 ≤ 1 4 and ∥vB∥2 = 1 4 such that Φ(ρ) = �1 2, vA, 2vA ⊗ vB, vB� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' (5) Proof: The first claim is an easy consequence of [17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let us now prove the second part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The bound on ∥vA∥ and the value of ∥vB∥ yield from the fact that Tr(TrBρ)2 = 1 2 + 2 �3 j=1(vA j )2 and Tr(TrAρ)2 = 1 2 + 2 �3 j=1(vB j )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Moreover 1 4 d dtTrρ2 B = ⟨vB,�hBvB⟩ + ⟨vB, �HIbvAB⟩ (6) = 3 � r,s=1 λrs⟨vB, Ts(vAB (r−1)+1, vAB (r−1)+2, vAB (r−1)+3)⟩, where we used equation (8) and the fact that �hB is antisym- metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' By hypothesis, Trρ2 B cannot increase with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' On the other hand, as {T1, T2, T3} constitute a ba- sis of so(3), and equation (6) holds for any HI, ⟨vB, A(vAB (r−1)+1, vAB (r−1)+2, vAB (r−1)+3)⟩ must be zero for ev- ery antisymmetric matrix A and for r = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Thus, the vectors (vAB (r−1)+1, vAB (r−1)+2, vAB (r−1)+3) must be collinear to vB, and ρ can be written as ρ = 1 4I4 + σA ⊗ I2 √ 2 + 1 2 � γσ1 + βσ2 + θσ3 + I2 � ⊗ � vB 1 σ1 + vB 2 σ2 + vB 3 σ3), for some σA = 1 √ 2 � vA 1 σ1+vA 2 σ2+vA 3 σ3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let {ψ1, ψ2} be the (normalized) eigenvectors of vB 1 σ1 +vB 2 σ2 +vB 3 σ3, rel- ative to the eigenvalues µ1 = − 1 2 and µ2 = 1 2, respectively, and let {φ1, φ2} be any (possibly different) orthonormal basis of C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' let λ be an eigenvalue of ρ, and ϕ a corresponding eigenvector, which can be written as ϕ = �2 i,j=1 aijφi ⊗ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Set Aj = σA √ 2 + µj 2 (γσ1+βσ2+θσ3+I2)+ 1 4I2, for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We remark that the spectrum of ρ is given by the union of the spectra of A1 and A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, the eigenvalues of A1 are given by ν± = ± � �vA 1 2 − γ 4 �2 + �vA 2 2 − β 4 �2 + �vA 3 2 − θ 4 �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As they belong also to the spectrum of ρ, which is positive semidefinite, then it must be vA 1 = γ 2 , vA 2 = β 2 , and vA 3 = θ 2, and the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Remark 1: Proposition 2 can be easily generalized to the case in which the subsystem A has (complex) dimension nA ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' FIRST ANSWERS TO QUESTION 1) Proposition 2 imposes a constraint on the structure of states ρ whose reduction to B is pure, and can be thus exploited to study the controllability of the reduced states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' First of all, together with Theorem 1, it yields the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Corollary 1: Let ρ0, ρT ∈ P such that Φ(ρ0) belongs to the interior of K and Tr(TrAρT )2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Then, there is no essentially bounded control function that can send ρ0 to ρT in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In other to give partial answers to Question 1, we study the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Problem 1: Let ρ0 such that Tr � TrAρ0)2 = 1, and call ρ(t) the solution at time t of (1), under the assumption in Section II-C, with ρ(0) = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Is it possible to find ǫ > 0 and a control function defined on [0, ǫ] such that Tr � TrAρ(t))2 = 1 for t ∈ [0, ǫ]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Assume that it is true, for some piecewise-C∞ con- trol function ˆu : [0, ǫ] → R3, Set Φ(ρ(t)) = � 1 2, vA(t), vAB(t), vB(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' By Proposition 2, if Trρ2 B(t) = 1 for t ∈ [0, ǫ], then vAB(t) = 2vA(t) ⊗ vB(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, for every k ≥ 1, we have dk dtk vAB(t) = 2 k � j=0 �k j � djvA dtj ⊗ dk−jvB dtk−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' (7) In the following, we will use equations (7) to find out the controls satisfying the claim (if any), for different expres- sions of the interaction HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In order to do it, we define the vector w(t) = ˙vAB(t) − 2 � j=0,1 � k j � djvA dtj ⊗ dk−jvB dtk−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Without loss of generality, we can choose bases on HA and HB such that HA0 = ωaσ3 HB0 = ωbσ3 Eventually performing a linear transformation in the control space, we also assume that HAi = σi for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' “Dispersive” coupling: HI = gσ3 ⊗ σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Computing the vector w, we notice that w3 = gvA 2 (1 − 4(vB 3 )2) w8 = gvB 2 (1 − 4(vA 3 )2), so that equation (7) is satisfied only if (vB 3 )2 is identically equal to 1 4 (which implies vB 1 ≡ vB 2 ≡ 0) or if (vA 3 )2 ≡ 1 4 (which implies vA 1 ≡ vA 2 ≡ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We remark that the first scenario corresponds to freezing ρB to the state 1 2 � I2+σ3 � or to the state 1 2 � I2−σ3 � , that is, ρB is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, by computation we notice that, for this choice of HI, equation (3) has the block-triangular form \uf8eb \uf8ed ˙z1 ˙z2 ˙vB 3 \uf8f6 \uf8f8 = \uf8eb \uf8ed C11 C12 C13 0 C22 0 0 0 0 \uf8f6 \uf8f8 \uf8eb \uf8ed z1 z2 vB 3 \uf8f6 \uf8f8 with z2 = (vAB 1 , vAB 2 , vAB 4 , vAB 5 , vAB 7 , vAB 8 , vB 1 , vB 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' If vB 3 = | 1 2| and ρB is pure at t = 0, then z2(0) = 0, so that z2(t) is zero for all t ≥ 0, and every value of the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In particular, the submanifolds of P {ρA ⊗ 1 2 � I2 + σ3 � : ρA = ρ† A, ρA ≥ 0, TrρA = 1} {ρA ⊗ 1 2 � I2 − σ3 � : ρA = ρ† A, ρA ≥ 0, TrρA = 1}, are invariant for equation (1), for every choice of the control and for any dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let us now consider the second case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' first of all, by (3), we notice that vA 3 can be constantly equal to | 1 2| only if v03 2 +vA 3 ˆd33 = 0, which restrict the class of dissipative terms L that allow such behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Some examples of such operators are the so-called amplitude damping channels ([20]), that is, associated respectively to the jump operators σ+ ⊗ I2 or σ− ⊗ I2, where σ± = σ1 ± iσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' On the other hand, taking into account the fact that the state is factorized, the equation for vB becomes \uf8eb \uf8ed ˙vB 1 ˙vB 2 ˙vB 3 \uf8f6 \uf8f8= \uf8eb \uf8ed 0 −ωb − 2gvA 3 0 ωb + 2gvA 3 0 0 0 0 0 \uf8f6 \uf8f8 \uf8eb \uf8ed vB 1 vB 2 vB 3 \uf8f6 \uf8f8, which is unaffected by the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' “Resonant” coupling: HI = g(σ+⊗σ−+σ−⊗σ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' First of all, we remark that we can write HI = g 2(σ1 ⊗σ1+σ2⊗σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As we did above, we try to find conditions that guarantee that w is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' As ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='w = g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='−4(vA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='3 vB ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='3 vB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='3 )) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='4(vA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='2 vB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='1 − vA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='1 vB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='2 )(vA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='3 − vB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='3 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 we deduce that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' if w is null,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' then vA 2 vB 1 = vA 1 vB 2 and/or vA 3 = vB 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In the first case, plugging the equality into the expression of w and setting it to zero, we obtain \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 vA 1 (1 − 4vA 3 vB 3 ) = 0 vA 2 (1 − 4vA 3 vB 3 ) = 0 vB 1 (1 − 4vA 3 vB 3 ) = 0 vB 2 (1 − 4vA 3 vB 3 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' These four equations are satisfied if vA 1 = vA 2 = vB 1 = vB 2 = 0, or if vA 3 vB 3 = 1/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' in both cases, due to the fact that ρB is pure and to the constraints on the length of vA and vB, we obtain that |vA 3 | = |vB 3 | = 1/2 and that also ρA is a pure state (and, as a consequence of the “factorized structure”, the whole state ρ is pure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' If instead vA 3 = vB 3 , setting to zero the second and the fourth components of w, we obtain that |vA 1 | = |vB 2 | and |vA 2 | = |vB 1 |, which again implies that ρA is a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Summing up, in presence of resonant coupling, it is not possible to keep the partial state ρB pure if the whole state ρ itself is not kept pure by the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' On the other hand, as d dtTrρ2 = 2Tr(ρL(ρ)), it follows that the state ρB can be kept pure only if ρ evolves in the set Tr(ρL(ρ)) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' for “factorized states” (that is, of the form (5)), this happens only for states such that v0 2 + ˆdvA = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' depending on the particular choice of L, this equation may not have solutions vA of norm 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' HI = gσ3 ⊗ σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We finally discuss a further case in which the interaction does not commute with the free Hamiltonian HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Setting w ≡ 0, we find the following constraints: either vB 2 ≡ vB 3 ≡ 0 and (vB 1 )2 ≡ 1/4, or vA 1 ≡ vA 2 ≡ 0 and (vA 3 )2 ≡ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' The first case does not correspond to an admissible so- lution of the control system: indeed, by computations, it is possible to see that no control can keep ρ in a state of the form ρA ⊗ 1 2 � I2 ± σ1 � on a nonzero time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Let us now look for admissible trajectories along which vA 1 ≡ vA 2 ≡ 0 and (vA 3 )2 ≡ 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' First of all, as ˙vA 3 = v03 2 +vA 3 ˆd33+vA 1 ( ˆd31−u2)+vA 2 ( ˆd32+u1), such trajectories are admissible only if vA 3 ˆd33 + v03 2 = 0, that is true for some particular dissipation terms only (as we already saw, the amplitude damping channels satisfy such a constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Inspecting the differential equations for vA 1 and vA 2 , we see that they stay constant only for the choice of the control u1 = v02 + 2vA 3 ˆd23 2vA 3 u2 = −v01 + 2vA 3 ˆd13 2vA 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Moreover, as this guarantees w ≡ 0, that is, vρ has the form (5), we can substitute the values of vAB into the differential equations for vB, getting \uf8eb \uf8ed ˙vB 1 ˙vB 2 ˙vB 3 \uf8f6 \uf8f8 = \uf8eb \uf8ed 0 −ωb 0 ωb 0 −2gvA 3 0 2gvA 3 0 \uf8f6 \uf8f8 \uf8eb \uf8ed vB 2 vB 2 vB 3 \uf8f6 \uf8f8 , that is, the dynamics of ρB are protected from dissipation, but the control does not affect them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' CONCLUSIONS In this paper, we presented a first analysis on the indirect controllability properties of a 2-qubit system, in the case in which the ancilla is subject to dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' First of all, we observed that states ρ such that their reduction TrAρ is pure are not reachable (in finite time) from the interior of the space P, thus obtaining a first obstruction to the indirect controllability of the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' it would be interesting to investigate if such states are reachable from any other point of the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We then focus on the possibility of preserve the subsystem B from dissipation, that is, to find admissible trajectories ρ(t) such that TrAρ(t) is pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' We investigated three par- ticular cases of interaction (among them, the well known dispersive and resonant couplings), and we found that the only admissible trajectories are either trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' their re- duction TrAρ(t) is constant) or are unaffected by the action of the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' In our opinion, similar results hold also for other interaction Hamiltonians HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' APPENDIX First of all we recall that the matrices T1 = � 0 0 0 0 0 −1 0 1 0 � T2 = � 0 0 1 0 0 0 −1 0 0 � T3 = � 0 −1 0 1 0 0 0 0 0 � are the representations on R3 of the operators −i[ σj 2 , ·], j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Then, if hA = �3 j=1 αj 2 σj and hB = �3 j=1 βj 2 σj (we recall that we assumed them traceless), the matrices ˆhA and ˆhB in (4) are simply ˆhA = �3 j=1 αjTj and ˆhB = �3 j=1 βjTj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Writing HI = 1 2 �3 i,j=1 λijσi ⊗ σj, long but easy com- putations give �HIt = 3 � j=1 Tj ⊗ (λj1, λj2, λj3) �HIb = 3 � j=1 (λ1j, λ2j, λ3j) ⊗ Tj (8) and the blocks �HIl and �HIr can be recovered by antisym- 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and Quantum Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} +page_content=' Cambridge University Press, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf'} diff --git a/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/2301.00412v1.pdf.txt b/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/2301.00412v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f4c5c5cb7ae5f43efc5408d1fc590594b7e4eb93 --- /dev/null +++ b/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/2301.00412v1.pdf.txt @@ -0,0 +1,2104 @@ +arXiv:2301.00412v1 [nlin.PS] 1 Jan 2023 +Whitham Shocks and Resonant Dispersive Shock Waves +Governed by the Higher Order Korteweg-de Vries Equation +Saleh Baqer, +Department of Mathematics, Faculty of Science, +Kuwait University, Kuwait City 13060, Kuwait +Noel F. Smyth, +School of Mathematics, University of Edinburgh, +Edinburgh, Scotland, EH9 3FD, U.K. +and +School of Mathematics and Applied Statistics, +University of Wollongong, +Northfields Avenue, Wollongong, New South Wales, Australia, 2522. +Abstract +The addition of higher order asymptotic corrections to the Korteweg-de Vries equation +results in the extended Korteweg-de Vries equation. These higher order terms destabilise +the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and +result in the emission of resonant radiation. +In broad terms, there are three possible +dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dis- +persive shock wave (CDSW) and travelling dispersive shock wave (TDSW). While there +are existing solutions for the RDSW and TDSW regimes obtained using modulation the- +ory, there is no existing solution for the CDSW regime. +Modulation theory and the +associated concept of a Whitham shock are used to obtain this CDSW solution. In addi- +tion, it is found that the resonant wavetrain emitted by the extended Korteweg-de Vries +equation with water wave coefficients has a minimal amplitude. This minimal amplitude +is explained based on the developed Whitham modulation theory. +This paper is dedicated to Gerald B. Whitham, FRS whose prophetic speculations on dis- +persive shocks in the 1960s and 1970s have been verified by this and other publications. +1 +Introduction +A solitary wave is the standard solution of nonlinear, dispersive wave equations, intensively +studied since the 1960s, when it was found that certain solitary wave supporting equations +are integrable via the method of inverse scattering [1]. A solitary wave is termed a soliton +as for such equations solitary waves interact cleanly, with no change of form (other than +a phase shift), the word soliton then chosen due to its connotation with interacting sub- +atomic particles. The dispersive shock wave (DSW), also termed an undular bore in fluid +mechanics, is another generic solution of nonlinear, dispersive wave equations which has been +receiving increased study. In its standard form a DSW is a non-steady modulated wavetrain, +consisting of solitary waves at one edge and linear dispersive waves at the opposite edge, +1 + +connecting two distinct flow states. The separation between these two edges continuously +increases. +DSWs are formed due to the dispersive resolution of a discontinuity, with the +simplest initial condition generating one being a step. Undular bores were first observed as +the tidal bores which occur in coastal areas of strong tide and appropriate topography, for +example in Australia, Brazil, Canada, China, France, the United Kingdom and the United +States. However, DSWs are more widely observed in nature, with applications to water waves +[2, 3], meteorology [4, 5, 6], oceanography [7], geophysics [8, 9, 10], solid mechanics [11], +nonlinear optics [12, 13, 14, 15, 16, 17, 18, 19], Bose-Einstein condensates [21] and Fermionic +fluids [22]. +DSW solutions are found using Whitham modulation theory [1, 23, 24]. Whitham mod- +ulation theory is an asymptotic technique used to study slowly varying periodic wave solu- +tions of dispersive wave equations and gives partial differential equations, termed modulation +equations, for the slowly varying parameters of the wavetrain, for instance its amplitude, +wavelength and mean height. In the case for which these modulation equations form a hy- +perbolic system the underlying periodic wavetrain is modulationally stable. A simple wave +solution of these hyperbolic modulation equations is then the DSW solution of the underlying +equation [26, 27]. Such simple wave solutions are easy to determine if the modulation equa- +tions can be set in Riemann invariant form, which is guaranteed if the underlying equation +is integrable [28], but is difficult for non-integrable equations. It has been found that in the +solitary wave and linear wave limits modulation equations have degenerate forms, from which +it was shown that the solitary wave and linear wave edges of a DSW can be determined +without knowledge of the full modulation equations or their Riemann invariant form [29]. An +extensive review of DSWs, their physical applications and the connections between Whitham +modulation equations and DSWs can be found in [30]. +This standard view of DSWs substantially alters for nonlinear, dispersive wave equations +with non-convex dispersion. This non-convex dispersion allows dispersive radiation to be in +resonance with solitary waves, so that the solitary wave sheds dispersive radiation and so +is nonlocal [31, 32, 33, 34, 35, 36]. The solitary wave then radiates away. As a DSW is a +modulated wavetrain with solitary waves at one edge, non-convex dispersion means that a +DSW can also be resonant, shedding radiation, with the individual waves of the DSW being in +resonance, not just the solitary wave at one edge. As the DSW connects distinct flow states, +it does not radiate away as the mass shed in radiation is replaced by that of the flow state into +which it expands. The addition of higher order dispersion to standard nonlinear, dispersive +wave equations, such as the Korteweg-de Vries equation [37, 38] and the nonlinear Schr¨odinger +(NLS) equation [39, 40, 41, 42, 43], results in their DSWs being resonant. In addition, the +DSW solution of the equations governing nonlinear optical beam propagation in nematic liquid +crystals can be resonant or non-resonant [16, 18, 19, 20], depending on the size of the jump of +beam power across it. The extended Korteweg-de Vries (eKdV) equation is an extension of +the standard Korteweg-de Vries (KdV) equation for which the next higher order dispersive, +nonlinear and nonlinear-dispersive terms are included in the asymptotic expansion [1] which +yields the KdV equation from more general equations, such as the water wave equations +[44]. The eKdV DSW solution, undular bore, is resonant [38], with the higher order, fifth +order, dispersion being a major driver of this [37]. This resonance has a profound effect on +the structure of the DSW, with the classical structure outlined above destroyed if the higher +order terms are strong enough. Three regimes have been identified, radiating dispersive shock +wave (RDSW), see Figure 1(c), crossover dispersive shock wave (CDSW), see Figure 1(b), and +travelling dispersive shock wave (TDSW), see Figure 1(d). An RDSW occurs when the effect +2 + + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 +-200 +-150 +-100 +-50 + 0 + 50 + 100 + 150 + 200 +u +x +(c) +-0.2 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 + 0 + 500 + 1000 + 1500 + 2000 + 2500 +u +x +(b) +-0.2 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 +-200 + 0 + 200 + 400 + 600 + 800 + 1000 +u +x +(c) +-0.2 +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0 + 200 + 400 + 600 + 800 + 1000 +u +x +(d) +Figure 1: DSW types. (a) classical KdV DSW with ci = 0, i = 1, . . . , 4, (b) non-classical +CDSW with c1 = −1, c2 = c3 = 1, c4 = 0.4, (c) non-classical RDSW with c1 = −1, +c2 = c3 = 1, c4 = 0.3, (d) non-classical TDSW with c1 = −1, c2 = c3 = 1, c4 = 2.0 . In these +figures, t = 50, ǫ = 0.15 with u− = 0.5 and u+ = 0. +of the higher order terms is weak and so is a perturbed, radiating version of the classical +DSW. Above a threshold the resonance completely destroys the classical DSW structure, +leaving a non-oscillatory jump between the initial levels of the step generating it. Essentially, +the classical DSW is radiated away by the dominant resonant wavetrain. +Between these +two regimes, there is the CDSW, illustrated in Figure 1(b). The classical DSW, illustrated in +Figure 1(a) by the KdV DSW, is destabilised by the shed resonant wavetrain, which results in +a structure similar to that for the focusing NLS equation with the amplitude ordered classical +DSW of Figure 1(a) replaced by an amplitude disordered CDSW of nearly constant average +amplitude at its leading edge, but with a rapid decrease to the level behind over its trailing +edge, as seen in Figure 1(b), so that a CDSW has a nearly constant amplitude over much +of its length [45]. That is, both the unstable focusing NLS DSW and the CDSW are large +genus wavetrains (multi-phase wavetrains) that result in the long term formation of a so-called +soliton gas (a large number of randomly interacting solitary waves) [45, 46, 47]. However, +the latter sheds resonant radiation due to the non-convex dispersion effect. The connection +between a soliton gas and a CDSW deserves more detailed study, which is beyond the present +work . Figure 1(d) displays the time evolution of the resonant wavetrain propagating ahead of +the CDSW to emphasise details of its modulational instability. This instability is analogous +to the Benjamin-Feir instability observed in water waves as sideband perturbations [1, 25], +see [48] for experimental photographs of this phenomenon. The underlying high frequency +resonant wavetrain with its unstable amplitude modulation is clear. +3 + +The eKdV equation RDSW can be found as a perturbation of the KdV equation DSW [49]. +The TDSW solution has also been found as a TDSW is essentially a jump between two levels +connected to the resonant wavetrain [37, 50, 51]. The present work will determine the eKdV +equation CDSW solution based on Whitham modulation theory and the associated concept of +a Whitham shock, which is a shock in the modulation variables [24, 1, 51] and links the CDSW +to the resonant wavetrain. When Whitham developed modulation theory he speculated on +the role of shocks in the case for which the modulation equations were hyperbolic, so that +the underlying periodic wavetrain is stable [24], but did not explore the topic in extensive +detail. The use of shocks in hyperbolic modulation equations has only recently been developed +[51, 52], with applications to the Kawahara equation [51], the fifth order KdV equation [51] +and the BBM equation [53]. Finally, it has been previously found that there is a resonant +wave amplitude minimum for the CDSW solution of the eKdV equation with water wave +coefficients [38]. This amplitude minimum is deduced as a result of the developed modulation +theory. +2 +Extended Korteweg-de Vries equation +The Korteweg-de Vries (KdV) equation arises as an asymptotic approximation to more general +equations in the long wavelength, weakly nonlinear limit when weak nonlinearity is balanced +with weak dispersion [1]. If this asymptotic expansion is taken to one order beyond the KdV +approximation, the result is the extended Korteweg-de Vries (eKdV) equation [44] +ut + 6uux + uxxx + ǫ +� +c1u2ux + c2uxuxx + c3uuxxx + c4uxxxxx +� += 0. +(1) +Here, ǫ is a parameter measuring the weak nonlinearity, which for water waves is the ratio of +the wave amplitude to the undisturbed fluid depth. In the particular case of surface water +waves the higher order coefficients are c1 = −3/2, c2 = 23/4, c3 = 5/2 and c4 = 19/40 +[44]. The eKdV equation also arises in the nonlinear optics of coherent beam propagation +in nematic liquid crystals [19], for which ǫ is the ratio of the optical beam amplitude above +a background level to this background level, and nonlinear elasticity [11]. The higher order +coefficients ci, i = 1, . . . , 4, are involved in this case and can be found in the original work. +The Kawahara equation is the special case for which there is only fifth order dispersion [31], +ci = 0, i = 1, 2, 3, and c4 ̸= 0, +ut + 6uux + uxxx + ǫc4uxxxxx = 0. +(2) +The Kawahara equation arises for gravity-capillary waves for which the Bond number is near +1/3 [37]. In the present work, to generate a DSW the step initial condition +u(x, 0) = +� u+, +x > 0 +u−, +x < 0 +(3) +will be used, with u− > u+. +The eKdV equation has the mass conservation equation +∂ +∂tu + ∂ +∂x +� +3u2 + uxx + ǫ +�1 +3c1u3 + c3uuxx + 1 +2 (c2 − c3) u2 +x + c4uxxxx +�� += 0. +(4) +4 + +The eKdV energy conservation is less straightforward to derive [19]. On multiplying the eKdV +equation (1) by u and integrating by parts gives +∂ +∂t +1 +2u2 + ∂ +∂x +� +2u3 + uuxx − 1 +2u2 +x + ǫ +�1 +4c1u4 + 1 +2c2u2uxx + c4uuxxxx − c4uxuxxx ++ 1 +2c4u2 +xx +�� ++ ǫ +� +c3 − 1 +2c2 +� +u2uxxx = 0. +(5) +To set the final term in conservation form we use the fact that ǫ is small. The first order +equation, the KdV equation (ǫ = 0), gives +∂ +∂tu3 = −3u2 (6uux + uxxx) = − ∂ +∂x +9 +2u4 − 3u2uxxx, +(6) +so that (5) becomes the eKdV energy conservation equation +∂ +∂t +�1 +2u2 − 1 +3ǫ +� +c3 − 1 +2c2 +� +u3 +� ++ ∂ +∂x +� +2u3 + uuxx − 1 +2u2 +x + ǫ +�1 +4c1u4 ++ 1 +2c2u2uxx + c4uuxxxx − c4uxuxxx + 1 +2c4u2 +xx − 3 +2 +� +c3 − 1 +2c2 +� +u4 +�� += 0, +(7) +which is valid to O(ǫ2). We note that if c2 = 2c3, then this energy conservation law is exact. +However, this relation does not hold for the eKdV equation with the water wave coefficients. +The mass (4) and energy (7) conservation equations are of the form +∂Pm +∂t ++ ∂Qm +∂x += 0, +∂Pe +∂t + ∂Qe +∂x = 0, +(8) +where Pm, Pe and Qm, Qe are the mass and energy densities and fluxes, +Pm += +u, +(9) +Pe += +1 +2u2 − 1 +3ǫ +� +c3 − 1 +2c2 +� +u3, +(10) +Qm += +3u2 + uxx + ǫ +�1 +3c1u3 + c3uuxx + 1 +2 (c2 − c3) u2 +x + c4uxxxx +� +, +(11) +Qe += +2u3 + uuxx − 1 +2u2 +x + ǫ +�1 +4c1u4 + 1 +2c2u2uxx + c4uuxxxx − c4uxuxxx + 1 +2c4u2 +xx ++ 3 +2 +�1 +2c2 − c3 +� +u4 +� +, +(12) +respectively. +3 +Resonant wave Stokes expansion +It can be seen from Figure 1 that the amplitude of the resonant wavetrain generated by +a CDSW is small, certainly smaller than the amplitude of the DSW itself. The resonant +wavetrain can then be taken as a Stokes wave expansion of the form +ur = ¯ur + ar cos θr + a2 +ru2 cos 2θr + O(a3 +r), +ωr(kr, ar) = ω0 + arω1 + a2 +rω2 + O(a3 +r), +(13) +5 + +where ¯ur is the mean height of the wavetrain. +Here, ar is the amplitude of the resonant +wavetrain and ωr and kr are its frequency and wavenumber, with θr = krx−ωrt. Substituting +this Stokes wave expansion into the eKdV equation (1), separating out the resulting equation +at O(ǫn), n = 0, 1, 2, and eliminating secular terms gives +ω0 = (6¯ur + ǫc1¯u2 +r)kr − (1 + ǫc3¯ur) k3 +r + ǫc4k5 +r, +ω1 = 0, +(14) +ω2 = 36 + 24ǫc1¯ur − ǫ (48c3 − 6c1 − 6c2) k2 +r +24kr − ǫ (120c4k3r − 24c3¯urkr) ++ O +� +ǫ2� += +3 +2kr ++ ǫ +�1 +4 (c1 + c2 − 8c3 + 30c4) kr + +� +c1 − 3 +2c3 +� ¯ur +kr +� ++ O +� +ǫ2� +(15) +and +u2 = +6 + 2ǫc1¯ur − ǫ(c2 + c3)k2 +r +12k2r + 12ǫ (c3¯ur − 5c4k2r) k2r += +1 +2k2r +− 1 +12ǫ +� +c2 + c3 − 30c4 − 2 (c1 − 3c3) ¯ur +k2r +� ++ O +� +ǫ2� +. +(16) +The reason that the Stokes wave coefficients are expanded for small ǫ in (14)–(16) is to make +the Whitham modulation jump conditions that connect the bore with the resonant radiation +ahead in the CDSW regime as simple as possible. +4 +CDSW equal amplitude approximation +As seen from the example shown in Figure 2 the eKdV CDSW is unstable, as is its generated +resonant wavetrain, see Figure 2(b), and does not exhibit the standard KdV DSW structure, +as discussed in Section 1. The amplitudes of the waves of the CDSW do not decrease mono- +tonically from the leading to the trailing edges. The amplitudes of the waves are random due +to the instability, but distributed around a constant mean, with a rapid decrease to u− at the +trailing edge [45]. This broad structure of a CDSW can be exploited to obtain an approximate +solution for it [18, 19, 54], which can then be linked to loss radiated in the resonant wavetrain +[18, 19]. In essence, the CDSW is approximated by a train of uniform solitary waves, with +mass and energy conservation used to determine the amplitude and spacing of these waves. +The basis of the equal amplitude approximation is the solitary wave solution of the eKdV +equation. While there is no exact solitary wave solution of this equation, there is a perturba- +tion solution based on ǫ small [49] +us = ¯us + +� +as + ǫc6a2 +s +� +sech2 wsθs + ǫc7a2 +s sech4 wsθs + O(ε2), +(17) +with phase θs = x − Ust, inverse width ws = +� +as/2 and velocity Us = 2as + 4ǫc4a2 +s + O(ǫ2), +for which the new constants c6 and c7 are +c6 = −1 +6c1 + 1 +6c2 + 2 +3c3 − 5c4, +c7 = 1 +12c1 − 1 +4c2 − 1 +2c3 + 15 +2 c4. +(18) +Let us assume that in the CDSW regime the solution consists of the level ahead u+, +joining to a resonant wavetrain of amplitude ar, wavenumber kr and mean height ¯ur (the +frequency ωr given by the Stokes wave dispersion relation (13)), followed by a uniform train +of N(t) equal solitary waves of amplitude as on a mean level ¯us, which then links to the level +u− behind. This matches the general form of the example CDSW solution shown in Figure +1. +6 + +-0.2 + 0 + 0.2 + 0.4 + 0.6 + 0.8 + 1 + 1.2 +-300 +-200 +-100 + 0 + 100 + 200 + 300 + 400 +u +x +(a) +(b) +Figure 2: Example CDSWs. (a) Detail of CDSW at t = 50, (b) evolution detail of resonant +wavetrain. In these figures, c1 = −1, c2 = c3 = 1, c4 = 0.4 with ǫ = 0.15 and u− = 0.5, +u+ = 0. +The eKdV mass and energy conservation equations (4) and (7) are now integrated in x +from the level behind u− to the trailing edge of the resonant wavetrain, giving +N +� ∞ +−∞ +us dx += +� +3u2 +− + 1 +3ǫc1u3 +− − ¯Qmr +� +t, +(19) +N +� ∞ +−∞ +�1 +2u2 +s − 1 +3ǫ +� +c2 − 1 +2c2 +� +u3 +s +� +dx += +� +2u3 +− + 1 +4ǫ (c1 + 3c2 − 6c3) u4 +− − ¯Qer +� +t. (20) +Here, ¯Qmr and ¯Qer are the mass and energy fluxes (11) and (12) averaged over the Stokes wave +(13), respectively. Dividing these mass and energy results and evaluating the mass and energy +fluxes over the resonant wavetrain gives a relation linking the solitary wave amplitude as and +mean height ¯us to the amplitude ar, wavenumber kr and mean height ¯ur of the resonant +wavetrain. +As this expression is involved, particularly the energy density, the relation is +detailed in Appendix A. Dividing the mass and energy results (19) and (20) gives the implicit +relation +� ∞ +−∞ us dx +� ∞ +−∞ +� 1 +2u2s − 1 +3ǫ +� +c2 − 1 +2c2 +� +u3s +� +dx = +3u2 +− + 1 +3ǫc1u3 +− − ¯Qmr +2u3 +− + 1 +4ǫ (c1 + 3c2 − 6c3) u4 +− − ¯Qer +, +(21) +which determines the amplitude of the CDSW. +7 + +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.1 +200 +0.05 +300 +0 +400 +-0.05 +500 +600 +10 +700 +20 +30 +800 +40 +x +900 +50 +t5 +Modulation theory for CDSW and resonant wavetrain +The resonance condition between the CDSW and the resonant wavetrain is that the solitary +wave velocity Us equals the Stokes wave velocity, so that +2as + 4ǫc4a2 +s = c = ω0 + a2 +rω2 +kr +, +(22) +where ω0 and ω2 are given by the dispersion relation (13). +The example numerical solution of the eKdV equation in the CDSW regime shown in +Figure 1 shows that it consists of five distinct regions. The initial level behind u− links to +an unstable DSW, which then shows a sharp jump to the shed resonant wavetrain, which is +here modelled by a Whitham shock. The resonant wavetrain then transitions to the initial +level ahead u+ via a “partial DSW” [50, 55]. A “full DSW,” as for the standard KdV DSW, +is a transition between two uniform levels via a modulated wavetrain with solitary waves at +one edge and linear waves at the opposite edge. A partial DSW links two uniform wavetrains +via a modulated wavetrain whose amplitude, wavenumber and mean height are continuous +at the two edges. A full DSW is a limit of a partial DSW with no steady wavetrains at its +edges. The partial DSW smoothly raises the initial level ahead u+ to the mean level of the +resonant wavetrain ¯ur, which then jumps discontinuously to the mean level of the CDSW +¯us via the Whitham shock joining the resonant wavetrain to the CDSW. To fully determine +the partial DSW the Whitham modulation equations for the resonant wavetrain need to be +calculated [55]. This results in an involved system of equations [55] when the jump conditions +across the Whitham shock are included. However, ¯ur is very close to u+ [50] and to a good +approximation can be set to u+ and the extra accuracy obtained by the inclusion of the +partial DSW is minimal. Indeed, the mean level will be found to have minimal change over +the Whitham shock. +It can be seen from Figure 1 that the DSW and its associated resonant wavetrain are +unstable in the CDSW regime. The Whitham modulation equations can then be assumed +to be elliptic [1, 24], even though they have not been calculated. Although this ellipticity +conclusion comes from numerical observations and not from an exact calculation of the as- +sociated Whitham modulation system, it was analytically verified in [50] that the Whitham +modulation system becomes fully elliptic when the stable Kawahara DSW regime approaches +the CDSW regime. Extending the modulation theory work of [50] to eKdV CDSWs is outside +the scope of the present study and will not be dealt with here. It is another topic for future +study. +In theory elliptic systems do not have discontinuous solutions with shock waves. However, +Figure 1 shows that there is a clear rapid connection between the CDSW and the resonant +wavetrain, which we shall model as a discontinuity. We shall then model this connection by +a Whitham shock based on the conservation of mass and energy across it as these quantities +have to be conserved for any valid solution. This approach was found to be successful in the +study of the DSW solution of the nematic equations [18, 19]. +Averaging the mass and energy conservation equations (8) over the resonant wavetrain +ahead of the Whitham shock and the CDSW behind the shock gives the jump conditions +across the shock. As the CDSW is led by solitary waves, the Whitham shock velocity is equal +to the solitary wave velocity Us [1]. The jump conditions are then +− Us +� ¯Pmcdsw − ¯Pmr +� ++ +� ¯Qmcdsw − ¯Qmr +� += 0, +−Us +� ¯Pecdsw − ¯Per +� ++ +� ¯Qecdsw − ¯Qer +� += 0, +(23) +8 + +where ¯Pmcdsw, ¯Pecdsw and ¯Pmr, ¯Per denote the mass and energy densities (9) and (10) averaged +over the CDSW regime and the Stokes wave, respectively. Similarly, ¯Qmcdsw, ¯Qecdsw and +¯Qecdsw, ¯Qer denote the mass and energy fluxes (11) and (12) averaged over the bore in the +CDSW regime and the Stokes wave, respectively. +Here, Us denotes the Whitham shock +velocity. +The CDSW amplitude relation (21), the resonance condition (22) and the jump conditions +(23) give four relations for the five unknowns as, ar, ¯us, ¯ur and kr. With the approximation +that ¯ur = u+, this gives a complete system of equations. This system of equations to determine +the resonant CDSW were solved numerically using Newton’s method. It was found to be +sufficient to keep terms up to the orders O(ǫ) and O(a2 +r) in the CDSW amplitude relation +(21), the resonance condition (22) and the mass jump condition of (23) in order to keep +these equations as simple as possible. We note that the eKdV equation (1) as a reduction +of the water wave equations is asymptotically valid to O(ǫ). However, it was found to be +essential to keep all terms in the Whitham energy jump condition of (23) in order to obtain +good agreement with numerical solutions. Indeed, Newton’s method had a tendency to not +converge if all energy terms were not included. Including higher order terms in the CDSW +amplitude relation (21), the resonance condition (22) and the mass jump condition of (23) +resulted in no graphical difference in the resulting solution. Indeed, if the theoretical resonant +wave amplitude is truncated to the linear value Ar = ar, there is no observable difference in the +comparisons with numerical solutions. As the full detailed jump conditions, particularly the +energy jump condition, are then extensive, they are given in Appendix B. A similar situation +was encountered in the study of nematic CDSWs when the nonlocality of the nematic medium +is decreased to the local optical limit [19]. +6 +Comparisons with numerical solutions +The eKdV equation (1) was solved numerically using the pseudo-spectral method of Fornberg +and Whitham [27] as extended to enhance stability at high wavenumbers, particularly due +to the higher order fifth order dispersion [56, 57]. The spatial derivatives were calculated +in Fourier space, with the equation propagated in time using the fourth order Runge-Kutta +method in Fourier space. As stated, to enhance stability linear dispersion was propagated +using an integrating factor [56, 57]. Numerical solutions generated using this numerical scheme +will now be compared with solutions of the modulation theory of Section 5. As can be seen +from Figure 1 that the resonant wavetrain is unstable. The resonant wave amplitude was +then calculated by averaging the amplitude over the resonant wavetrain up to its front. +Figure 3 shows comparisons between full numerical solutions of the Kawahara equation +(2) and modulation theory for the resonant wavetrain amplitude +Ar = ar + a2 +ru2, +(24) +see (13), the CDSW solitary wave amplitude +As = as + ǫ (c6 + c7) a2 +s, +(25) +see (17), the resonant wavetrain wavenumber kr and the Whitham shock velocity Us. +In +general, the agreement between modulation theory and numerical solutions is excellent across +the CDSW regime. The agreement even extends beyond the CDSW regime into the RDSW +9 + + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.12 + 0.14 + 0.16 + 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 +Ar +ε +(a) +CDSW +RDSW +TDSW + 0.6 + 0.65 + 0.7 + 0.75 + 0.8 + 0.85 + 0.9 + 0.95 + 1 + 0 + 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 +As +ε +(b) +CDSW +RDSW +TDSW + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 0 + 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 +kr +ε +(c) +CDSW +RDSW +TDSW + 1.2 + 1.3 + 1.4 + 1.5 + 1.6 + 1.7 + 1.8 + 1.9 + 2 + 0 + 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 +Us +ε +(d) +CDSW +RDSW +TDSW +Figure 3: Comparison between numerical solutions of the Kawahara equation (2) and modu- +lation theory. Numerical solution: ◦ (red circle); modulation theory solution: red (solid) line; +boundaries between DSW regimes: green (dashed) line. (a) resonant wave amplitude Ar, (b) +solitary wave amplitude of CDSW As, (c) wavenumber of resonant wavetrain kr, (d) velocity +of Whitham shock Us. Here, c4 = 1, u− = 0.5 and u+ = 0. +and TDSW regimes, particularly for the wavenumber of the resonant wavetrain kr. +We +note that the resonant radiation is an unstable wavetrain (multi-phase wavetrain), so there +exists no single dominant wavenumber. The numerical values of kr were then determined by +averaging over the resonant wavetrain. It was found that averaging over 10 to 20 crests was +sufficient. The agreement is good in the TDSW regime, which is expected as modulation +theory in the TDSW regime is the limit of the present modulation theory if the amplitude +As of the solitary waves vanishes and the mean level becomes the level behind u− [51]. In +this context, it should be noted that in the present work the solitary wave amplitude As is +measured from the level ahead u+ = 0 as the equal amplitude approximation is based on the +leading edge of the DSW, so that in the TDSW limit As → u−, while in the work of Sprenger +and Hoefer [51] the wave amplitude is measured from the local mean level. The agreement +for the solitary wave amplitude As is less good in the RDSW regime as the DSW cannot +be approximated by a train of equal amplitude solitary waves since the DSW is a perturbed +KdV DSW in this regime [44]. +The present modulation theory gives that the amplitude Ar of the resonant wavetrain +rapidly approaches zero in the RDSW regime, as expected, but less rapidly than the numerical +amplitude. +Similarly, the modulation and numerical Whitham shock velocities Us are in +excellent agreement in the CDSW regime, and even in the RDSW regime. +The latter is +10 + +ǫ = 0.01 +ǫ = 0.03 +ǫ = 0.05 +ǫ = 0.1 +ǫ = 0.2 +¯us = 5.03 × 10−6 +¯us = 9.6 × 10−5 +¯us = 3.05 × 10−4 +¯us = 1.1 × 10−3 +¯us = 3.20 × 10−3 +Table 1: Modulation mean level ¯us of CDSW for Kawahara equation (2). +Here, c4 = 1, +u− = 0.5 and u+ = 0. +c1 = 0.006 +c1 = 0.660 +c1 = 1.333 +c1 = 2.000 +c1 = 2.266 +¯us = 3.4 × 10−4 +¯us = 5.3 × 10−4 +¯us = 9.3 × 10−4 +¯us = 1.58 × 10−3 +¯us = 1.96 × 10−3 +Table 2: Modulation mean level ¯us of the CDSW for the eKdV equation (1). The coefficients +c1, c2 and c3 are taken equal, while c4 is fixed at c4 = 0.351. Here, ǫ = 0.15, u− = 0.5 and +u+ = 0. +expected as the RDSW solution is a perturbation of the generic DSW solution. In the TDSW +regime, the modulation Whitham shock velocity differs slightly from the numerical velocity. +Table 1 displays the modulation theory solution for the mean level ¯us of the Kawahara +CDSW for a range of ǫ. As stated above, the Whitham shock results in a very small change +in mean level from that ahead, u+, which validates the assumption above used to solve the +system of modulation equations for the CDSW. +Table 2 shows the modulation solution for the mean height ¯us of the CDSW solitary waves +for a general eKdV equation. As for the Kawahara equation results of Table 1, the jump in +mean height across the Whitham shock is minimal, so much so that it is difficult to measure +from numerical solutions. We then conclude that the resonant wavetrain is again essentially +linear. As the mean level and the wave propagating on this mean level do not couple at +the linear level [1], the minimal variation of ¯ur from u+, as assumed in order to solve the +modulation equations, is then clear. +Figure 4 shows similar comparisons as Figure 3 for the Kawahara equation, but for a +general eKdV equation with c1, c2 and c3 non-zero. The overall agreement is similar to that +for the Kawahara equation in the CDSW regime. +The modulation theory resonant wave +amplitude Ar is in excellent agreement with the numerical amplitude, with reasonable agree- +ment even in the RDSW regime, as for the Kawahara equation. In contrast, the modulation +solitary wave amplitude As is in excellent agreement with the numerical amplitude, even in +the RDSW regime, for which good agreement is not expected, as discussed above for the +Kawahara equation results. Figures 4(c) and (d) show similar excellent agreement for the +resonant wavetrain wavenumber kr and the Whitham shock velocity Us, with the excellent +agreement holding into the RDSW regime. The Whitham shock velocity Us and the resonant +wavenumber kr are connected through the resonance condition (22), so this similar agreement +with numerical solutions is expected. +To additionally ensure the validity of the equal amplitude approximation used for the +unstable bore of the CDSW, we report a few comparison cases for the number of leading, +randomly distributed solitary waves N, on neglecting the descending waves near the trailing +edge which take the solution down to the initial level behind u−, as compared with numerical +results. In theory, N is given by equation (19) or equation (20). For example, with ǫ = 0.04, +c4 = 1 (ǫ = 0.14, c4 = 1) at t = 30, Kawahara CDSW modulation theory gives N = 8.6580 ≈ 9 +(N = 10.16 ≈ 10) and numerical solutions give N ≈ 9 (N ≈ 11). Similarly, with ǫ = 0.15, +c1 = c2 = c3 = 0.006 (c1 = c2 = c3 = 1.333), c4 = 0.3513 at t = 30, eKdV CDSW modulation +theory gives N = 8.9041 ≈ 9 (N = 8.463 ≈ 8), and numerical solutions give N ≈ 10 (N ≈ 8). +11 + + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +Ar +c1=c2=c3 +(a) +RDSW +CDSW + 0.84 + 0.86 + 0.88 + 0.9 + 0.92 + 0.94 + 0.96 + 0.98 + 1 + 1.02 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +As +c1=c2=c3 +(b) +RDSW +CDSW + 4.53 + 4.535 + 4.54 + 4.545 + 4.55 + 4.555 + 4.56 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +kr +c1=c2=c3 +(c) +RDSW +CDSW + 1.65 + 1.7 + 1.75 + 1.8 + 1.85 + 1.9 + 1.95 + 2 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +Us +c1=c2=c3 +(d) +RDSW +CDSW +Figure 4: Comparison between numerical solutions of the eKdV equation (1) with c1 = c2 = c3 +and c4 = 0.351 with modulation theory. Numerical solution: ◦ (red circle); modulation theory +solution: red (solid) line; boundaries between DSW regimes: green (dashed) line. (a) resonant +wave amplitude Ar, (b) solitary wave amplitude of CDSW As, (c) wavenumber of resonant +wavetrain kr, (d) velocity of Whitham shock Us. Here, ε = 0.15, u− = 0.5 and u+ = 0. +7 +Water Waves +One of the motivations behind the present work is the observed resonant wave amplitude +minimum for the eKdV equation (1) with the water wave coefficients [38]. This is illustrated +in Figure 5 based on two values of the nonlinearity parameter ǫ. Figure 5(a) displays the +water wave bore for ǫ = 0.15. The bore is in the RDSW regime and the resonant wavetrain +has amplitude ∼ 6 × 10−3. To illustrate the effect of the higher order coefficients on the +bore structure, Figure 5(b) displays the bore solution for the water wave coefficients c1, c2 +and c4, but with c3 = 0. The coefficient c3 was varied as this coefficient was found to have +the greatest effect on the resonant wave amplitude. The bore has become unstable and is +bordering on the CDSW regime with the resonant wavetrain having amplitude ∼ 2 × 10−2. +The increase of ǫ to 0.3 shows the same overall behaviour. The bore with the water wave +coefficients, Figure 5(c), is bordering on the CDSW regime, with the resonant wavetrain still +having minimal amplitude ∼ 1 × 10−2. In contrast with c3 = 0, Figure 5(d), the bore is +bordering on the TDSW regime, with the waves of the bore having much reduced amplitude +and extent, and the resonant wavetrain having amplitude ∼ 4 × 10−2. As well as greatly +reducing the resonant wave amplitude, the water wave coefficients delay the onset of the +transition between the bore regimes, RDSW to CDSW to TDSW. As the resonant wavetrain +12 + +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 250 + 300 +u +x +(a) +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 250 + 300 +u +x +(b) +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 250 + 300 +u +x +(c) +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 250 + 300 +u +x +(d) +Figure 5: Dependence of bore structure on higher order parameters with c1 = −3/2, c2 = 23/4 +and c4 = 19/40. (a) c3 = 5/2, ǫ = 0.15, (b) c3 = 0, ǫ = 0.15, (c) c3 = 5/2, ǫ = 0.3, (d) c3 = 0, +ǫ = 0.3. Here, u− = 0.5, u+ = 0 and t = 10. +amplitude increases on transition from RDSW to CDSW to TDSW, these two effects are +connected. The modulation theory of the present work will now be used to analyse the effect +of the values of ci, i = 1, . . . , 4, on the bore structure. +The modulation equations with the Whitham shock jump conditions for the eKdV equa- +tion do not have a (real) solution when the coefficients ci, i = 1, . . . , 4, take the water wave +values. However, a solution of the modulation equations does exist for ranges of the higher +order coefficients. To be specific, the dependence of the existence of modulation theory so- +lutions on the higher order coefficients will be explored by varying the nonlinearity ǫ with +c1, c2 and c4 taking the water wave values and determining the existence ranges in the final +higher order coefficient c3. Figure 6(a) shows the existence range of the modulation theory +solution for a range of the nonlinearity ǫ, up to high values which are outside the range of +asymptotic validity of the eKdV equation, noting that the water wave value is c3 = 5/2. +Figure 6(b) shows a bore for c3 = 3.5 and ǫ = 0.3, which is in the region for which the +modulation equations do not have a solution. The location of this c3 value is shown by the +upper black dot in Figure 6(a). Comparing with Figure 5(d) it can be seen that the reso- +nant wavetrain has greatly reduced amplitude, in agreement with modulation theory. The +location of the c3 value used for Figure 5(d) is shown by the lower black dot in Figure 6(a). +The resonant wave amplitude of Figure 6(b) is of the same order as the example shown in +Figure 5(c), which is for the water wave coefficients for the same value of ǫ. While this is not +conclusive justification for the low resonant wave amplitudes seen in Figures 5(a) and (c) and +13 + +6(b) as the bore for ǫ = 0.15 is in the RDSW regime and for ǫ = 0.3 is just in the CDSW +regime and the present modulation theory is for the CDSW regime, it is consistent with the +observed numerical results and provides justification to some degree. A final observation from +these modulation theory results is that the minimal resonant wave amplitude exists for an +unbounded range of c3. This is in contrast to the conclusion of [38] that an amplitude node +exists for a discrete combination of the higher order coefficients ci, i = 1, . . . , 4, with small +amplitude in a neighbourhood of this node. This conclusion was based on previous results for +resonant solitary waves governed by the eKdV equation, as detailed in this work. It can then +be concluded that results for solitary waves do not necessarily transfer to bores governed by +the same equation. +-2 +-1 + 0 + 1 + 2 + 3 + 4 + 5 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 +c3 +ε +(a) +-0.1 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 + 1 +-100 +-50 + 0 + 50 + 100 + 150 + 200 + 250 + 300 +u +x +(b) +Figure 6: (a) Existence interval for the modulation solution as the nonlinearity ǫ and c3 vary. +The modulation solution exists in the red (shaded) region. (b) Example of bore for c3 = 3.5 +and ǫ = 0.3 at t = 10. The other coefficients are the water wave values c1 = −3/2, c2 = 23/4 +and c4 = 19/40. Here, u− = 0.5 and u+ = 0. + 0 + 0.02 + 0.04 + 0.06 + 0.08 + 0.1 + 0.12 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 +Ar +ε +(a) + 0 + 0.005 + 0.01 + 0.015 + 0.02 + 0.025 + 0.03 + 0.035 + 0.04 + 0.045 + 0.05 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +Ar +c3 +(b) +Figure 7: (a) Resonant wave amplitude Ar just below the existence borderline of Figure 6. +(b) Resonant wave amplitude Ar as c3 varies with ǫ = 0.13. Modulation theory amplitude: +red (solid) line; numerical amplitude: ◦ red circle. The other coefficients are the water wave +values c1 = −3/2, c2 = 23/4 and c4 = 19/40. Here, u− = 0.5 and u+ = 0. +Figure 7(a) shows the resonant wave amplitude Ar just below the modulation theory cutoff +of Figure 6(a) as given by modulation theory and full numerical solutions. It can be seen +14 + +that the agreement between theory and numerical solutions is excellent and that the resonant +wave amplitude is small, even up to very large values of the nonlinearity parameter ǫ. Figure +7(b) details the rapid decrease of the resonant wave amplitude Ar as the modulation theory +borderline of Figure 6(a) is crossed. For this figure the higher order coefficient c3 was varied, +while c1, c2 and c4 were kept at their water wave values. The nonlinearity parameter used +was ǫ = 0.13, for which modulation theory gives the borderline c3 = 2.1. The rapid decrease +of both the numerical and modulation theory amplitudes as the theoretical borderline is +approached as c3 increases is clearly visible, with the numerical amplitude being very small +above c3 = 2.1. +Indeed, as c3 increases above 2.1, the numerical amplitude continues to +decrease. +Finally, the agreement between the numerical and modulation theory resonant +wave amplitudes is good up to the cut-off. +8 +Conclusions +Whitham modulation theory has been developed to obtain the cross-over dispersive shock +wave (CDSW) solution of the extended Korteweg-de Vries equation. The DSW itself in this +regime is unstable and has a different structure to the standard Korteweg-de Vries DSW, +consisting of a train of solitary waves of equal amplitude on average, instead of a modulated +train of waves of nearly linearly decreasing amplitude from leading to trailing edges. This +non-standard structure has been exploited to obtain an approximate solution for the DSW. +The resonant wavetrain was obtained as a Stokes wave. +The key concept of a Whitham +shock, a jump in the modulated parameters of a wavetrain, was used to link the CDSW +and the resonant wavetrain. It was found that this combination of modulation theory and +approximate theory gave solutions in excellent agreement with full numerical solutions of the +eKdV equation. +The modulation theory developed in the present work was used to successfully explain the +numerically observed minimal resonant wave amplitude when the higher order coefficients in +the eKdV equation (1) take the water wave values. Previous work to explain this minimal +amplitude based on resonant solitary wave theory predicted that this amplitude vanishes for a +fixed combination of the higher order coefficients, with minimal amplitude in a neighbourhood +of this node [38]. However, the present modulation theory predicts that the resonant wave +amplitude is minimal for regions in the higher order coefficient parameter space, as also shown +by full numerical solutions of the eKdV equation. +The combination of modulation theory with the concept of a Whitham shock and approx- +imate theory can be used for other problems which involve resonant dispersive shock waves. +These include resonant optical dispersive shock waves in nematic liquid crystals [18, 19, 20]. +These are governed by the eKdV equation for small steps in the optical beam power which +generates them, but for general initial steps the governing equations are more complicated, +consisting of a nonlinear Schr¨odinger-type equation for the optical field and an elliptic equa- +tion for the nematic response [58]. +Acknowledgement +S.B. would like to thank Patrick Sprenger with whom this research work stimulated extensive +discussions. S.B. is grateful for the hospitality of Westminster College arranged by the Isaac +Newton Institute (INI) for Mathematical Sciences, University of Cambridge, where part of +15 + +this work was reported and discussed in the Physical Applications Workshop HY2W05. The +authors are thankful for the referees’ comments and suggestions which greatly improved the +manuscript. +A +CDSW equal amplitude relation +The averaged mass and energy densities for the eKdV solitary wave are, respectively, +� ∞ +−∞ +us dx = ¯us + 2 +√ +2√as + ǫ +√ +2a3/2 +s +� +2c6 + 4 +3c7 +� +, +(26) +� ∞ +−∞ +�1 +2u2 +s − 1 +3ǫ +� +c2 − 1 +2c2 +� +u3 +s +� +dx = 1 +2 ¯u2 +s + 2 +√ +2¯us +√as + 2 +√ +2 +3 a3/2 +s ++ ǫ +� +1 +4 (c2 − 2c3) ¯u3 +s + 3 +√ +2 +2 +√as (c2 − 2c3) ¯u2 +s + +√ +2a3/2 +s +� +c2 − 2c3 + 2c6 + 4 +3c7 +� +¯us ++ 4 +√ +2 +15 (5c6 + 4c7) a5/2 +s +� +. +(27) +The averaged mass and energy fluxes for the resonant Stokes wave are, respectively, +¯Qmr = 3¯u2 +r + 3 +2a2 +r + ǫ +�1 +3c1¯u3 +r + 1 +2c1¯ura2 +r + 1 +4c2k2 +r − 3 +4c3k2 +r +� +, +(28) +¯Qer += +2¯u3 +r + 3 +� +¯ur − 1 +4k2 +r +� +a2 +r + ǫ +�1 +4 (c1 + 3c2 − 6c3) ¯u4 +r ++ 1 +4 +� +(3c1 + 9c2 − 18c3) ¯u2 +r − 2c2¯urk2 +r + 5c4k4 +r +� +a2 +r +� +. +(29) +B +Modulation theory jump conditions +The averaged mass density for the resonant Stokes wave and the bore in the CDSW regime +to O(ǫ) are, respectively, +¯Pmr = ¯ur, +(B 1) +¯Pmcdsw = ¯us + 2 +√ +2√as + 2 +√ +2ǫ +� +c6 + 2 +3c7 +� +a3/2 +s +. +(B 2) +The averaged mass flux for the resonant Stokes wave and the bore in the CDSW regime to +O(ǫ) are, respectively, +¯Qmr = 3¯u2 +r + 3 +2a2 +r + ǫ +�1 +3c1¯u3 +r + 1 +2c1¯ura2 +r + 1 +4c2k2 +r − 3 +4c3k2 +r +� +(B 3) +and +¯Qmcdsw = 3¯u2 +s + 12 +√ +2¯us +√as + 4 +√ +2a3/2 +s ++ ǫ +�1 +3 ¯u3 +sc1 + 2 +√ +2c1¯u2 +s +√as + 4 +√ +2 +�1 +3c1 ++ 3c6 + 2c7) ¯usa3/2 +s ++ 4 +√ +2 +� 1 +15c2 − 1 +5c3 + 2c6 + 8 +5c7 +� +a5/2 +s +� +. +(B 4) +16 + +The averaged energy density for the resonant Stokes wave and the bore in the CDSW +regime, with all terms taken into account for the CDSW, as discussed in Section 5, are, +respectively, +¯Per += +1 +2 ¯u2 +r + 1 +4a2 +r + 1 +4u2 +2a4 +r + 1 +6ǫ (c2 − 2c3) ¯u3 +r + 1 +4ǫ (c2 − 2c3) ¯ura2 +r ++ 1 +8ǫ +� +c2u2 + 2c2u2 +2¯ur − 2c3u2 − 4c3u2 +2¯ur +� +a4 +r +(B 5) +and +¯Pecdsw = ¯Pecdsw,0 + ǫ ¯Pecdsw,1 + ǫ2 ¯Pecdsw,2 + ǫ3 ¯Pecdsw,3 + ǫ4 ¯Pecdsw,4, +(B 6) +where +¯Pecdsw,0 = 1 +2 ¯u2 +s + 2 +√ +2¯us +√as + 2 +3 +√ +2a3/2 +s +, +¯Pecdsw,1 = 1 +6 (c2 − 2c3) ¯u3 +s + +√ +2 (c2 − 2c3) ¯u2 +s +√as ++ 2 +3 +√ +2 (c2 − 2c3 + 3c6 + 2c7) ¯usa3/2 +s ++ 4 +15 +√ +2 (5c6 + 4c7) a5/2 +s +, +¯Pecdsw,2 = +√ +2 +� +c2c6 + 2 +3c2c7 − 2c3c6 − 4 +3c3c7 +� +¯u2 +sa3/2 +s ++ 4 +15 +√ +2 (5c2c6 + 4c2c7 +− 10c3c6 − 8c3c7) ¯usa5/2 +s ++ 2 +√ +2 +� 4 +15c2c6 + 8 +35c2c7 − 8 +15c3c6 − 16 +35c3c7 + 1 +3c2 +6 ++ 8 +15c6c7 + 4 +17c2 +7 +� +a7/2 +s +, +(B 7) +¯Pecdsw,3 = 2 +√ +2 +�1 +3c2c2 +6 + 8 +15c2c6c7 + 8 +35c2c2 +7 − 16 +15c3c6c7 − 2 +3c3c2 +6 +− 16 +35c3c2 +7 +� +¯usa7/2 +s ++ 8 +5 +√ +2 +�1 +3c2c2 +6 − 2 +3c3c2 +6 + 4 +7c2c6c7 − 8 +7c3c6c7 +� +a9/2 +s +, +¯Pecdsw,4 = 8 +√ +2 +� 1 +45c2c3 +6 + 2 +35c2c2 +6c7 + 16 +315c2c6c2 +7 + +32 +2079c2c3 +7 − 2 +45c3c3 +6 − 4 +35c3c2 +6c7 +− 32 +315c3c6c2 +7 − +64 +2079c3c3 +7 +� +a11/2 +s +. +The averaged energy flux for the resonant Stokes wave and the bore in the CDSW regime, +with all terms taken into account for the CDSW, as discussed in Section 5, are, respectively, +¯Qer += +2¯u3 +r + 3 +� +¯ur − 1 +4k2 +r +� +a2 +r + 3 +�1 +2 − k2 +ru2 + u2¯ur +� +u2a4 +r + 1 +4ǫ +� +3c1¯u2 +r + 9c2¯u2 +r +−2c2¯urk2 +r − 18c3¯u2 +r + 5c4k4 +r +� +a2 +r + 1 +32ǫ +� +3c1 + 24c1u2¯ur + 24c1u2 +2¯u2 +r + 9c2 ++ 72c2u2 +2¯u2 +r + 72c2u2¯ur − 24c2u2k2 +r − 64c2k2 +ru2 +2¯ur − 18c3 − 144c3u2 +2¯u2 +r − 144c3u2¯ur ++ 640c4k4 +ru2 +2 +� +a4 +r + 3 +8ǫ (c1 + 3c2 − 6c3) u2 +2a6 +r + 3 +32ǫ (c1 + 3c2 − 6c3) u4 +2a8 +r, +(B 8) +and +¯Qecdsw = ¯Qecdsw,0 + ǫ ¯Qecdsw,1 + ǫ2 ¯Qecdsw,2 + ǫ3 ¯Qecdsw,3 + ǫ4 ¯Qecdsw,4 + ǫ5 ¯Qecdsw,5, +(B 9) +17 + +where +¯Qecdsw,0 = 2¯u3 +s + 12 +√ +2¯u2 +s +√as + 8 +√ +2¯usa3/2 +s +− 4 +5 +√ +2a5/2 +s +, +¯Qecdsw,1 = 1 +4 (c1 + 3c2 − 6c3) ¯u4 +s + 2 +√ +2 (c1 + 3c2 − 6c3) ¯u3 +s +√as + 2 +√ +2 (c1 + 3c2 − 6c3 ++ 6c6 + 4c7) ¯u2 +sa3/2 +s ++ 8 +√ +2 +� +− 1 +15c2 + 2c6 + 8 +5c7 +� +¯usa5/2 +s ++ 8 +√ +2 +� 1 +21c4 + 3 +5c6 + 16 +35c7 +� +a7/2 +s +, +¯Qecdsw,2 = 2 +√ +2 +�2 +3c1c7 + c1c6 + 2c2c7 + 3c2c6 − 4c3c7 − 6c3c6 +� +¯u3 +sa3/2 +s ++ 4 +√ +2 +� +c1c6 + 4 +5c1c7 + 3c2c6 + 12 +5 c2c7 − 6c3c6 − 24 +5 c3c7 +� +¯u2 +sa5/2 +s ++ 8 +5 +√ +2 +� +2c1c6 + 12 +7 c1c7 + 16 +3 c2c6 + 92 +21c2c7 − 72 +7 c3c7 − 12c3c6 + 5c2 +6 + 8c6c7 ++ 24 +7 c2 +7 +� +¯usa7/2 +s ++ 2 +5 +√ +2 +� +5c1c2 +6 + 24 +7 c1c2 +7 + 24c2c6c7 + 72 +7 c2c2 +7 + 15c2c2 +6 − 48c3c6c7 +− 144 +7 c3c2 +7 − 30c3c2 +6 + 8c1c6c7 +� +¯u2 +sa7/2 +s ++ 4 +√ +2 +� +− 8 +35c2c6 − 16 +63c2c7 + 20 +21c4c6 + 32 +21c4c7 ++ 7 +5c2 +6 + 16 +7 c6c7 − 32 +105c2 +7 +� +a9/2 +s +¯u2 +s, +¯Qecdsw,3 = 8 +√ +2 +�2 +5c1c2 +6 + 32 +105c1c2 +7 + 24 +35c1c6c7 + 17 +15c2c2 +6 + 40 +21c2c6c7 + 256 +315c2c2 +7 +− 144 +35 c3c6c7 − 12 +5 c3c2 +6 − 64 +35c3c2 +7 +� +¯usa9/2 +s ++ 8 +√ +2 +� 6 +35c1c2 +6 + 32 +231c1c2 +7 + 32 +105c1c6c7 + 2 +5c2c2 +6 ++ 208 +315c2c6c7 + 928 +3465c2c2 +7 − 36 +35c3c2 +6 − 64 +77c3c2 +7 −64 +35c3c6c7 + 5 +21c4c2 +6 + 512 +693c4c2 +7 ++ 16 +21c4c6c7 + 4 +15c3 +6 + 24 +35c2 +6c7 + 64 +105c6c2 +7 + 128 +693c3 +7 +� +a11/2 +s +, +¯Qecdsw,4 = 16 +√ +2 +� 1 +15c1c3 +6 + 16 +35c2c6c2 +7 + 16 +105c1c6c2 +7 + 6 +35c1c2 +6c7 + 32 +693c1c3 +7 + 1 +5c2c3 +6 ++ 18 +35c2c2 +6c7 + 32 +231c2c3 +7 − 2 +5c3c3 +6 − 64 +231c3c3 +7 − 32 +35c3c6c2 +7 − 36 +35c3c2 +6c7 +� +¯usa11/2 +s ++ 32 +√ +2 +� 1 +35c1c3 +6 + +8 +105c1c2 +6c7 + 16 +315c1c3 +6c7 + +64 +3003c1c3 +7 + +8 +105c2c3 +6 + +64 +1287c2c3 +7 ++ 62 +315c2c2 +6c7 + 16 +105c2c3 +6c7 − 128 +3465c2c6c2 +7 − 6 +35c3c3 +6 − 16 +35c3c2 +6c7 − 32 +105c3c3 +6c7 +− 128 +1001c3c3 +7 +� +a13/2 +s +, +¯Qecdsw,5 = 8 +√ +2 +� 1 +35c1c4 +6 + 32 +315c1c3 +6c7 + 32 +231c1c2 +6c2 +7 + 256 +3003c1c6c3 +7 + 128 +6435c1c4 +7 + 3 +35c2c4 +6 ++ 32 +105c2c3 +6c7 + 32 +77c2c2 +6c2 +7 + 256 +1001c2c6c3 +7 + 128 +2145c2c4 +7 − 6 +35c3c4 +6 − 64 +105c3c3 +6c7 +− 64 +77c3c2 +6c2 +7 − 512 +1001c3c6c3 +7 − 256 +2145c3c4 +7 +� +a15/2 +s +. +(B 10) +18 + +References +[1] G.B. 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Rep., 516, 147–208. +22 + diff --git a/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/load_file.txt b/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d47273b7eacf06bf23ad250ee7d96a7d6334e0c --- /dev/null +++ b/ddAyT4oBgHgl3EQfjPi0/content/tmp_files/load_file.txt @@ -0,0 +1,1180 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf,len=1179 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='00412v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='PS] 1 Jan 2023 Whitham Shocks and Resonant Dispersive Shock Waves Governed by the Higher Order Korteweg-de Vries Equation Saleh Baqer, Department of Mathematics, Faculty of Science, Kuwait University, Kuwait City 13060, Kuwait Noel F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Smyth, School of Mathematics, University of Edinburgh, Edinburgh, Scotland, EH9 3FD, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' and School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong, New South Wales, Australia, 2522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Abstract The addition of higher order asymptotic corrections to the Korteweg-de Vries equation results in the extended Korteweg-de Vries equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' These higher order terms destabilise the dispersive shock wave solution, also termed an undular bore in fluid dynamics, and result in the emission of resonant radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In broad terms, there are three possible dispersive shock wave regimes: radiating dispersive shock wave (RDSW), cross-over dis- persive shock wave (CDSW) and travelling dispersive shock wave (TDSW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' While there are existing solutions for the RDSW and TDSW regimes obtained using modulation the- ory, there is no existing solution for the CDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Modulation theory and the associated concept of a Whitham shock are used to obtain this CDSW solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In addi- tion, it is found that the resonant wavetrain emitted by the extended Korteweg-de Vries equation with water wave coefficients has a minimal amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This minimal amplitude is explained based on the developed Whitham modulation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This paper is dedicated to Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Whitham, FRS whose prophetic speculations on dis- persive shocks in the 1960s and 1970s have been verified by this and other publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 1 Introduction A solitary wave is the standard solution of nonlinear, dispersive wave equations, intensively studied since the 1960s, when it was found that certain solitary wave supporting equations are integrable via the method of inverse scattering [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A solitary wave is termed a soliton as for such equations solitary waves interact cleanly, with no change of form (other than a phase shift), the word soliton then chosen due to its connotation with interacting sub- atomic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The dispersive shock wave (DSW), also termed an undular bore in fluid mechanics, is another generic solution of nonlinear, dispersive wave equations which has been receiving increased study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In its standard form a DSW is a non-steady modulated wavetrain, consisting of solitary waves at one edge and linear dispersive waves at the opposite edge, 1 connecting two distinct flow states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The separation between these two edges continuously increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' DSWs are formed due to the dispersive resolution of a discontinuity, with the simplest initial condition generating one being a step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Undular bores were first observed as the tidal bores which occur in coastal areas of strong tide and appropriate topography, for example in Australia, Brazil, Canada, China, France, the United Kingdom and the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, DSWs are more widely observed in nature, with applications to water waves [2, 3], meteorology [4, 5, 6], oceanography [7], geophysics [8, 9, 10], solid mechanics [11], nonlinear optics [12, 13, 14, 15, 16, 17, 18, 19], Bose-Einstein condensates [21] and Fermionic fluids [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' DSW solutions are found using Whitham modulation theory [1, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Whitham mod- ulation theory is an asymptotic technique used to study slowly varying periodic wave solu- tions of dispersive wave equations and gives partial differential equations, termed modulation equations, for the slowly varying parameters of the wavetrain, for instance its amplitude, wavelength and mean height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In the case for which these modulation equations form a hy- perbolic system the underlying periodic wavetrain is modulationally stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A simple wave solution of these hyperbolic modulation equations is then the DSW solution of the underlying equation [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Such simple wave solutions are easy to determine if the modulation equa- tions can be set in Riemann invariant form, which is guaranteed if the underlying equation is integrable [28], but is difficult for non-integrable equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It has been found that in the solitary wave and linear wave limits modulation equations have degenerate forms, from which it was shown that the solitary wave and linear wave edges of a DSW can be determined without knowledge of the full modulation equations or their Riemann invariant form [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' An extensive review of DSWs, their physical applications and the connections between Whitham modulation equations and DSWs can be found in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This standard view of DSWs substantially alters for nonlinear, dispersive wave equations with non-convex dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This non-convex dispersion allows dispersive radiation to be in resonance with solitary waves, so that the solitary wave sheds dispersive radiation and so is nonlocal [31, 32, 33, 34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The solitary wave then radiates away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As a DSW is a modulated wavetrain with solitary waves at one edge, non-convex dispersion means that a DSW can also be resonant, shedding radiation, with the individual waves of the DSW being in resonance, not just the solitary wave at one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As the DSW connects distinct flow states, it does not radiate away as the mass shed in radiation is replaced by that of the flow state into which it expands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The addition of higher order dispersion to standard nonlinear, dispersive wave equations, such as the Korteweg-de Vries equation [37, 38] and the nonlinear Schr¨odinger (NLS) equation [39, 40, 41, 42, 43], results in their DSWs being resonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In addition, the DSW solution of the equations governing nonlinear optical beam propagation in nematic liquid crystals can be resonant or non-resonant [16, 18, 19, 20], depending on the size of the jump of beam power across it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The extended Korteweg-de Vries (eKdV) equation is an extension of the standard Korteweg-de Vries (KdV) equation for which the next higher order dispersive, nonlinear and nonlinear-dispersive terms are included in the asymptotic expansion [1] which yields the KdV equation from more general equations, such as the water wave equations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The eKdV DSW solution, undular bore, is resonant [38], with the higher order, fifth order, dispersion being a major driver of this [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This resonance has a profound effect on the structure of the DSW, with the classical structure outlined above destroyed if the higher order terms are strong enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Three regimes have been identified, radiating dispersive shock wave (RDSW), see Figure 1(c), crossover dispersive shock wave (CDSW), see Figure 1(b), and travelling dispersive shock wave (TDSW), see Figure 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' An RDSW occurs when the effect 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1 200 150 100 50 0 50 100 150 200 u x (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0 500 1000 1500 2000 2500 u x (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 200 0 200 400 600 800 1000 u x (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0 200 400 600 800 1000 u x (d) Figure 1: DSW types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (a) classical KdV DSW with ci = 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' , 4, (b) non-classical CDSW with c1 = −1, c2 = c3 = 1, c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4, (c) non-classical RDSW with c1 = −1, c2 = c3 = 1, c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3, (d) non-classical TDSW with c1 = −1, c2 = c3 = 1, c4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In these figures, t = 50, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15 with u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' of the higher order terms is weak and so is a perturbed, radiating version of the classical DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Above a threshold the resonance completely destroys the classical DSW structure, leaving a non-oscillatory jump between the initial levels of the step generating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Essentially, the classical DSW is radiated away by the dominant resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Between these two regimes, there is the CDSW, illustrated in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The classical DSW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' illustrated in Figure 1(a) by the KdV DSW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' is destabilised by the shed resonant wavetrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' which results in a structure similar to that for the focusing NLS equation with the amplitude ordered classical DSW of Figure 1(a) replaced by an amplitude disordered CDSW of nearly constant average amplitude at its leading edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' but with a rapid decrease to the level behind over its trailing edge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' as seen in Figure 1(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' so that a CDSW has a nearly constant amplitude over much of its length [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' That is, both the unstable focusing NLS DSW and the CDSW are large genus wavetrains (multi-phase wavetrains) that result in the long term formation of a so-called soliton gas (a large number of randomly interacting solitary waves) [45, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, the latter sheds resonant radiation due to the non-convex dispersion effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The connection between a soliton gas and a CDSW deserves more detailed study, which is beyond the present work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 1(d) displays the time evolution of the resonant wavetrain propagating ahead of the CDSW to emphasise details of its modulational instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This instability is analogous to the Benjamin-Feir instability observed in water waves as sideband perturbations [1, 25], see [48] for experimental photographs of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The underlying high frequency resonant wavetrain with its unstable amplitude modulation is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 3 The eKdV equation RDSW can be found as a perturbation of the KdV equation DSW [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The TDSW solution has also been found as a TDSW is essentially a jump between two levels connected to the resonant wavetrain [37, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The present work will determine the eKdV equation CDSW solution based on Whitham modulation theory and the associated concept of a Whitham shock, which is a shock in the modulation variables [24, 1, 51] and links the CDSW to the resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' When Whitham developed modulation theory he speculated on the role of shocks in the case for which the modulation equations were hyperbolic, so that the underlying periodic wavetrain is stable [24], but did not explore the topic in extensive detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The use of shocks in hyperbolic modulation equations has only recently been developed [51, 52], with applications to the Kawahara equation [51], the fifth order KdV equation [51] and the BBM equation [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Finally, it has been previously found that there is a resonant wave amplitude minimum for the CDSW solution of the eKdV equation with water wave coefficients [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This amplitude minimum is deduced as a result of the developed modulation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 2 Extended Korteweg-de Vries equation The Korteweg-de Vries (KdV) equation arises as an asymptotic approximation to more general equations in the long wavelength, weakly nonlinear limit when weak nonlinearity is balanced with weak dispersion [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' If this asymptotic expansion is taken to one order beyond the KdV approximation, the result is the extended Korteweg-de Vries (eKdV) equation [44] ut + 6uux + uxxx + ǫ � c1u2ux + c2uxuxx + c3uuxxx + c4uxxxxx � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (1) Here, ǫ is a parameter measuring the weak nonlinearity, which for water waves is the ratio of the wave amplitude to the undisturbed fluid depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In the particular case of surface water waves the higher order coefficients are c1 = −3/2, c2 = 23/4, c3 = 5/2 and c4 = 19/40 [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The eKdV equation also arises in the nonlinear optics of coherent beam propagation in nematic liquid crystals [19], for which ǫ is the ratio of the optical beam amplitude above a background level to this background level, and nonlinear elasticity [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The higher order coefficients ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' , 4, are involved in this case and can be found in the original work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The Kawahara equation is the special case for which there is only fifth order dispersion [31], ci = 0, i = 1, 2, 3, and c4 ̸= 0, ut + 6uux + uxxx + ǫc4uxxxxx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (2) The Kawahara equation arises for gravity-capillary waves for which the Bond number is near 1/3 [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In the present work, to generate a DSW the step initial condition u(x, 0) = � u+, x > 0 u−, x < 0 (3) will be used, with u− > u+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The eKdV equation has the mass conservation equation ∂ ∂tu + ∂ ∂x � 3u2 + uxx + ǫ �1 3c1u3 + c3uuxx + 1 2 (c2 − c3) u2 x + c4uxxxx �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (4) 4 The eKdV energy conservation is less straightforward to derive [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' On multiplying the eKdV equation (1) by u and integrating by parts gives ∂ ∂t 1 2u2 + ∂ ∂x � 2u3 + uuxx − 1 2u2 x + ǫ �1 4c1u4 + 1 2c2u2uxx + c4uuxxxx − c4uxuxxx + 1 2c4u2 xx �� + ǫ � c3 − 1 2c2 � u2uxxx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (5) To set the final term in conservation form we use the fact that ǫ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The first order equation, the KdV equation (ǫ = 0), gives ∂ ∂tu3 = −3u2 (6uux + uxxx) = − ∂ ∂x 9 2u4 − 3u2uxxx, (6) so that (5) becomes the eKdV energy conservation equation ∂ ∂t �1 2u2 − 1 3ǫ � c3 − 1 2c2 � u3 � + ∂ ∂x � 2u3 + uuxx − 1 2u2 x + ǫ �1 4c1u4 + 1 2c2u2uxx + c4uuxxxx − c4uxuxxx + 1 2c4u2 xx − 3 2 � c3 − 1 2c2 � u4 �� = 0, (7) which is valid to O(ǫ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' We note that if c2 = 2c3, then this energy conservation law is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, this relation does not hold for the eKdV equation with the water wave coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The mass (4) and energy (7) conservation equations are of the form ∂Pm ∂t + ∂Qm ∂x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ∂Pe ∂t + ∂Qe ∂x = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (8) where Pm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Pe and Qm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Qe are the mass and energy densities and fluxes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Pm = u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (9) Pe = 1 2u2 − 1 3ǫ � c3 − 1 2c2 � u3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (10) Qm = 3u2 + uxx + ǫ �1 3c1u3 + c3uuxx + 1 2 (c2 − c3) u2 x + c4uxxxx � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (11) Qe = 2u3 + uuxx − 1 2u2 x + ǫ �1 4c1u4 + 1 2c2u2uxx + c4uuxxxx − c4uxuxxx + 1 2c4u2 xx + 3 2 �1 2c2 − c3 � u4 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (12) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 3 Resonant wave Stokes expansion It can be seen from Figure 1 that the amplitude of the resonant wavetrain generated by a CDSW is small, certainly smaller than the amplitude of the DSW itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The resonant wavetrain can then be taken as a Stokes wave expansion of the form ur = ¯ur + ar cos θr + a2 ru2 cos 2θr + O(a3 r), ωr(kr, ar) = ω0 + arω1 + a2 rω2 + O(a3 r), (13) 5 where ¯ur is the mean height of the wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, ar is the amplitude of the resonant wavetrain and ωr and kr are its frequency and wavenumber, with θr = krx−ωrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Substituting this Stokes wave expansion into the eKdV equation (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' separating out the resulting equation at O(ǫn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' and eliminating secular terms gives ω0 = (6¯ur + ǫc1¯u2 r)kr − (1 + ǫc3¯ur) k3 r + ǫc4k5 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ω1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (14) ω2 = 36 + 24ǫc1¯ur − ǫ (48c3 − 6c1 − 6c2) k2 r 24kr − ǫ (120c4k3r − 24c3¯urkr) + O � ǫ2� = 3 2kr + ǫ �1 4 (c1 + c2 − 8c3 + 30c4) kr + � c1 − 3 2c3 � ¯ur kr � + O � ǫ2� (15) and u2 = 6 + 2ǫc1¯ur − ǫ(c2 + c3)k2 r 12k2r + 12ǫ (c3¯ur − 5c4k2r) k2r = 1 2k2r − 1 12ǫ � c2 + c3 − 30c4 − 2 (c1 − 3c3) ¯ur k2r � + O � ǫ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (16) The reason that the Stokes wave coefficients are expanded for small ǫ in (14)–(16) is to make the Whitham modulation jump conditions that connect the bore with the resonant radiation ahead in the CDSW regime as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 4 CDSW equal amplitude approximation As seen from the example shown in Figure 2 the eKdV CDSW is unstable, as is its generated resonant wavetrain, see Figure 2(b), and does not exhibit the standard KdV DSW structure, as discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The amplitudes of the waves of the CDSW do not decrease mono- tonically from the leading to the trailing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The amplitudes of the waves are random due to the instability, but distributed around a constant mean, with a rapid decrease to u− at the trailing edge [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This broad structure of a CDSW can be exploited to obtain an approximate solution for it [18, 19, 54], which can then be linked to loss radiated in the resonant wavetrain [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In essence, the CDSW is approximated by a train of uniform solitary waves, with mass and energy conservation used to determine the amplitude and spacing of these waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The basis of the equal amplitude approximation is the solitary wave solution of the eKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' While there is no exact solitary wave solution of this equation, there is a perturba- tion solution based on ǫ small [49] us = ¯us + � as + ǫc6a2 s � sech2 wsθs + ǫc7a2 s sech4 wsθs + O(ε2), (17) with phase θs = x − Ust, inverse width ws = � as/2 and velocity Us = 2as + 4ǫc4a2 s + O(ǫ2), for which the new constants c6 and c7 are c6 = −1 6c1 + 1 6c2 + 2 3c3 − 5c4, c7 = 1 12c1 − 1 4c2 − 1 2c3 + 15 2 c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (18) Let us assume that in the CDSW regime the solution consists of the level ahead u+, joining to a resonant wavetrain of amplitude ar, wavenumber kr and mean height ¯ur (the frequency ωr given by the Stokes wave dispersion relation (13)), followed by a uniform train of N(t) equal solitary waves of amplitude as on a mean level ¯us, which then links to the level u− behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This matches the general form of the example CDSW solution shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 300 200 100 0 100 200 300 400 u x (a) (b) Figure 2: Example CDSWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (a) Detail of CDSW at t = 50, (b) evolution detail of resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In these figures, c1 = −1, c2 = c3 = 1, c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15 and u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5, u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The eKdV mass and energy conservation equations (4) and (7) are now integrated in x from the level behind u− to the trailing edge of the resonant wavetrain, giving N � ∞ −∞ us dx = � 3u2 − + 1 3ǫc1u3 − − ¯Qmr � t, (19) N � ∞ −∞ �1 2u2 s − 1 3ǫ � c2 − 1 2c2 � u3 s � dx = � 2u3 − + 1 4ǫ (c1 + 3c2 − 6c3) u4 − − ¯Qer � t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (20) Here, ¯Qmr and ¯Qer are the mass and energy fluxes (11) and (12) averaged over the Stokes wave (13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Dividing these mass and energy results and evaluating the mass and energy fluxes over the resonant wavetrain gives a relation linking the solitary wave amplitude as and mean height ¯us to the amplitude ar, wavenumber kr and mean height ¯ur of the resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As this expression is involved, particularly the energy density, the relation is detailed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Dividing the mass and energy results (19) and (20) gives the implicit relation � ∞ −∞ us dx � ∞ −∞ � 1 2u2s − 1 3ǫ � c2 − 1 2c2 � u3s � dx = 3u2 − + 1 3ǫc1u3 − − ¯Qmr 2u3 − + 1 4ǫ (c1 + 3c2 − 6c3) u4 − − ¯Qer , (21) which determines the amplitude of the CDSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 300 0 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 500 600 10 700 20 30 800 40 x 900 50 t5 Modulation theory for CDSW and resonant wavetrain The resonance condition between the CDSW and the resonant wavetrain is that the solitary wave velocity Us equals the Stokes wave velocity, so that 2as + 4ǫc4a2 s = c = ω0 + a2 rω2 kr , (22) where ω0 and ω2 are given by the dispersion relation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The example numerical solution of the eKdV equation in the CDSW regime shown in Figure 1 shows that it consists of five distinct regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The initial level behind u− links to an unstable DSW, which then shows a sharp jump to the shed resonant wavetrain, which is here modelled by a Whitham shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The resonant wavetrain then transitions to the initial level ahead u+ via a “partial DSW” [50, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A “full DSW,” as for the standard KdV DSW, is a transition between two uniform levels via a modulated wavetrain with solitary waves at one edge and linear waves at the opposite edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A partial DSW links two uniform wavetrains via a modulated wavetrain whose amplitude, wavenumber and mean height are continuous at the two edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A full DSW is a limit of a partial DSW with no steady wavetrains at its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The partial DSW smoothly raises the initial level ahead u+ to the mean level of the resonant wavetrain ¯ur, which then jumps discontinuously to the mean level of the CDSW ¯us via the Whitham shock joining the resonant wavetrain to the CDSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' To fully determine the partial DSW the Whitham modulation equations for the resonant wavetrain need to be calculated [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This results in an involved system of equations [55] when the jump conditions across the Whitham shock are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, ¯ur is very close to u+ [50] and to a good approximation can be set to u+ and the extra accuracy obtained by the inclusion of the partial DSW is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Indeed, the mean level will be found to have minimal change over the Whitham shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It can be seen from Figure 1 that the DSW and its associated resonant wavetrain are unstable in the CDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The Whitham modulation equations can then be assumed to be elliptic [1, 24], even though they have not been calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Although this ellipticity conclusion comes from numerical observations and not from an exact calculation of the as- sociated Whitham modulation system, it was analytically verified in [50] that the Whitham modulation system becomes fully elliptic when the stable Kawahara DSW regime approaches the CDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Extending the modulation theory work of [50] to eKdV CDSWs is outside the scope of the present study and will not be dealt with here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It is another topic for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In theory elliptic systems do not have discontinuous solutions with shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, Figure 1 shows that there is a clear rapid connection between the CDSW and the resonant wavetrain, which we shall model as a discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' We shall then model this connection by a Whitham shock based on the conservation of mass and energy across it as these quantities have to be conserved for any valid solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This approach was found to be successful in the study of the DSW solution of the nematic equations [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Averaging the mass and energy conservation equations (8) over the resonant wavetrain ahead of the Whitham shock and the CDSW behind the shock gives the jump conditions across the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As the CDSW is led by solitary waves, the Whitham shock velocity is equal to the solitary wave velocity Us [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The jump conditions are then − Us � ¯Pmcdsw − ¯Pmr � + � ¯Qmcdsw − ¯Qmr � = 0, −Us � ¯Pecdsw − ¯Per � + � ¯Qecdsw − ¯Qer � = 0, (23) 8 where ¯Pmcdsw, ¯Pecdsw and ¯Pmr, ¯Per denote the mass and energy densities (9) and (10) averaged over the CDSW regime and the Stokes wave, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Similarly, ¯Qmcdsw, ¯Qecdsw and ¯Qecdsw, ¯Qer denote the mass and energy fluxes (11) and (12) averaged over the bore in the CDSW regime and the Stokes wave, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, Us denotes the Whitham shock velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The CDSW amplitude relation (21), the resonance condition (22) and the jump conditions (23) give four relations for the five unknowns as, ar, ¯us, ¯ur and kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' With the approximation that ¯ur = u+, this gives a complete system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This system of equations to determine the resonant CDSW were solved numerically using Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It was found to be sufficient to keep terms up to the orders O(ǫ) and O(a2 r) in the CDSW amplitude relation (21), the resonance condition (22) and the mass jump condition of (23) in order to keep these equations as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' We note that the eKdV equation (1) as a reduction of the water wave equations is asymptotically valid to O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, it was found to be essential to keep all terms in the Whitham energy jump condition of (23) in order to obtain good agreement with numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Indeed, Newton’s method had a tendency to not converge if all energy terms were not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Including higher order terms in the CDSW amplitude relation (21), the resonance condition (22) and the mass jump condition of (23) resulted in no graphical difference in the resulting solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Indeed, if the theoretical resonant wave amplitude is truncated to the linear value Ar = ar, there is no observable difference in the comparisons with numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As the full detailed jump conditions, particularly the energy jump condition, are then extensive, they are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A similar situation was encountered in the study of nematic CDSWs when the nonlocality of the nematic medium is decreased to the local optical limit [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 6 Comparisons with numerical solutions The eKdV equation (1) was solved numerically using the pseudo-spectral method of Fornberg and Whitham [27] as extended to enhance stability at high wavenumbers, particularly due to the higher order fifth order dispersion [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The spatial derivatives were calculated in Fourier space, with the equation propagated in time using the fourth order Runge-Kutta method in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As stated, to enhance stability linear dispersion was propagated using an integrating factor [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Numerical solutions generated using this numerical scheme will now be compared with solutions of the modulation theory of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As can be seen from Figure 1 that the resonant wavetrain is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The resonant wave amplitude was then calculated by averaging the amplitude over the resonant wavetrain up to its front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 3 shows comparisons between full numerical solutions of the Kawahara equation (2) and modulation theory for the resonant wavetrain amplitude Ar = ar + a2 ru2, (24) see (13), the CDSW solitary wave amplitude As = as + ǫ (c6 + c7) a2 s, (25) see (17), the resonant wavetrain wavenumber kr and the Whitham shock velocity Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In general, the agreement between modulation theory and numerical solutions is excellent across the CDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The agreement even extends beyond the CDSW regime into the RDSW 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 Ar ε (a) CDSW RDSW TDSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 As ε (b) CDSW RDSW TDSW 2 3 4 5 6 7 8 9 10 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 kr ε (c) CDSW RDSW TDSW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 Us ε (d) CDSW RDSW TDSW Figure 3: Comparison between numerical solutions of the Kawahara equation (2) and modu- lation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Numerical solution: ◦ (red circle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' modulation theory solution: red (solid) line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' boundaries between DSW regimes: green (dashed) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (a) resonant wave amplitude Ar, (b) solitary wave amplitude of CDSW As, (c) wavenumber of resonant wavetrain kr, (d) velocity of Whitham shock Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, c4 = 1, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' and TDSW regimes, particularly for the wavenumber of the resonant wavetrain kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' We note that the resonant radiation is an unstable wavetrain (multi-phase wavetrain), so there exists no single dominant wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The numerical values of kr were then determined by averaging over the resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It was found that averaging over 10 to 20 crests was sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The agreement is good in the TDSW regime, which is expected as modulation theory in the TDSW regime is the limit of the present modulation theory if the amplitude As of the solitary waves vanishes and the mean level becomes the level behind u− [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In this context, it should be noted that in the present work the solitary wave amplitude As is measured from the level ahead u+ = 0 as the equal amplitude approximation is based on the leading edge of the DSW, so that in the TDSW limit As → u−, while in the work of Sprenger and Hoefer [51] the wave amplitude is measured from the local mean level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The agreement for the solitary wave amplitude As is less good in the RDSW regime as the DSW cannot be approximated by a train of equal amplitude solitary waves since the DSW is a perturbed KdV DSW in this regime [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The present modulation theory gives that the amplitude Ar of the resonant wavetrain rapidly approaches zero in the RDSW regime, as expected, but less rapidly than the numerical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Similarly, the modulation and numerical Whitham shock velocities Us are in excellent agreement in the CDSW regime, and even in the RDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The latter is 10 ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='01 ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='03 ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ¯us = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='03 × 10−6 ¯us = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 × 10−5 ¯us = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 × 10−4 ¯us = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 × 10−3 ¯us = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='20 × 10−3 Table 1: Modulation mean level ¯us of CDSW for Kawahara equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, c4 = 1, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='006 c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='660 c1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='333 c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='000 c1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='266 ¯us = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 × 10−4 ¯us = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 × 10−4 ¯us = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 × 10−4 ¯us = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='58 × 10−3 ¯us = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='96 × 10−3 Table 2: Modulation mean level ¯us of the CDSW for the eKdV equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The coefficients c1, c2 and c3 are taken equal, while c4 is fixed at c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' expected as the RDSW solution is a perturbation of the generic DSW solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In the TDSW regime, the modulation Whitham shock velocity differs slightly from the numerical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Table 1 displays the modulation theory solution for the mean level ¯us of the Kawahara CDSW for a range of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As stated above, the Whitham shock results in a very small change in mean level from that ahead, u+, which validates the assumption above used to solve the system of modulation equations for the CDSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Table 2 shows the modulation solution for the mean height ¯us of the CDSW solitary waves for a general eKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As for the Kawahara equation results of Table 1, the jump in mean height across the Whitham shock is minimal, so much so that it is difficult to measure from numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' We then conclude that the resonant wavetrain is again essentially linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As the mean level and the wave propagating on this mean level do not couple at the linear level [1], the minimal variation of ¯ur from u+, as assumed in order to solve the modulation equations, is then clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 4 shows similar comparisons as Figure 3 for the Kawahara equation, but for a general eKdV equation with c1, c2 and c3 non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The overall agreement is similar to that for the Kawahara equation in the CDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The modulation theory resonant wave amplitude Ar is in excellent agreement with the numerical amplitude, with reasonable agree- ment even in the RDSW regime, as for the Kawahara equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In contrast, the modulation solitary wave amplitude As is in excellent agreement with the numerical amplitude, even in the RDSW regime, for which good agreement is not expected, as discussed above for the Kawahara equation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figures 4(c) and (d) show similar excellent agreement for the resonant wavetrain wavenumber kr and the Whitham shock velocity Us, with the excellent agreement holding into the RDSW regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The Whitham shock velocity Us and the resonant wavenumber kr are connected through the resonance condition (22), so this similar agreement with numerical solutions is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' To additionally ensure the validity of the equal amplitude approximation used for the unstable bore of the CDSW, we report a few comparison cases for the number of leading, randomly distributed solitary waves N, on neglecting the descending waves near the trailing edge which take the solution down to the initial level behind u−, as compared with numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In theory, N is given by equation (19) or equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' For example, with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04, c4 = 1 (ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='14, c4 = 1) at t = 30, Kawahara CDSW modulation theory gives N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6580 ≈ 9 (N = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='16 ≈ 10) and numerical solutions give N ≈ 9 (N ≈ 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Similarly, with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15, c1 = c2 = c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='006 (c1 = c2 = c3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='333), c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3513 at t = 30, eKdV CDSW modulation theory gives N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9041 ≈ 9 (N = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='463 ≈ 8), and numerical solutions give N ≈ 10 (N ≈ 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 Ar c1=c2=c3 (a) RDSW CDSW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='98 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 As c1=c2=c3 (b) RDSW CDSW 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='535 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='545 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='555 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='56 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 kr c1=c2=c3 (c) RDSW CDSW 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='95 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 Us c1=c2=c3 (d) RDSW CDSW Figure 4: Comparison between numerical solutions of the eKdV equation (1) with c1 = c2 = c3 and c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='351 with modulation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Numerical solution: ◦ (red circle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' modulation theory solution: red (solid) line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' boundaries between DSW regimes: green (dashed) line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (a) resonant wave amplitude Ar, (b) solitary wave amplitude of CDSW As, (c) wavenumber of resonant wavetrain kr, (d) velocity of Whitham shock Us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 7 Water Waves One of the motivations behind the present work is the observed resonant wave amplitude minimum for the eKdV equation (1) with the water wave coefficients [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This is illustrated in Figure 5 based on two values of the nonlinearity parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 5(a) displays the water wave bore for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The bore is in the RDSW regime and the resonant wavetrain has amplitude ∼ 6 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' To illustrate the effect of the higher order coefficients on the bore structure, Figure 5(b) displays the bore solution for the water wave coefficients c1, c2 and c4, but with c3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The coefficient c3 was varied as this coefficient was found to have the greatest effect on the resonant wave amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The bore has become unstable and is bordering on the CDSW regime with the resonant wavetrain having amplitude ∼ 2 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The increase of ǫ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 shows the same overall behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The bore with the water wave coefficients, Figure 5(c), is bordering on the CDSW regime, with the resonant wavetrain still having minimal amplitude ∼ 1 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' In contrast with c3 = 0, Figure 5(d), the bore is bordering on the TDSW regime, with the waves of the bore having much reduced amplitude and extent, and the resonant wavetrain having amplitude ∼ 4 × 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As well as greatly reducing the resonant wave amplitude, the water wave coefficients delay the onset of the transition between the bore regimes, RDSW to CDSW to TDSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' As the resonant wavetrain 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 1 100 50 0 50 100 150 200 250 300 u x (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 1 100 50 0 50 100 150 200 250 300 u x (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 1 100 50 0 50 100 150 200 250 300 u x (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 100 50 0 50 100 150 200 250 300 u x (d) Figure 5: Dependence of bore structure on higher order parameters with c1 = −3/2, c2 = 23/4 and c4 = 19/40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (a) c3 = 5/2, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15, (b) c3 = 0, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15, (c) c3 = 5/2, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3, (d) c3 = 0, ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5, u+ = 0 and t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' amplitude increases on transition from RDSW to CDSW to TDSW, these two effects are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The modulation theory of the present work will now be used to analyse the effect of the values of ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' , 4, on the bore structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The modulation equations with the Whitham shock jump conditions for the eKdV equa- tion do not have a (real) solution when the coefficients ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' , 4, take the water wave values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, a solution of the modulation equations does exist for ranges of the higher order coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' To be specific, the dependence of the existence of modulation theory so- lutions on the higher order coefficients will be explored by varying the nonlinearity ǫ with c1, c2 and c4 taking the water wave values and determining the existence ranges in the final higher order coefficient c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 6(a) shows the existence range of the modulation theory solution for a range of the nonlinearity ǫ, up to high values which are outside the range of asymptotic validity of the eKdV equation, noting that the water wave value is c3 = 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 6(b) shows a bore for c3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3, which is in the region for which the modulation equations do not have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The location of this c3 value is shown by the upper black dot in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Comparing with Figure 5(d) it can be seen that the reso- nant wavetrain has greatly reduced amplitude, in agreement with modulation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The location of the c3 value used for Figure 5(d) is shown by the lower black dot in Figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The resonant wave amplitude of Figure 6(b) is of the same order as the example shown in Figure 5(c), which is for the water wave coefficients for the same value of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' While this is not conclusive justification for the low resonant wave amplitudes seen in Figures 5(a) and (c) and 13 6(b) as the bore for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15 is in the RDSW regime and for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 is just in the CDSW regime and the present modulation theory is for the CDSW regime, it is consistent with the observed numerical results and provides justification to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A final observation from these modulation theory results is that the minimal resonant wave amplitude exists for an unbounded range of c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This is in contrast to the conclusion of [38] that an amplitude node exists for a discrete combination of the higher order coefficients ci, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' , 4, with small amplitude in a neighbourhood of this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This conclusion was based on previous results for resonant solitary waves governed by the eKdV equation, as detailed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It can then be concluded that results for solitary waves do not necessarily transfer to bores governed by the same equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 2 1 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 c3 ε (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 1 100 50 0 50 100 150 200 250 300 u x (b) Figure 6: (a) Existence interval for the modulation solution as the nonlinearity ǫ and c3 vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The modulation solution exists in the red (shaded) region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (b) Example of bore for c3 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 at t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The other coefficients are the water wave values c1 = −3/2, c2 = 23/4 and c4 = 19/40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='9 Ar ε (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 Ar c3 (b) Figure 7: (a) Resonant wave amplitude Ar just below the existence borderline of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (b) Resonant wave amplitude Ar as c3 varies with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Modulation theory amplitude: red (solid) line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' numerical amplitude: ◦ red circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The other coefficients are the water wave values c1 = −3/2, c2 = 23/4 and c4 = 19/40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Here, u− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 and u+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 7(a) shows the resonant wave amplitude Ar just below the modulation theory cutoff of Figure 6(a) as given by modulation theory and full numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It can be seen 14 that the agreement between theory and numerical solutions is excellent and that the resonant wave amplitude is small, even up to very large values of the nonlinearity parameter ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Figure 7(b) details the rapid decrease of the resonant wave amplitude Ar as the modulation theory borderline of Figure 6(a) is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' For this figure the higher order coefficient c3 was varied, while c1, c2 and c4 were kept at their water wave values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The nonlinearity parameter used was ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='13, for which modulation theory gives the borderline c3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The rapid decrease of both the numerical and modulation theory amplitudes as the theoretical borderline is approached as c3 increases is clearly visible, with the numerical amplitude being very small above c3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Indeed, as c3 increases above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1, the numerical amplitude continues to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Finally, the agreement between the numerical and modulation theory resonant wave amplitudes is good up to the cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' 8 Conclusions Whitham modulation theory has been developed to obtain the cross-over dispersive shock wave (CDSW) solution of the extended Korteweg-de Vries equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The DSW itself in this regime is unstable and has a different structure to the standard Korteweg-de Vries DSW, consisting of a train of solitary waves of equal amplitude on average, instead of a modulated train of waves of nearly linearly decreasing amplitude from leading to trailing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' This non-standard structure has been exploited to obtain an approximate solution for the DSW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The resonant wavetrain was obtained as a Stokes wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The key concept of a Whitham shock, a jump in the modulated parameters of a wavetrain, was used to link the CDSW and the resonant wavetrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' It was found that this combination of modulation theory and approximate theory gave solutions in excellent agreement with full numerical solutions of the eKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The modulation theory developed in the present work was used to successfully explain the numerically observed minimal resonant wave amplitude when the higher order coefficients in the eKdV equation (1) take the water wave values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Previous work to explain this minimal amplitude based on resonant solitary wave theory predicted that this amplitude vanishes for a fixed combination of the higher order coefficients, with minimal amplitude in a neighbourhood of this node [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' However, the present modulation theory predicts that the resonant wave amplitude is minimal for regions in the higher order coefficient parameter space, as also shown by full numerical solutions of the eKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The combination of modulation theory with the concept of a Whitham shock and approx- imate theory can be used for other problems which involve resonant dispersive shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' These include resonant optical dispersive shock waves in nematic liquid crystals [18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' These are governed by the eKdV equation for small steps in the optical beam power which generates them, but for general initial steps the governing equations are more complicated, consisting of a nonlinear Schr¨odinger-type equation for the optical field and an elliptic equa- tion for the nematic response [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' Acknowledgement S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' would like to thank Patrick Sprenger with whom this research work stimulated extensive discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' is grateful for the hospitality of Westminster College arranged by the Isaac Newton Institute (INI) for Mathematical Sciences, University of Cambridge, where part of 15 this work was reported and discussed in the Physical Applications Workshop HY2W05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The authors are thankful for the referees’ comments and suggestions which greatly improved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' A CDSW equal amplitude relation The averaged mass and energy densities for the eKdV solitary wave are, respectively, � ∞ −∞ us dx = ¯us + 2 √ 2√as + ǫ √ 2a3/2 s � 2c6 + 4 3c7 � , (26) � ∞ −∞ �1 2u2 s − 1 3ǫ � c2 − 1 2c2 � u3 s � dx = 1 2 ¯u2 s + 2 √ 2¯us √as + 2 √ 2 3 a3/2 s + ǫ � 1 4 (c2 − 2c3) ¯u3 s + 3 √ 2 2 √as (c2 − 2c3) ¯u2 s + √ 2a3/2 s � c2 − 2c3 + 2c6 + 4 3c7 � ¯us + 4 √ 2 15 (5c6 + 4c7) a5/2 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (27) The averaged mass and energy fluxes for the resonant Stokes wave are, respectively, ¯Qmr = 3¯u2 r + 3 2a2 r + ǫ �1 3c1¯u3 r + 1 2c1¯ura2 r + 1 4c2k2 r − 3 4c3k2 r � , (28) ¯Qer = 2¯u3 r + 3 � ¯ur − 1 4k2 r � a2 r + ǫ �1 4 (c1 + 3c2 − 6c3) ¯u4 r + 1 4 � (3c1 + 9c2 − 18c3) ¯u2 r − 2c2¯urk2 r + 5c4k4 r � a2 r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (29) B Modulation theory jump conditions The averaged mass density for the resonant Stokes wave and the bore in the CDSW regime to O(ǫ) are, respectively, ¯Pmr = ¯ur, (B 1) ¯Pmcdsw = ¯us + 2 √ 2√as + 2 √ 2ǫ � c6 + 2 3c7 � a3/2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 2) The averaged mass flux for the resonant Stokes wave and the bore in the CDSW regime to O(ǫ) are, respectively, ¯Qmr = 3¯u2 r + 3 2a2 r + ǫ �1 3c1¯u3 r + 1 2c1¯ura2 r + 1 4c2k2 r − 3 4c3k2 r � (B 3) and ¯Qmcdsw = 3¯u2 s + 12 √ 2¯us √as + 4 √ 2a3/2 s + ǫ �1 3 ¯u3 sc1 + 2 √ 2c1¯u2 s √as + 4 √ 2 �1 3c1 + 3c6 + 2c7) ¯usa3/2 s + 4 √ 2 � 1 15c2 − 1 5c3 + 2c6 + 8 5c7 � a5/2 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 4) 16 The averaged energy density for the resonant Stokes wave and the bore in the CDSW regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' with all terms taken into account for the CDSW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' as discussed in Section 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Per = 1 2 ¯u2 r + 1 4a2 r + 1 4u2 2a4 r + 1 6ǫ (c2 − 2c3) ¯u3 r + 1 4ǫ (c2 − 2c3) ¯ura2 r + 1 8ǫ � c2u2 + 2c2u2 2¯ur − 2c3u2 − 4c3u2 2¯ur � a4 r (B 5) and ¯Pecdsw = ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='0 + ǫ ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 + ǫ2 ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 + ǫ3 ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 + ǫ4 ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 6) where ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='0 = 1 2 ¯u2 s + 2 √ 2¯us √as + 2 3 √ 2a3/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 = 1 6 (c2 − 2c3) ¯u3 s + √ 2 (c2 − 2c3) ¯u2 s √as + 2 3 √ 2 (c2 − 2c3 + 3c6 + 2c7) ¯usa3/2 s + 4 15 √ 2 (5c6 + 4c7) a5/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 = √ 2 � c2c6 + 2 3c2c7 − 2c3c6 − 4 3c3c7 � ¯u2 sa3/2 s + 4 15 √ 2 (5c2c6 + 4c2c7 − 10c3c6 − 8c3c7) ¯usa5/2 s + 2 √ 2 � 4 15c2c6 + 8 35c2c7 − 8 15c3c6 − 16 35c3c7 + 1 3c2 6 + 8 15c6c7 + 4 17c2 7 � a7/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 7) ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 = 2 √ 2 �1 3c2c2 6 + 8 15c2c6c7 + 8 35c2c2 7 − 16 15c3c6c7 − 2 3c3c2 6 − 16 35c3c2 7 � ¯usa7/2 s + 8 5 √ 2 �1 3c2c2 6 − 2 3c3c2 6 + 4 7c2c6c7 − 8 7c3c6c7 � a9/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Pecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 = 8 √ 2 � 1 45c2c3 6 + 2 35c2c2 6c7 + 16 315c2c6c2 7 + 32 2079c2c3 7 − 2 45c3c3 6 − 4 35c3c2 6c7 − 32 315c3c6c2 7 − 64 2079c3c3 7 � a11/2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' The averaged energy flux for the resonant Stokes wave and the bore in the CDSW regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' with all terms taken into account for the CDSW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' as discussed in Section 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qer = 2¯u3 r + 3 � ¯ur − 1 4k2 r � a2 r + 3 �1 2 − k2 ru2 + u2¯ur � u2a4 r + 1 4ǫ � 3c1¯u2 r + 9c2¯u2 r −2c2¯urk2 r − 18c3¯u2 r + 5c4k4 r � a2 r + 1 32ǫ � 3c1 + 24c1u2¯ur + 24c1u2 2¯u2 r + 9c2 + 72c2u2 2¯u2 r + 72c2u2¯ur − 24c2u2k2 r − 64c2k2 ru2 2¯ur − 18c3 − 144c3u2 2¯u2 r − 144c3u2¯ur + 640c4k4 ru2 2 � a4 r + 3 8ǫ (c1 + 3c2 − 6c3) u2 2a6 r + 3 32ǫ (c1 + 3c2 − 6c3) u4 2a8 r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 8) and ¯Qecdsw = ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='0 + ǫ ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 + ǫ2 ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 + ǫ3 ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 + ǫ4 ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 + ǫ5 ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 9) 17 where ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='0 = 2¯u3 s + 12 √ 2¯u2 s √as + 8 √ 2¯usa3/2 s − 4 5 √ 2a5/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1 = 1 4 (c1 + 3c2 − 6c3) ¯u4 s + 2 √ 2 (c1 + 3c2 − 6c3) ¯u3 s √as + 2 √ 2 (c1 + 3c2 − 6c3 + 6c6 + 4c7) ¯u2 sa3/2 s + 8 √ 2 � − 1 15c2 + 2c6 + 8 5c7 � ¯usa5/2 s + 8 √ 2 � 1 21c4 + 3 5c6 + 16 35c7 � a7/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3c1c7 + c1c6 + 2c2c7 + 3c2c6 − 4c3c7 − 6c3c6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯u3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='sa3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='c1c6 + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5c1c7 + 3c2c6 + 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 c2c7 − 6c3c6 − 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 c3c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='sa5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2c1c6 + 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c1c7 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 c2c6 + 92 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='21c2c7 − 72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c3c7 − 12c3c6 + 5c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + 8c6c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯usa7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5c1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + 24c2c6c7 + 72 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + 15c2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 − 48c3c6c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='− 144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 − 30c3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + 8c1c6c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='sa7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='− 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c2c6 − 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='63c2c7 + 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='21c4c6 + 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='21c4c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 c6c7 − 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='a9/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3 = 8 √ 2 �2 5c1c2 6 + 32 105c1c2 7 + 24 35c1c6c7 + 17 15c2c2 6 + 40 21c2c6c7 + 256 315c2c2 7 − 144 35 c3c6c7 − 12 5 c3c2 6 − 64 35c3c2 7 � ¯usa9/2 s + 8 √ 2 � 6 35c1c2 6 + 32 231c1c2 7 + 32 105c1c6c7 + 2 5c2c2 6 + 208 315c2c6c7 + 928 3465c2c2 7 − 36 35c3c2 6 − 64 77c3c2 7 −64 35c3c6c7 + 5 21c4c2 6 + 512 693c4c2 7 + 16 21c4c6c7 + 4 15c3 6 + 24 35c2 6c7 + 64 105c6c2 7 + 128 693c3 7 � a11/2 s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='4 = 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='15c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c2c6c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c1c6c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 + 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='693c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5c2c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 + 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='231c2c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5c3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 − 64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='231c3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 − 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c3c6c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 − 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='¯usa11/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c1c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='315c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3003c1c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c2c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1287c2c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='+ 62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='315c2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c2c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 − 128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='3465c2c6c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6 − 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='35c3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 − 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='105c3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='6c7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='− 128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='1001c3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='a13/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' ¯Qecdsw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='5 = 8 √ 2 � 1 35c1c4 6 + 32 315c1c3 6c7 + 32 231c1c2 6c2 7 + 256 3003c1c6c3 7 + 128 6435c1c4 7 + 3 35c2c4 6 + 32 105c2c3 6c7 + 32 77c2c2 6c2 7 + 256 1001c2c6c3 7 + 128 2145c2c4 7 − 6 35c3c4 6 − 64 105c3c3 6c7 − 64 77c3c2 6c2 7 − 512 1001c3c6c3 7 − 256 2145c3c4 7 � a15/2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content=' (B 10) 18 References [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAyT4oBgHgl3EQfjPi0/content/2301.00412v1.pdf'} +page_content='B.' metadata={'source': 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+gonul.ayci@boun.edu.tr +Pınar Yolum +Utrecht University +The Netherlands +p.yolum@uu.nl +Arzucan Özgür +Bogazici University +Turkey +arzucan.ozgur@boun.edu.tr +Murat Şensoy +Ozyegin University +Turkey +drmuratsensoy@gmail.com +ABSTRACT +Privacy assistants help users manage their privacy online. Their +tasks could vary from detecting privacy violations to recommending +sharing actions for content that the user intends to share. Recent +work on these tasks are promising and show that privacy assistants +can successfully tackle them. However, for such privacy assistants +to be employed by users, it is important that these assistants can +explain their decisions to users. Accordingly, this paper develops a +methodology to create explanations of privacy. The methodology +is based on identifying important topics in a domain of interest, +providing explanation schemes for decisions, and generating them +automatically. We apply our proposed methodology on a real-world +privacy data set, which contains images labeled as private or public +to explain the labels. We evaluate our approach on a user study that +depicts what factors are influential for users to find explanations +useful. +KEYWORDS +Privacy, explainability, online social networks +1 +INTRODUCTION +Managing privacy online is becoming more and more challenging. +On one hand, people use systems, such as online social networks +or Internet of Things applications heavily as these systems provide +useful services. For example, it is common to share a document with +co-authors over a Cloud service or make use of home entertainment +systems that communicate with each other. On the other hand, +people are worried about their privacy and think twice before +using these systems. It is common for people to delete content after +sharing or self-censoring themselves [6]. The problem is getting +more difficult to handle as people are constantly in a situation to +decide whether they would be willing to share a piece of content or +not. Since the amount of content is high, people easily make errors +in their decisions. Even worse, due to decision fatigue, people do +not spend the time to make an informed decision. Various recent +surveys conducted with users of online social networks indicate that +people do not even read the privacy policies that they accept [1, 26]. +Privacy assistants that work side by side with humans in a decen- +tralized manner could serve to address this problem. Privacy assis- +tants have been developed for various privacy assistance including +checking for privacy violations [11], resolving privacy conflicts +among humans [20, 25], recommending sharing policies [8, 23], +and signaling if a piece of content is private [12, 24]. While doing +these tasks, it is important for the privacy assistant to be able to +explain its decisions to the user. +This paper considers a personal assistant that helps its user de- +cide if a given image is private or not and proposes a methodology +and a system to explain this decision to the user. One of the im- +portant works of explainability in conjunction with privacy is by +Mosca and Such [20], where they develop an agent that uses compu- +tational argumentation to resolve disputes and propose a text-based +description of the outcome generated by the system. However, to +the best of our knowledge, there does not exist any methodology +to generate explanations as to why a given content is private or +public. +Existing work on explanation for binary classifications generally +consider what features of the classification have been influential +for the classification. Using these saliency methods, for example, +heat maps can be generated such that parts of an image are high- +lighted to demonstrate its effect on the decision [5]. Lundberg et +al. [17] propose a model-agnostic feature relevance explanation +model, SHAP (SHapley Additive exPlanations), that is based on a +game theoretically Shapley values [22]. This method computes the +contribution of each feature to the prediction output. Lundberg +et al. [16] also propose TreeExplainer that explains predictions of +tree-based machine learning models. TreeExplainer is a variant +of SHAP, which provides the computation of local explanations +based on Shapley values in polynomial time. These approaches are +important because they provide interpretability for the underlying +classifier. However, they are not meant to provide explanations to +the end user as we aim here. +In order to address this problem, we propose a new representa- +tion for explaining why an image is considered private or public. +Our representation is made up of visually exhibiting one or more +topics that the image is associated with while emphasizing im- +portant keywords that put the image in a given topic. A natural +language description accompanies the visuals to describe the rela- +tion between the topics. We provide a methodology to derive these +explanations from a dataset where images are labeled as private +or public. We implement our methodology and apply it to a well- +known image dataset for privacy. We then perform a user study to +measure if users actually find these explanations useful and what +factors of the explanation or the image affect users’ understanding +of the decision. +arXiv:2301.02079v1 [cs.AI] 5 Jan 2023 + +The rest of this paper is organized as follows. Section 2 explains +our understanding of explanation, its formalization, and its rela- +tion to topic modelling. Section 3 develops our methodology into a +system that can be readily used to explain privacy labels of images +and evaluates the effectiveness of the extracted topics. Section 4 +presents how to generate explanations from topics. Section 5 evalu- +ates our system through a user study. Section 6 discusses our work +in relation to related work. Finally, Section 7 concludes our work +and provides future directions. +2 +METHODOLOGY FOR EXPLAINING +PRIVACY +Given an image that is classified as private or public, we would like +to generate an explanation as to why this is so. +2.1 +Understanding Explanation +The explanations that we are interested in generating are meant +for end users. Hence, even if our explanations are influenced by the +features that are used for classification, our aim is not to educate +the user about how the underlying classifier works. Hence, the +explanation should not be too technical. At the same time, given that +many users do not read long texts on privacy policies for example, +we would like the explanation to be visually understandable and +supported by a short text. +Based on these constraints, we propose to formulate an explana- +tion as to whether an image is private or public by a set of topics +that the image belongs to. These topics are shown as a circle and +labeled by the topic name. Each image can have one or more topics. +Additionally, we identify one or more keywords that link this image +to each topic and denote them in the corresponding topic circle. +The intended understanding of this representation is that the image +is private or public, because it can be described with these topics +and keywords. This visual representation is augmented with a short +description that falls into a predetermined language structure to +explain the visual representation. The text is thus supplementary +and does not provide addition information. Figure 1 shows an ex- +ample image, which is annotated as public by annotators. Figure 2 +shows the explanation for the image in the proposed explanation +schema. The explanation provides information that the image is +classified as public because it is associated with topic Business with +the specific keyword "sign". +Figure 1: Example image annotated as public +Figure 2: The explanation for the image in Figure 1 +2.2 +Understanding Topics +In order to realize the above explanations, we need to understand +how we can associate images with topics. Machine Learning al- +gorithms are mostly black-box models and use a large number +of features while making predictions. Thus, the models are not +straightforwardly understandable for humans and are not able to +make explainable predictions. Motivated by this observation, our +aim is to understand the model and its predictions and develop a +methodology to generate explanations for privacy decisions. Thus, +for a prediction of a single instance, we need to extract the most +important and relevant features from all the features in the deci- +sion. For this purpose, we propose to uncover groups of keywords +(i.e., latent topics) from a collection of textual information that best +represents the information in the collection. A topic consists of +relevant descriptive keywords. Each image is associated with topics +based on its keywords. +Topics should be meaningful and interpretable for humans. One +way of realizing this during computation of topics is to ensure +that the topics are coherent. Each topic should pertain to images +that could be described with similar keywords. At the same time, +each topic is relatively different from each other. We can measure +coherence based on two different criteria as follows: +(i) Intra-topic similarity: The average semantic similarity be- +tween all pairs of the most associated N keywords in the +same topic. +(ii) Inter-topic similarity: The average semantic similarity of the +most associated N keywords from different topics. +That is, we can calculate how close the keywords that describe +a topic are semantically using intra-topic similarity and how far +the topics are semantically apart using inter-topic similarity. For a +good topic modelling, we would want to maximize the intra-topic +similarity and minimize inter-topic similarity. +3 +GENERATING TOPICS +Topic Modelling is a technique that discovers latent topics within a +collection of textual information. It allows us to extract different +topics (features) from keyword sets. +3.1 +Topic Modelling +We use a widely used topic modelling technique, namely, Non- +negative Matrix Factorization (NMF) [14]. NMF is an approximation +to factorize a non-negative matrix of a non-negative image-keyword +matrix X, into non-negative matrices W and H as in Figure 3. W + +Business +sign(features) matrix stores how much each image belongs to a topic +and H (components) matrix stores how much each keyword belongs +to each topic. The W and H matrices are initialized randomly. NMF +algorithm runs iteratively until it finds a W and H that minimize the +Frobenius norm of the matrix, that is, ∥𝑋 −𝑊 × 𝐻 ∥𝐹 . NMF is suit- +able for interpretability (components are non-negative) and works +better and faster for short texts (a set of keywords) as compared to +alternatives such as Latent Dirichlet allocation (LDA) [4]. In this +study, we make use of the term weighting method, namely, the +Term Frequency - Inverse Document Frequency (TF-IDF) model to +transform keywords into numerical vectors in order to construct an +image-keyword (X) matrix. TF-IDF assigns weights based on how +relevant a keyword is to a given collection of keyword sets. We +build the NMF model for a different number of topic (k) values, +which generates an image-topic (W) matrix and a topic-keyword +(H) matrix, and then we use the Random Forest algorithm to make +predictions. +Figure 3: Non-negative Matrix Factorization (NMF) concept +3.2 +Evaluation of Topics +We use a balanced subset of the publicly available PicAlert dataset [27]. +The PicAlert is a well-known and widely used dataset for the pri- +vacy prediction problem for images [3, 12, 23, 24]. It contains Flickr +images that are labeled as private or public by annotators. These +images are labeled by 81 users between 10 and 59 years of age with +different backgrounds. We consider an image as private if at least +one annotator has annotated it as private and public if all the an- +notators have annotated it as public. The balanced subset we work +with contains 32𝐾 samples, including 27𝐾 Train and 5𝐾 Test, which +are labeled as private or public. Then, we automatically generate 20 +different descriptive keywords for each image using Clarifai API 1. +In the NMF model, we set the number of topics based on the +model performance in terms of coherence. While calculating intra- +topic similarity and inter-topic similarity, each keyword is repre- +sented by word embedding vectors, namely, word2vec [18]. The +similarity between two keyword vectors is measured by the Cosine- +Similarity metric. Semantically similar tags tend to be close to each +other in the semantic space. Intra-topic similarity values for 20 +topics and 10 topics are 0.20 and 0.18, respectively. Additionally, +inter-topic similarity values for 20 topics and 10 topics are 0.43 +and 0.48. This indicates that topics are more distinguishable from +other topics for 𝑘 = 20 as compared to 𝑘 = 10, while also keywords +in the topics are closer to each other. We represent keywords as +300−dimensional vectors of the word2vec model trained on Google +News when calculating coherence. Note that the cosine-similarity +values between two vectors for this model are generally low (e.g., +1https://www.clarifai.com/ +the similarity between "person" and "people" is 0.51 and "tree" and +"park" is 0.23). +We named 20 topics that we discover using NMF. Figure 4 shows +keyword clouds for five different topics (i.e., Nature, Child, Per- +formance, Business, and Fashion) with the top 20 keywords that +describe each topic. The font size is sensitive to relative signifi- +cance. That is, the most descriptive keyword is displayed as the +largest. For instance, the top five descriptive keywords of the topic +Nature are {𝑡𝑟𝑒𝑒, 𝑝𝑎𝑟𝑘,𝑤𝑜𝑜𝑑,𝑛𝑎𝑡𝑢𝑟𝑒,𝑜𝑢𝑡𝑑𝑜𝑜𝑟𝑠}. Figure 5 shows the +percentage of each topic being associated with private and public +images. Some topics such as the topic People are associated more +frequently with the private class, whereas some of them such as +Sky are associated more frequently with the public class. Note that +although some topics are associated more frequently with private +and some with public images, the topics do not have an explicit class +to which they belong. Therefore, the topic itself does not directly +signal a certain class and thus, it is not straightforward to generate +an explanation for the decision only by looking at its class. +To evaluate the representation of the images with the topics +extracted using NMF, we trained a Random Forest classifier where +the images are represented as TF-IDF vectors of these topics. The +classifier yields an accuracy of 88.5% on the test set, indicating that +the NMF-extracted topics are effective for privacy prediction. It +achieves state-of-the-art performance for image privacy prediction +compared to existing approaches [3, 24]. +4 +GENERATING EXPLANATIONS FROM +TOPICS +The TreeExplainer [16] model provides the contributions of each +feature in terms of Shapley values, which affect the model output +of tree-based algorithms such as Random Forest. Not all features +have equal contribution to a class prediction: a feature can push the +prediction higher (positive Shapley value) or lower (negative Shap- +ley value). The machine learning model concludes its prediction by +taking into account the contribution of each feature. This is useful +in interpreting how the classifier works. One way to create expla- +nations would be to display all these values to the user. However, +as the number of features increases, it would be cumbersome and +confusing to show them all to the end user. Therefore, we start from +the TreeExplainer idea, but modify it to match our expectations for +explanations, as described in Section 2. +In this study, each feature corresponds to a topic. We are inter- +ested in identifying topics that are useful in explaining the content +of the image at hand. For example, for a given image, a large posi- +tive Shapley value might be assigned to a topic because the image +is related to that topic. But, it might also be the case that a large +negative value is assigned to a topic that is unrelated to the topic. +The second category shows that the classifier made a decision based +on the fact that the image did not exhibit the properties associated +with this topic. While useful to understand the classifier, this in- +formation is difficult and possibly unnecessary to show to the user. +Hence, we need to carefully decide how to use the Shapley values +when creating the explanations. +Our methodology generates human-understandable explana- +tions through topic reduction in the output of the TreeExplainer + +keyword 1 keyword 2 keyword 3 keyword 4 +topic 1 +topic 2 +keyword 1 keyword 2 keyword 3 keyword 4 +image 1 +image 1 +topic 1 +x +2 +image 2 +image 2 +topic 2 +image 3 +image 3 +X +W +H(a) Nature +(b) Child +(c) Performance +(d) Business +(e) Fashion +Figure 4: Keyword clouds for Topics Nature, Child, Performance, Business, and Fashion +Figure 5: Percentage of occurrence of each topic in private +and public images +model. Regarding topic reduction, we divide images into four cate- +gories: Dominant, Conflicting, Collaborative, and Vague, in terms of +the contribution of topics to the decision. +Dominant: An image belongs to the Dominant category when the +contribution of one topic is decisive for the class prediction. That +is, a topic makes a relatively high contribution compared to other +topics of the image. +Figure 6 shows an example image in the Dominant category +that has been identified as private by annotators. The generated +explanation for this image being assigned to the private class is +that it is relevant to the topic Child with the keywords including +{𝑐ℎ𝑖𝑙𝑑, 𝑓𝑢𝑛,𝑠𝑜𝑛,ℎ𝑎𝑝𝑝𝑖𝑛𝑒𝑠𝑠, 𝑓 𝑎𝑚𝑖𝑙𝑦}. +Algorithm 1: Find Dominant Topics +Input +:𝑃 ∈ N, the number of images +𝑇 ∈ N, the number of topics +𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 , stores the normalized +Shapley values of the associated topics for each image +𝑑_𝑢𝑏 ∈ [0, 1], the upper bound +Output:𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡, the set of the indexes of images +that have Dominant Topics +1 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ← {∅} +2 for p = 1 to P do +3 +for t = 1 to T do +4 +if 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡] ≥ 𝑑_𝑢𝑏 then +5 +𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ← 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ∪ {𝑝} +else +6 +do nothing +Figure 6: Example image annotated as private and its gener- +ated explanation with the topic Child (Dominant category) +Algorithm 1 describes how to find images belonging to the dom- +inant category, that is, the images with dominant topics. In this +algorithm, 𝑃 and 𝑇 correspond to the number of images and top- +ics, respectively. 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 is the matrix that contains +normalized Shapley values of topics for each image. 𝑑_𝑢𝑏 is the +upper bound with respect to deciding a Dominant topic. First, we +initialize the output of the Algorithm 1, 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡, to store the +indexes of images with dominant topics (line 1). We normalized the +Shapley values by dividing each Shapley value by the sum of the +absolute values of all topics of the image. The algorithm enlarges +the 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 set when the normalized Shapley values of the +associated topic with the image are greater than or equal to the +threshold (lines 4 − 5). For the evaluations, we set this threshold to +0.7. +Collaborative: An image belongs to the Collaborative category +when the contributions of its topics arrive at a consensus about +the decision. That is, the images in this category do not have a + +The generated explanation for this image being assigned to the private class is +that it is related to the topic Child with these specific keywords. +Child +son +peoplehappiness +child +enjoyment +fun +togetherness +boy +familyfair weather +nature +Wood +environment +tr +travel +wildlife +landscape +ee +fall +wild +daylight +scenic +sky +summer +K +bar +no_person +leaf +outdoorsbabyfun +togetherness +portrait +happiness +enjoymeni +LOVe +leisure +1 +peopl +ch +d +boy +1 +toddler +oy +cute +family +little +innocenceband +audience +microphone +Concert +stage +guitarlst +people +songwriter +stringed_instrument +muslclan +nightclub +pop_musicfestiValinstrumenttextpaper +Isignalise +moneyeducation- +internet +e +communication +buslness +no_person +data +symbol +technologyface +elegant +skin +girl +modes +eye +dress +sexy +beautltul +woman +portraitstyle +S +pretty40 +Public +Private +: +21 +Topicssingle decisive topic as in Dominant, but have topics that support +each other collaboratively. Figure 7 shows an example image that +has been identified as private. The generated explanation by our +algorithm for this image being assigned to the private class is that +it is relevant to the topics People, Fashion, and Room with certain +keywords shown in the topic circles. All three topics push the +prediction higher. The Algorithm 1 extended by N topics is also +applicable to finding such images that belong to the collaborative +category. That is, the total contributions of N topics are decisive. +Figure 7: Example image annotated as private and its gener- +ated explanation with the topics People, Fashion, and Room +(Collaborative category) +Conflicting: The topics associated with an image do not always +agree on whether the image should be private or public. In such +situations, the explanation should indicate this. An image belongs to +the Conflicting category if the image has topics whose magnitudes +are almost equal but the contributions to a class prediction are in +the opposite direction. Making a decision can be difficult when +an image has conflicting topics that have opposing forces in the +decision. +Figure 8 shows an example image that has been identified as +public by annotators. The generated explanation for this image is +that even though it is relevant to the topic People with the specific +keywords (i.e., "wear", "man", "people"), it is also relevant to the +topic Art/Vintage that pushes the prediction higher and for that +reason, it is classified as public. +Algorithm 2 describes the process of finding images with conflict- +ing topics. In this algorithm, 𝑃 and 𝑇 correspond to the number of +images and topics, respectively. We normalized the Shapley values +by dividing each Shapley value by the sum of the absolute values of +all topics of the image. 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥 is the matrix that contains the +Shapley values of topics for each image and 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 +stores the normalization of 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥. 𝑐_𝑢𝑏 is the upper bound +Figure 8: Example image annotated as public and its gen- +erated explanation with the topics Art/Vintage and People +(Conflicting category) +with respect to deciding a Conflicting topic. First, we initialize the +output of Algorithm 2, 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡, to store the indexes of images +with conflicting topics (line 1). For each image, the algorithm tries to +find topics that push predictions high and low and also magnitudes +of these contribution are greater than or equal to the threshold, +0.2 (lines 1 − 9). The algorithm enlarges the 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 set when +there exist at least two such topics in the image (lines 10 − 12). +Vague: It is also possible that an image belongs to many topics with +a low confidence. Thus, it would not fall into any of the above three +categories. Therefore, its class cannot be explained as clearly as the +others. We call this category Vague and generate an explanation +that contains the top topics. That is, if there are topics whose con- +tributions are relatively small, we ignore them. In doing so, our aim +is to generate explanations with the most relevant and influential +topics for the decision. +Figure 9 shows an example image that has been identified as +private by annotators. The generated explanation for this image is +that even though it is related to the topic Urban with the specific +keywords (i.e. "urban", "city", "street"), it is also relevant to the topics +People and Offense and for that reason, it is classified as private. +5 +EVALUATION +We performed an online user study to evaluate our proposed expla- +nation model in terms of sufficiency, satisfaction, and understand- +ing. We conducted a pilot study (with 𝑛 = 5 users) before the real +study to test whether the study is understandable. Based on the +comments during the pilot, we improved the initial description of +the study and reworded one question. + +The generated explanation for this image being assigned to the private class is +that it is related to the topics People, Fashion, and Room with these specific +keywords. +People +Fashion +Room +people +portrait +window +adult +fashion +furniture +room +1ght +man +model +indoors +woman +contemporaryThe generated explanation for this is that even though it is related to the +topic People with the specific keywords below (which signals the private class): +it is also related to the topic Art/Vintage and for that reason, it is classified as +public. +Art/Vintage +People +ancient +retro +people +vintage +wear +man +o1d +wall +desktop +antiqueAlgorithm 2: Find Conflicting Topics +Input +:𝑃 ∈ N, the number of images +𝑇 ∈ N, the number of topics +𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 , stores the Shapley values of +the associated topics for each image +𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 , stores the normalized +Shapley values of the associated topics for each image +𝑐_𝑢𝑏 ∈ [0, 1], the upper bound +Output:𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡, the set of the indexes of images that +have Conflicting Topics +1 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ← {∅} +2 for p = 1 to P do +𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 0 +𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 0 +3 +for t = 1 to T do +4 +if 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡] ≥ 𝑐_𝑢𝑏 then +5 +if 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡]) > 0 then +6 +𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 + 1 +7 +else if 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡]) < 0 then +8 +𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 + 1 +9 +else +10 +do nothing +11 +if 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 > 0 and 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 > 0 then +12 +𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ← 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ∪ {𝑝} +else +13 +do nothing +5.1 +User Study +Our user study has three phases. In the first phase, we present a +plain language statement that describes the study and a consent +form. The second phase is meant to explains the study over an +example, wherein we show an image, its generated explanation, +and the three questions that will be asked to the participant. Finally, +in the third phase, each participant is exposed to 16 images with +generated explanations in a random order. Two of these images +deliberately provide irrelevant explanations so that we can differ- +entiate the participants that are attentive during the survey. Thus, +these questions are meant to filter out the participants who are not +focused. Such users are removed from the analysis. +In order to examine our explanation model, we personalize the +Explanation Satisfaction Scale proposed by Hoffman et al. [10]. We +ask participants to rank the following questions: +(1) This explanation that the algorithm produces has SUFFI- +CIENT DETAIL. +(2) This explanation produced by the algorithm is SATISFYING. +(3) From this explanation, I UNDERSTAND why an image has +been identified as private or public. +Each factor is accompanied by a 5−point Likert scale (Strongly +agree = 5, Somewhat agree = 4, Neither agree nor disagree = 3, +Somewhat disagree = 2, Strongly disagree = 1). In the final phase, +participants responded to anonymously collected demographic +Figure 9: Example image annotated as private and its gener- +ated explanation with the topics People, Urban, and Offense +(Vague category) +questions (age, gender, and education level) and optionally pro- +vided free-form text for comments/feedback. We designed our user +study using the Qualtrics online survey tool 2. +5.2 +Participants +A total of 57 participants responded to questions but we excluded +12 of them who did not catch the check questions properly. 64% of +the remaining 45 participants were male and 36% were female. 26 +participants were between 25-34 years old, 14 were between 18-24, +4 were between 35-44, and 1 was between 55-64. In terms of the +highest degree of education, 19 of them had a Master’s degree, 11 of +them had Bachelor’s degree, 6 of them were High school graduates, +5 of them attended Some college (1-4 years, no degree), 2 of them +had Doctorate degree, and 2 of them had Professional school degree +(MD, DDC, JD, etc). +5.3 +Results +The performed user study shows that generated explanations are +useful for and make sense to humans. Table 1 demonstrates how +confidence levels change based on intervals of mean value. Suffi- +cient, Satisfying, and Understandable in Figures 10 and 11 correspond +to the question 1, 2, and 3 in Section 5.1, respectively. Our results +indicate that participants were very confident that explanations +were sufficiently detailed (𝑀 = 3.88,𝑆𝐷 = 1.12), found the explana- +tions satisfactory (𝑀 = 3.62,𝑆𝐷 = 1.28), and understood why the +images were labeled as private or public (𝑀 = 3.8,𝑆𝐷 = 1.33). We +2https://www.qualtrics.com + +The generated explanation for this is that even though it is related to the +topic Urban with the specific keywords below (which signals the public class), it +is also related to the topics People and Offense and for that reason, it is +classified as private. +People +Urban +Offense +man +city +lid people +police +wea oy +administration +urban +adult +school +offense +street +portraitInterval Level +Confidence Level +1.00 - 1.79 +Not at all confident +1.80 - 2.59 +Slightly confident +2.60 - 3.39 +Moderately confident +3.40 - 4.19 +Very confident +4.20 - 5.00 +Extremely confident +Table 1: Confidence Levels based on Mean +evaluate the performance of our methodology based on the private +and public classes and different image categories such as dominant, +collaborative, conflicting, and vague. +(a) Private +(b) Public +Figure 10: Distribution of answers with respect to the classes. +The explanations for images that are labeled as public have +been found to be more sufficient, more satisfying and more +understandable compared to the images labeled as private. +Figure 10 shows the distributions of answers for the survey +questions with respect to the private and public classes. Figure 10(a) +indicates that participants were very confident that the generated +explanations for the private class images are sufficient, satisfying, +and understandable. For instance, from the explanations for private +images, they understood why images have been identified as private +(𝑀 = 3.51,𝑆𝐷 = 1.39). However, 10(b) indicates that participants +found the explanations for public images to be more sufficient, +more satisfying, and more understandable compared to the images +labeled as private. For instance, participants understand better why +images have been identified as public (𝑀 = 4.1,𝑆𝐷 = 1.2). This +is inline with recent work [3], which has shown that privacy is +inherently ambiguous and their personal privacy assistant yields +better performance for the public class. +Figure 11 shows the distributions of answers to assess suffi- +ciency, satisfaction, and understandability with respect to different +categories (i.e., Dominant, Conflicting, Collaborative, and Vague). +Figure 11(a) and 11(b) demonstrate when an explanation has a de- +cisive topic or is composed of like-minded topics in the decision, +participants are very confident that the explanations of images +belonging to such categories are sufficiently detailed and satisfying. +Additionally, participants are confident about understanding why +an image is identified as belonging to a certain class (private or +public). On the other hand, compared to the Dominant and Col- +laborative categories, Figure 11(c) shows that participants are less +confident (𝑀 = 3.47,𝑆𝐷 = 1.45) about understanding the decision +when an explanation has topics that have opposing forces in the +decision. The images in this category have conflicting topics in +terms of the contribution to the decisions. Thus, making a decision +is not straightforward for the images whose explanations belong +to the Conflicting category as compared to the Dominant and Col- +laborative categories. Moreover, Figure 11(d) shows the results for +the explanations of the images belonging to the Vague category. +Even if participants are moderately confident (𝑀 = 3.27,𝑆𝐷 = 1.22) +that the explanations are satisfying, they are very confident about +the sufficiency of the explanations and understandability of a class +decision based on the explanations. +6 +DISCUSSION +In the literature, several studies on image privacy prediction make +use of descriptive keywords (tags) and visual features. Squicciarini +et al. [23] present a Tag-To-Protect (T2P) system that automatically +recommends privacy policies using the image tags. Their exper- +iment shows that prediction accuracy decreases when there are +large tag sets and when the number of tags per image increases. +Tonge and Caragea [24] use deep visual semantic (i.e., deep tags) +and textual features (i.e., user tags) to develop a model to predict +the privacy of images as private or public. They use Support Vec- +tor Machine (SVM) classifiers with pre-trained CNN architectures +such as AlexNet, GoogLeNet, VGG-16, and ResNet to extract fea- +tures (tags). Deep tags of images are the top k predicted object +categories extracted from pre-trained models. Using user-created +tags, they create deep visual features by adding highly correlated +tags to visual features extracted from the fully connected layer +of the pre-trained models. They find that a combination of user +tags and deep visual features from ResNet with the top 350 corre- +lated tags performs the best. Kurtan and Yolum [12] propose an +agent-based approach, namely PELTE, which addresses the same +problem with automatically generated image tags. The internal tag +table stores the data of privacy labels collected from images shared +by the user itself. The external tag table stores the data of images +shared by the user’s friends. Their proposed system performs well +in predicting privacy, even though the personal assistant only has +access to a small amount of data. Ayci et al. [3] propose a personal +privacy assistant called PURE to preserve the privacy of its user. +PURE is aware of uncertainty by generating an uncertainty value +for each prediction of a given image, informing its user about it, +and delegating decisions back to the user if it is uncertain about its +predictions. PURE is able to make personalized predictions by using +the personal data of its user. It is also risk-averse, by incorporating +the user’s risk of misclassification. Their experiments are fruitful in +analyzing the link between uncertainty and misclassification. They +show that PURE captures uncertainty well and performs better +compared to alternative models for quantifying uncertainty (i.e., +Monte Carlo dropout [9] and Deep Ensemble [13]). Although they +demonstrate the success of using descriptive keywords and visual +features to predict image privacy, neither of these approaches ad- +dress capturing the explanations for the privacy predictions as we +have done here. However, explaining the model predictions is criti- +cal to understanding people’s privacy expectations and preferences. +In this study, we propose a novel methodology that uses descriptive +keywords to explore latent topics by topic modelling and provides +explanation schemes for predictions. + +10 +Strongly agree +Somewhat agree +Neither agree nor disagree +Somewhat disagree +Strongly disagree +L +20 +Sufficient +Satisfying +Understandable140 +Strongly agree +Somewhat agree +81 +Neither agree nor disagree +Somewhat disagree +Strongly disagree +IL +24 +Sufficient +Satisfying +Understandable(a) Dominant +(b) Collaborative +(c) Conflicting +(d) Vague +Figure 11: The answers for the questions with respect to the categories +Dammu et al. [7] develop a personalized privacy prediction sys- +tem that is personalizable, explainable, configurable, and comes +with customizable privacy labels. The system consists of four mod- +ules: object detection, location detection, object localization, and +explicit content extraction. The decision network aggregates the +modules’ outputs for personalized privacy predictions. This ap- +proach enables personalized image predictions by incorporating +user feedback. However, it is not yet clear how this approach can +scale in applications that use large image sets. Miller [19] examine +studies of explainability within the scope of philosophy, social and +cognitive psychology, and cognitive science. Their study provides +various definitions of explainability, criteria for selecting explana- +tions, evaluating explanations, and useful insights for Explainable +Artificial Intelligence (XAI). They define interpretability of a model +as the degree to which the cause of a prediction can be understood. +Explainability is defined by the interpretability that one can adopt +and the understanding of the explanation obtained. Justification +is provided by explaining why a decision is good. Responsibility +is one of the criteria for selecting explanations, indicating what +caused an event to occur and the minimum number of changes that +must be made to prevent that event from occurring. Arrieta et al. [2] +provide an overview from a broad perspective on XAI by defining +interpretability and explainability. They define interpretability as +the ability to explain meaning in a form that people can understand. +They associate explainability with explanation as the interface be- +tween a human and a decision maker. They provide a taxonomy +for explainability techniques in machine learning (ML) models. +They examine XAI in ML, which captures transparent models (i.e., +linear regression or Bayesian models) and post-hoc explainability +techniques that can be both model-agnostic and model-specific. In +general, these techniques include model simplification (e.g., rule +extraction methods), feature relevance explanation, and visual ex- +planation. While they use all features in the explanations, this is not +always straightforwardly understandable. We develop a powerful +methodology that is capable of generating explanations with only +relevant topics. +Orekondy et al. [21] present a model for the privacy risk pre- +diction task for images and provide 68 privacy attributes such as +nudity, passport and religion. Li et al. [15] propose a method to find +out what kind of visual content is private. They develop a taxonomy +with 28 categories such as nudity/sexual, irresponsible to child and +bad characters/unlawful/criminal. Zhao et al [28] define a privacy +taxonomy with 10 categories with the most commonly used descrip- +tive keywords for a certain category. For example, the descriptive +keywords of the category religion/culture include culture, religion +and spiritual. Even though they propose inspiring taxonomies for +privacy by synthesizing existing literature, their approaches do not +provide explanations for a particular image as to why the image +is labeled private or public, as we have done here. Moreover, we +use topic modelling to explore hidden topics that are not associated +only with a particular class. +7 +CONCLUSION +In this paper, we propose a novel methodology to understand why a +given image is private or public. Our method is able to explore latent +topics using topic modelling from descriptive keywords of images. +It makes privacy predictions based on the relationship between +images and their associated topics, and automatically generates +explanations for privacy decisions. The privacy classifier achieves +high accuracy, demonstrating the effectiveness of the topic-based +representation of images. Based on a user study, we show that the +generated explanations make sense to people and that participants +find the explanations sufficient, satisfying, and understandable. An +important direction for future work is to be able to get feedback +from people and update the explanations. Another interesting di- +rection would be to incorporate prediction uncertainty [3] into our +proposed methodology. In this way, the uncertainty identified in +the images can be explained to the user. Further feedback from +the user could help the system decrease the uncertainty for future +predictions. +REFERENCES +[1] Alessandro Acquisti and Jens Grossklags. 2005. Privacy and rationality in indi- +vidual decision making. IEEE security & privacy 3, 1 (2005), 26–33. +[2] Alejandro Barredo Arrieta, Natalia Díaz-Rodríguez, Javier Del Ser, Adrien Ben- +netot, Siham Tabik, Alberto Barbado, Salvador García, Sergio Gil-López, Daniel +Molina, Richard Benjamins, et al. 2020. Explainable Artificial Intelligence (XAI): +Concepts, taxonomies, opportunities and challenges toward responsible AI. In- +formation fusion 58 (2020), 82–115. +[3] Gönül Aycı, Murat Şensoy, Arzucan Özgür, and Pınar Yolum. 2022. Uncertainty- +Aware Personal Assistant for Making Personalized Privacy Decisions. ACM +Transactions on Internet Technology (August 2022). +[4] David M Blei, Andrew Y Ng, and Michael I Jordan. 2003. 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In Pro- +ceedings of the International AAAI Conference on Web and Social Media, Vol. 16. +1352–1361. + diff --git a/jdA0T4oBgHgl3EQfIv9Y/content/tmp_files/load_file.txt b/jdA0T4oBgHgl3EQfIv9Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0125d990b0b7b4409a55c21595c5613590f4577 --- /dev/null +++ b/jdA0T4oBgHgl3EQfIv9Y/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf,len=673 +page_content='Explain to Me: Towards Understanding Privacy Decisions Gönül Aycı Bogazici University Turkey gonul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='ayci@boun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='tr Pınar Yolum Utrecht University The Netherlands p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='yolum@uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='nl Arzucan Özgür Bogazici University Turkey arzucan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='ozgur@boun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='tr Murat Şensoy Ozyegin University Turkey drmuratsensoy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='com ABSTRACT Privacy assistants help users manage their privacy online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Their tasks could vary from detecting privacy violations to recommending sharing actions for content that the user intends to share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Recent work on these tasks are promising and show that privacy assistants can successfully tackle them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, for such privacy assistants to be employed by users, it is important that these assistants can explain their decisions to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Accordingly, this paper develops a methodology to create explanations of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The methodology is based on identifying important topics in a domain of interest, providing explanation schemes for decisions, and generating them automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We apply our proposed methodology on a real-world privacy data set, which contains images labeled as private or public to explain the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We evaluate our approach on a user study that depicts what factors are influential for users to find explanations useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' KEYWORDS Privacy, explainability, online social networks 1 INTRODUCTION Managing privacy online is becoming more and more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' On one hand, people use systems, such as online social networks or Internet of Things applications heavily as these systems provide useful services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For example, it is common to share a document with co-authors over a Cloud service or make use of home entertainment systems that communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' On the other hand, people are worried about their privacy and think twice before using these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It is common for people to delete content after sharing or self-censoring themselves [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The problem is getting more difficult to handle as people are constantly in a situation to decide whether they would be willing to share a piece of content or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Since the amount of content is high, people easily make errors in their decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Even worse, due to decision fatigue, people do not spend the time to make an informed decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Various recent surveys conducted with users of online social networks indicate that people do not even read the privacy policies that they accept [1, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Privacy assistants that work side by side with humans in a decen- tralized manner could serve to address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Privacy assis- tants have been developed for various privacy assistance including checking for privacy violations [11], resolving privacy conflicts among humans [20, 25], recommending sharing policies [8, 23], and signaling if a piece of content is private [12, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' While doing these tasks, it is important for the privacy assistant to be able to explain its decisions to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This paper considers a personal assistant that helps its user de- cide if a given image is private or not and proposes a methodology and a system to explain this decision to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' One of the im- portant works of explainability in conjunction with privacy is by Mosca and Such [20], where they develop an agent that uses compu- tational argumentation to resolve disputes and propose a text-based description of the outcome generated by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, to the best of our knowledge, there does not exist any methodology to generate explanations as to why a given content is private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Existing work on explanation for binary classifications generally consider what features of the classification have been influential for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Using these saliency methods, for example, heat maps can be generated such that parts of an image are high- lighted to demonstrate its effect on the decision [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Lundberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [17] propose a model-agnostic feature relevance explanation model, SHAP (SHapley Additive exPlanations), that is based on a game theoretically Shapley values [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This method computes the contribution of each feature to the prediction output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Lundberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [16] also propose TreeExplainer that explains predictions of tree-based machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' TreeExplainer is a variant of SHAP, which provides the computation of local explanations based on Shapley values in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' These approaches are important because they provide interpretability for the underlying classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, they are not meant to provide explanations to the end user as we aim here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In order to address this problem, we propose a new representa- tion for explaining why an image is considered private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Our representation is made up of visually exhibiting one or more topics that the image is associated with while emphasizing im- portant keywords that put the image in a given topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' A natural language description accompanies the visuals to describe the rela- tion between the topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We provide a methodology to derive these explanations from a dataset where images are labeled as private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We implement our methodology and apply it to a well- known image dataset for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We then perform a user study to measure if users actually find these explanations useful and what factors of the explanation or the image affect users’ understanding of the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='02079v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='AI] 5 Jan 2023 The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Section 2 explains our understanding of explanation, its formalization, and its rela- tion to topic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Section 3 develops our methodology into a system that can be readily used to explain privacy labels of images and evaluates the effectiveness of the extracted topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Section 4 presents how to generate explanations from topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Section 5 evalu- ates our system through a user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Section 6 discusses our work in relation to related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Finally, Section 7 concludes our work and provides future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2 METHODOLOGY FOR EXPLAINING PRIVACY Given an image that is classified as private or public, we would like to generate an explanation as to why this is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1 Understanding Explanation The explanations that we are interested in generating are meant for end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Hence, even if our explanations are influenced by the features that are used for classification, our aim is not to educate the user about how the underlying classifier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Hence, the explanation should not be too technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' At the same time, given that many users do not read long texts on privacy policies for example, we would like the explanation to be visually understandable and supported by a short text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Based on these constraints, we propose to formulate an explana- tion as to whether an image is private or public by a set of topics that the image belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' These topics are shown as a circle and labeled by the topic name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Each image can have one or more topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Additionally, we identify one or more keywords that link this image to each topic and denote them in the corresponding topic circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The intended understanding of this representation is that the image is private or public, because it can be described with these topics and keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This visual representation is augmented with a short description that falls into a predetermined language structure to explain the visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The text is thus supplementary and does not provide addition information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 1 shows an ex- ample image, which is annotated as public by annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 2 shows the explanation for the image in the proposed explanation schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The explanation provides information that the image is classified as public because it is associated with topic Business with the specific keyword "sign".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 1: Example image annotated as public Figure 2: The explanation for the image in Figure 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2 Understanding Topics In order to realize the above explanations, we need to understand how we can associate images with topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Machine Learning al- gorithms are mostly black-box models and use a large number of features while making predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Thus, the models are not straightforwardly understandable for humans and are not able to make explainable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Motivated by this observation, our aim is to understand the model and its predictions and develop a methodology to generate explanations for privacy decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Thus, for a prediction of a single instance, we need to extract the most important and relevant features from all the features in the deci- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For this purpose, we propose to uncover groups of keywords (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', latent topics) from a collection of textual information that best represents the information in the collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' A topic consists of relevant descriptive keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Each image is associated with topics based on its keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Topics should be meaningful and interpretable for humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' One way of realizing this during computation of topics is to ensure that the topics are coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Each topic should pertain to images that could be described with similar keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' At the same time, each topic is relatively different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We can measure coherence based on two different criteria as follows: (i) Intra-topic similarity: The average semantic similarity be- tween all pairs of the most associated N keywords in the same topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' (ii) Inter-topic similarity: The average semantic similarity of the most associated N keywords from different topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, we can calculate how close the keywords that describe a topic are semantically using intra-topic similarity and how far the topics are semantically apart using inter-topic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For a good topic modelling, we would want to maximize the intra-topic similarity and minimize inter-topic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 3 GENERATING TOPICS Topic Modelling is a technique that discovers latent topics within a collection of textual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It allows us to extract different topics (features) from keyword sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1 Topic Modelling We use a widely used topic modelling technique, namely, Non- negative Matrix Factorization (NMF) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' NMF is an approximation to factorize a non-negative matrix of a non-negative image-keyword matrix X, into non-negative matrices W and H as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' W Business sign(features) matrix stores how much each image belongs to a topic and H (components) matrix stores how much each keyword belongs to each topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The W and H matrices are initialized randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' NMF algorithm runs iteratively until it finds a W and H that minimize the Frobenius norm of the matrix, that is, ∥𝑋 −𝑊 × 𝐻 ∥𝐹 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' NMF is suit- able for interpretability (components are non-negative) and works better and faster for short texts (a set of keywords) as compared to alternatives such as Latent Dirichlet allocation (LDA) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this study, we make use of the term weighting method, namely, the Term Frequency - Inverse Document Frequency (TF-IDF) model to transform keywords into numerical vectors in order to construct an image-keyword (X) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' TF-IDF assigns weights based on how relevant a keyword is to a given collection of keyword sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We build the NMF model for a different number of topic (k) values, which generates an image-topic (W) matrix and a topic-keyword (H) matrix, and then we use the Random Forest algorithm to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 3: Non-negative Matrix Factorization (NMF) concept 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2 Evaluation of Topics We use a balanced subset of the publicly available PicAlert dataset [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The PicAlert is a well-known and widely used dataset for the pri- vacy prediction problem for images [3, 12, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It contains Flickr images that are labeled as private or public by annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' These images are labeled by 81 users between 10 and 59 years of age with different backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We consider an image as private if at least one annotator has annotated it as private and public if all the an- notators have annotated it as public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The balanced subset we work with contains 32𝐾 samples, including 27𝐾 Train and 5𝐾 Test, which are labeled as private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Then, we automatically generate 20 different descriptive keywords for each image using Clarifai API 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In the NMF model, we set the number of topics based on the model performance in terms of coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' While calculating intra- topic similarity and inter-topic similarity, each keyword is repre- sented by word embedding vectors, namely, word2vec [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The similarity between two keyword vectors is measured by the Cosine- Similarity metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Semantically similar tags tend to be close to each other in the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Intra-topic similarity values for 20 topics and 10 topics are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='20 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='18, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Additionally, inter-topic similarity values for 20 topics and 10 topics are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='43 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This indicates that topics are more distinguishable from other topics for 𝑘 = 20 as compared to 𝑘 = 10, while also keywords in the topics are closer to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We represent keywords as 300−dimensional vectors of the word2vec model trained on Google News when calculating coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Note that the cosine-similarity values between two vectors for this model are generally low (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='clarifai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='com/ the similarity between "person" and "people" is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='51 and "tree" and "park" is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We named 20 topics that we discover using NMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 4 shows keyword clouds for five different topics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', Nature, Child, Per- formance, Business, and Fashion) with the top 20 keywords that describe each topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The font size is sensitive to relative signifi- cance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, the most descriptive keyword is displayed as the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For instance, the top five descriptive keywords of the topic Nature are {𝑡𝑟𝑒𝑒, 𝑝𝑎𝑟𝑘,𝑤𝑜𝑜𝑑,𝑛𝑎𝑡𝑢𝑟𝑒,𝑜𝑢𝑡𝑑𝑜𝑜𝑟𝑠}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 5 shows the percentage of each topic being associated with private and public images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Some topics such as the topic People are associated more frequently with the private class, whereas some of them such as Sky are associated more frequently with the public class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Note that although some topics are associated more frequently with private and some with public images, the topics do not have an explicit class to which they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Therefore, the topic itself does not directly signal a certain class and thus, it is not straightforward to generate an explanation for the decision only by looking at its class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' To evaluate the representation of the images with the topics extracted using NMF, we trained a Random Forest classifier where the images are represented as TF-IDF vectors of these topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The classifier yields an accuracy of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='5% on the test set, indicating that the NMF-extracted topics are effective for privacy prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It achieves state-of-the-art performance for image privacy prediction compared to existing approaches [3, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 4 GENERATING EXPLANATIONS FROM TOPICS The TreeExplainer [16] model provides the contributions of each feature in terms of Shapley values, which affect the model output of tree-based algorithms such as Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Not all features have equal contribution to a class prediction: a feature can push the prediction higher (positive Shapley value) or lower (negative Shap- ley value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The machine learning model concludes its prediction by taking into account the contribution of each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This is useful in interpreting how the classifier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' One way to create expla- nations would be to display all these values to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, as the number of features increases, it would be cumbersome and confusing to show them all to the end user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Therefore, we start from the TreeExplainer idea, but modify it to match our expectations for explanations, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this study, each feature corresponds to a topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We are inter- ested in identifying topics that are useful in explaining the content of the image at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For example, for a given image, a large posi- tive Shapley value might be assigned to a topic because the image is related to that topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' But, it might also be the case that a large negative value is assigned to a topic that is unrelated to the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The second category shows that the classifier made a decision based on the fact that the image did not exhibit the properties associated with this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' While useful to understand the classifier, this in- formation is difficult and possibly unnecessary to show to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Hence, we need to carefully decide how to use the Shapley values when creating the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Our methodology generates human-understandable explana- tions through topic reduction in the output of the TreeExplainer keyword 1 keyword 2 keyword 3 keyword 4 topic 1 topic 2 keyword 1 keyword 2 keyword 3 keyword 4 image 1 image 1 topic 1 x 2 image 2 image 2 topic 2 image 3 image 3 X W H(a) Nature (b) Child (c) Performance (d) Business (e) Fashion Figure 4: Keyword clouds for Topics Nature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Child,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Business,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' and Fashion Figure 5: Percentage of occurrence of each topic in private and public images model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Regarding topic reduction, we divide images into four cate- gories: Dominant, Conflicting, Collaborative, and Vague, in terms of the contribution of topics to the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Dominant: An image belongs to the Dominant category when the contribution of one topic is decisive for the class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, a topic makes a relatively high contribution compared to other topics of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 6 shows an example image in the Dominant category that has been identified as private by annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The generated explanation for this image being assigned to the private class is that it is relevant to the topic Child with the keywords including {𝑐ℎ𝑖𝑙𝑑, 𝑓𝑢𝑛,𝑠𝑜𝑛,ℎ𝑎𝑝𝑝𝑖𝑛𝑒𝑠𝑠, 𝑓 𝑎𝑚𝑖𝑙𝑦}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Algorithm 1: Find Dominant Topics Input :𝑃 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the number of images 𝑇 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the number of topics 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' stores the normalized Shapley values of the associated topics for each image 𝑑_𝑢𝑏 ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the upper bound Output:𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the set of the indexes of images that have Dominant Topics 1 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ← {∅} 2 for p = 1 to P do 3 for t = 1 to T do 4 if 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡] ≥ 𝑑_𝑢𝑏 then 5 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ← 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 ∪ {𝑝} else 6 do nothing Figure 6: Example image annotated as private and its gener- ated explanation with the topic Child (Dominant category) Algorithm 1 describes how to find images belonging to the dom- inant category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the images with dominant topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this algorithm, 𝑃 and 𝑇 correspond to the number of images and top- ics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 is the matrix that contains normalized Shapley values of topics for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 𝑑_𝑢𝑏 is the upper bound with respect to deciding a Dominant topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' First, we initialize the output of the Algorithm 1, 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡, to store the indexes of images with dominant topics (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We normalized the Shapley values by dividing each Shapley value by the sum of the absolute values of all topics of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The algorithm enlarges the 𝑖𝑑𝑥_𝑑𝑜𝑚𝑖𝑛𝑎𝑛𝑡 set when the normalized Shapley values of the associated topic with the image are greater than or equal to the threshold (lines 4 − 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For the evaluations, we set this threshold to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Collaborative: An image belongs to the Collaborative category when the contributions of its topics arrive at a consensus about the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, the images in this category do not have a The generated explanation for this image being assigned to the private class is that it is related to the topic Child with these specific keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Child ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='son ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='peoplehappiness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='child ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='enjoyment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='fun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='togetherness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='boy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='familyfair weather ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='nature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Wood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='tr ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='woman ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='portraitstyle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='pretty40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Public ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Private ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=': ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Topicssingle decisive topic as in Dominant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' but have topics that support each other collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 7 shows an example image that has been identified as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The generated explanation by our algorithm for this image being assigned to the private class is that it is relevant to the topics People, Fashion, and Room with certain keywords shown in the topic circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' All three topics push the prediction higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The Algorithm 1 extended by N topics is also applicable to finding such images that belong to the collaborative category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, the total contributions of N topics are decisive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 7: Example image annotated as private and its gener- ated explanation with the topics People, Fashion, and Room (Collaborative category) Conflicting: The topics associated with an image do not always agree on whether the image should be private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In such situations, the explanation should indicate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' An image belongs to the Conflicting category if the image has topics whose magnitudes are almost equal but the contributions to a class prediction are in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Making a decision can be difficult when an image has conflicting topics that have opposing forces in the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 8 shows an example image that has been identified as public by annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The generated explanation for this image is that even though it is relevant to the topic People with the specific keywords (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', "wear", "man", "people"), it is also relevant to the topic Art/Vintage that pushes the prediction higher and for that reason, it is classified as public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Algorithm 2 describes the process of finding images with conflict- ing topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this algorithm, 𝑃 and 𝑇 correspond to the number of images and topics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We normalized the Shapley values by dividing each Shapley value by the sum of the absolute values of all topics of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥 is the matrix that contains the Shapley values of topics for each image and 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 stores the normalization of 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 𝑐_𝑢𝑏 is the upper bound Figure 8: Example image annotated as public and its gen- erated explanation with the topics Art/Vintage and People (Conflicting category) with respect to deciding a Conflicting topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' First, we initialize the output of Algorithm 2, 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡, to store the indexes of images with conflicting topics (line 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For each image, the algorithm tries to find topics that push predictions high and low and also magnitudes of these contribution are greater than or equal to the threshold, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2 (lines 1 − 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The algorithm enlarges the 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 set when there exist at least two such topics in the image (lines 10 − 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Vague: It is also possible that an image belongs to many topics with a low confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Thus, it would not fall into any of the above three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Therefore, its class cannot be explained as clearly as the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We call this category Vague and generate an explanation that contains the top topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' That is, if there are topics whose con- tributions are relatively small, we ignore them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In doing so, our aim is to generate explanations with the most relevant and influential topics for the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 9 shows an example image that has been identified as private by annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The generated explanation for this image is that even though it is related to the topic Urban with the specific keywords (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' "urban", "city", "street"), it is also relevant to the topics People and Offense and for that reason, it is classified as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 5 EVALUATION We performed an online user study to evaluate our proposed expla- nation model in terms of sufficiency, satisfaction, and understand- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We conducted a pilot study (with 𝑛 = 5 users) before the real study to test whether the study is understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Based on the comments during the pilot, we improved the initial description of the study and reworded one question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The generated explanation for this image being assigned to the private class is that it is related to the topics People, Fashion, and Room with these specific keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' People Fashion Room people portrait window adult fashion furniture room 1ght man model indoors woman contemporaryThe generated explanation for this is that even though it is related to the topic People with the specific keywords below (which signals the private class): it is also related to the topic Art/Vintage and for that reason, it is classified as public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Art/Vintage People ancient retro people vintage wear man o1d wall desktop antiqueAlgorithm 2: Find Conflicting Topics Input :𝑃 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the number of images 𝑇 ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the number of topics 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' stores the Shapley values of the associated topics for each image 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥 ∈ R𝑃×𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' stores the normalized Shapley values of the associated topics for each image 𝑐_𝑢𝑏 ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the upper bound Output:𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' the set of the indexes of images that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='have Conflicting Topics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ← {∅} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2 for p = 1 to P do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='for t = 1 to T do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='if 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡] ≥ 𝑐_𝑢𝑏 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='if 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡]) > 0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='else if 𝑡𝑜𝑝𝑖𝑐_𝑚𝑎𝑡𝑟𝑖𝑥[𝑝][𝑡]) < 0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 ← 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='do nothing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='if 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 > 0 and 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒_𝑐𝑜𝑢𝑛𝑡 > 0 then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ← 𝑖𝑑𝑥_𝑐𝑜𝑛𝑓 𝑙𝑖𝑐𝑡 ∪ {𝑝} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='else ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='do nothing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1 User Study Our user study has three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In the first phase, we present a plain language statement that describes the study and a consent form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The second phase is meant to explains the study over an example, wherein we show an image, its generated explanation, and the three questions that will be asked to the participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Finally, in the third phase, each participant is exposed to 16 images with generated explanations in a random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Two of these images deliberately provide irrelevant explanations so that we can differ- entiate the participants that are attentive during the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Thus, these questions are meant to filter out the participants who are not focused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Such users are removed from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In order to examine our explanation model, we personalize the Explanation Satisfaction Scale proposed by Hoffman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We ask participants to rank the following questions: (1) This explanation that the algorithm produces has SUFFI- CIENT DETAIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' (2) This explanation produced by the algorithm is SATISFYING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' (3) From this explanation, I UNDERSTAND why an image has been identified as private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Each factor is accompanied by a 5−point Likert scale (Strongly agree = 5, Somewhat agree = 4, Neither agree nor disagree = 3, Somewhat disagree = 2, Strongly disagree = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In the final phase, participants responded to anonymously collected demographic Figure 9: Example image annotated as private and its gener- ated explanation with the topics People, Urban, and Offense (Vague category) questions (age, gender, and education level) and optionally pro- vided free-form text for comments/feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We designed our user study using the Qualtrics online survey tool 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2 Participants A total of 57 participants responded to questions but we excluded 12 of them who did not catch the check questions properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 64% of the remaining 45 participants were male and 36% were female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 26 participants were between 25-34 years old, 14 were between 18-24, 4 were between 35-44, and 1 was between 55-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In terms of the highest degree of education, 19 of them had a Master’s degree, 11 of them had Bachelor’s degree, 6 of them were High school graduates, 5 of them attended Some college (1-4 years, no degree), 2 of them had Doctorate degree, and 2 of them had Professional school degree (MD, DDC, JD, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='3 Results The performed user study shows that generated explanations are useful for and make sense to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Table 1 demonstrates how confidence levels change based on intervals of mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Suffi- cient, Satisfying, and Understandable in Figures 10 and 11 correspond to the question 1, 2, and 3 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Our results indicate that participants were very confident that explanations were sufficiently detailed (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='88,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='12), found the explana- tions satisfactory (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='62,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='28), and understood why the images were labeled as private or public (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='8,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='qualtrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='com The generated explanation for this is that even though it is related to the topic Urban with the specific keywords below (which signals the public class), it is also related to the topics People and Offense and for that reason, it is classified as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' People Urban Offense man city lid people police wea oy administration urban adult school offense street portraitInterval Level Confidence Level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='00 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='79 Not at all confident 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='80 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='59 Slightly confident 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='60 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='39 Moderately confident 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='40 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='19 Very confident 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='20 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='00 Extremely confident Table 1: Confidence Levels based on Mean evaluate the performance of our methodology based on the private and public classes and different image categories such as dominant, collaborative, conflicting, and vague.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' (a) Private (b) Public Figure 10: Distribution of answers with respect to the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The explanations for images that are labeled as public have been found to be more sufficient, more satisfying and more understandable compared to the images labeled as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 10 shows the distributions of answers for the survey questions with respect to the private and public classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 10(a) indicates that participants were very confident that the generated explanations for the private class images are sufficient, satisfying, and understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For instance, from the explanations for private images, they understood why images have been identified as private (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='51,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, 10(b) indicates that participants found the explanations for public images to be more sufficient, more satisfying, and more understandable compared to the images labeled as private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For instance, participants understand better why images have been identified as public (𝑀 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='1,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This is inline with recent work [3], which has shown that privacy is inherently ambiguous and their personal privacy assistant yields better performance for the public class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 11 shows the distributions of answers to assess suffi- ciency, satisfaction, and understandability with respect to different categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', Dominant, Conflicting, Collaborative, and Vague).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Figure 11(a) and 11(b) demonstrate when an explanation has a de- cisive topic or is composed of like-minded topics in the decision, participants are very confident that the explanations of images belonging to such categories are sufficiently detailed and satisfying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Additionally, participants are confident about understanding why an image is identified as belonging to a certain class (private or public).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' On the other hand, compared to the Dominant and Col- laborative categories, Figure 11(c) shows that participants are less confident (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='47,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='45) about understanding the decision when an explanation has topics that have opposing forces in the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The images in this category have conflicting topics in terms of the contribution to the decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Thus, making a decision is not straightforward for the images whose explanations belong to the Conflicting category as compared to the Dominant and Col- laborative categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Moreover, Figure 11(d) shows the results for the explanations of the images belonging to the Vague category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Even if participants are moderately confident (𝑀 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='27,𝑆𝐷 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='22) that the explanations are satisfying, they are very confident about the sufficiency of the explanations and understandability of a class decision based on the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 6 DISCUSSION In the literature, several studies on image privacy prediction make use of descriptive keywords (tags) and visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Squicciarini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [23] present a Tag-To-Protect (T2P) system that automatically recommends privacy policies using the image tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Their exper- iment shows that prediction accuracy decreases when there are large tag sets and when the number of tags per image increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Tonge and Caragea [24] use deep visual semantic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', deep tags) and textual features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', user tags) to develop a model to predict the privacy of images as private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They use Support Vec- tor Machine (SVM) classifiers with pre-trained CNN architectures such as AlexNet, GoogLeNet, VGG-16, and ResNet to extract fea- tures (tags).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Deep tags of images are the top k predicted object categories extracted from pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Using user-created tags, they create deep visual features by adding highly correlated tags to visual features extracted from the fully connected layer of the pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They find that a combination of user tags and deep visual features from ResNet with the top 350 corre- lated tags performs the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Kurtan and Yolum [12] propose an agent-based approach, namely PELTE, which addresses the same problem with automatically generated image tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The internal tag table stores the data of privacy labels collected from images shared by the user itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The external tag table stores the data of images shared by the user’s friends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Their proposed system performs well in predicting privacy, even though the personal assistant only has access to a small amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Ayci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [3] propose a personal privacy assistant called PURE to preserve the privacy of its user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' PURE is aware of uncertainty by generating an uncertainty value for each prediction of a given image, informing its user about it, and delegating decisions back to the user if it is uncertain about its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' PURE is able to make personalized predictions by using the personal data of its user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It is also risk-averse, by incorporating the user’s risk of misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Their experiments are fruitful in analyzing the link between uncertainty and misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They show that PURE captures uncertainty well and performs better compared to alternative models for quantifying uncertainty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', Monte Carlo dropout [9] and Deep Ensemble [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Although they demonstrate the success of using descriptive keywords and visual features to predict image privacy, neither of these approaches ad- dress capturing the explanations for the privacy predictions as we have done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, explaining the model predictions is criti- cal to understanding people’s privacy expectations and preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this study, we propose a novel methodology that uses descriptive keywords to explore latent topics by topic modelling and provides explanation schemes for predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 10 Strongly agree Somewhat agree Neither agree nor disagree Somewhat disagree Strongly disagree L 20 Sufficient Satisfying Understandable140 Strongly agree Somewhat agree 81 Neither agree nor disagree Somewhat disagree Strongly disagree IL 24 Sufficient Satisfying Understandable(a) Dominant (b) Collaborative (c) Conflicting (d) Vague Figure 11: The answers for the questions with respect to the categories Dammu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [7] develop a personalized privacy prediction sys- tem that is personalizable, explainable, configurable, and comes with customizable privacy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The system consists of four mod- ules: object detection, location detection, object localization, and explicit content extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The decision network aggregates the modules’ outputs for personalized privacy predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' This ap- proach enables personalized image predictions by incorporating user feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' However, it is not yet clear how this approach can scale in applications that use large image sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Miller [19] examine studies of explainability within the scope of philosophy, social and cognitive psychology, and cognitive science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Their study provides various definitions of explainability, criteria for selecting explana- tions, evaluating explanations, and useful insights for Explainable Artificial Intelligence (XAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They define interpretability of a model as the degree to which the cause of a prediction can be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Explainability is defined by the interpretability that one can adopt and the understanding of the explanation obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Justification is provided by explaining why a decision is good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Responsibility is one of the criteria for selecting explanations, indicating what caused an event to occur and the minimum number of changes that must be made to prevent that event from occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Arrieta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [2] provide an overview from a broad perspective on XAI by defining interpretability and explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They define interpretability as the ability to explain meaning in a form that people can understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They associate explainability with explanation as the interface be- tween a human and a decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They provide a taxonomy for explainability techniques in machine learning (ML) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They examine XAI in ML, which captures transparent models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', linear regression or Bayesian models) and post-hoc explainability techniques that can be both model-agnostic and model-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In general, these techniques include model simplification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=', rule extraction methods), feature relevance explanation, and visual ex- planation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' While they use all features in the explanations, this is not always straightforwardly understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' We develop a powerful methodology that is capable of generating explanations with only relevant topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Orekondy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [21] present a model for the privacy risk pre- diction task for images and provide 68 privacy attributes such as nudity, passport and religion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [15] propose a method to find out what kind of visual content is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' They develop a taxonomy with 28 categories such as nudity/sexual, irresponsible to child and bad characters/unlawful/criminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Zhao et al [28] define a privacy taxonomy with 10 categories with the most commonly used descrip- tive keywords for a certain category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' For example, the descriptive keywords of the category religion/culture include culture, religion and spiritual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Even though they propose inspiring taxonomies for privacy by synthesizing existing literature, their approaches do not provide explanations for a particular image as to why the image is labeled private or public, as we have done here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Moreover, we use topic modelling to explore hidden topics that are not associated only with a particular class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 7 CONCLUSION In this paper, we propose a novel methodology to understand why a given image is private or public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Our method is able to explore latent topics using topic modelling from descriptive keywords of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' It makes privacy predictions based on the relationship between images and their associated topics, and automatically generates explanations for privacy decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' The privacy classifier achieves high accuracy, demonstrating the effectiveness of the topic-based representation of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Based on a user study, we show that the generated explanations make sense to people and that participants find the explanations sufficient, satisfying, and understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' An important direction for future work is to be able to get feedback from people and update the explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Another interesting di- rection would be to incorporate prediction uncertainty [3] into our proposed methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' In this way, the uncertainty identified in the images can be explained to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Further feedback from the user could help the system decrease the uncertainty for future predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' REFERENCES [1] Alessandro Acquisti and Jens Grossklags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Privacy and rationality in indi- vidual decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' IEEE security & privacy 3, 1 (2005), 26–33.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Latent dirichlet allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Journal of machine Learning research 3, Jan (2003), 993–1022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [5] Ali Borji, Ming-Ming Cheng, Qibin Hou, Huaizu Jiang, and Jia Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Salient object detection: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Computational visual media 5, 2 (2019), 117–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [6] Philip Cook and Conrad Heilmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Two types of self-censorship: Public and private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Political studies 61, 1 (2013), 178–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [7] Preetam Prabhu Srikar Dammu, Srinivasa Rao Chalamala, and Ajeet Kumar Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Explainable and Personalized Privacy Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Strongly agree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} 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+page_content='Strongly disagree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Sufficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Satisfying ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='UnderstandableIDD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Strongly agree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Somewhat agree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Neither agree nor disagree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Somewhat disagree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Strongly disagree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Sufficient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Satisfying ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='Understandable[8] Ricard L Fogues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Pradeep K Murukannaiah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Jose M Such,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' and Munindar P Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Sosharp: Recommending sharing policies in multiuser privacy 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' PMLR, 1050–1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [10] Robert R Hoffman, Shane T Mueller, Gary Klein, and Jordan Litman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' Metrics for explainable AI: Challenges and prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' arXiv preprint arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content='04608 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [11] Nadin Kökciyan and Pınar Yolum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' P ri g uard: A semantic approach to detect privacy violations in online social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' IEEE Transactions on Knowledge and Data Engineering 28, 10 (2016), 2724–2737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf'} +page_content=' [12] A Kurtan and Pınar Yolum.' metadata={'source': 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a/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/2301.00444v1.pdf.txt b/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/2301.00444v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fe0a4a2c564b94f153cd9b888caddf8c68c418e --- /dev/null +++ b/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/2301.00444v1.pdf.txt @@ -0,0 +1,1029 @@ +IMC: A Classification of Identity Management +Approaches +Daniela P¨ohn1 and Wolfgang Hommel1 +Bundeswehr Universit¨at M¨unchen, Research Institute CODE, Munich, Germany +{daniela.poehn,wolfgang.hommel}@unibw.de +Abstract. This paper presents a comprehensive classification of iden- +tity management approaches. The classification makes use of three axes: +topology, type of user, and type of environment. The analysis of ex- +isting approaches using the resulting identity management cube (IMC) +highlights the trade-off between user control and trust in attributes. A +comparative analysis of IMC and established models identifies missing +links between the approaches. The IMC is extended by a morphology +of identity management, describing characteristics of cooperation. The +morphology is then mapped to the life cycle of users and identity man- +agement in a further step. These classifications are practically underlined +with current approaches. Both methods combined provide a comprehen- +sive characterization of identity management approaches. The methods +help to choose suited approaches and implement needed tools. +Keywords: Security · Identity Management · Model · Taxonomy +1 +Introduction +Thousands of web applications around the world provide different services via the +internet. These services require the user to present an identity for authentication, +otherwise the user is not able to access them. To manage different users with +their identities, identity management (IdM) was introduced as a paradigm more +than two decades ago. It focuses on managing usernames, which are used as +identifier assigned to users, some sort of credential, usually a password, and +further information, like email address and postal address, called user attributes. +Different evolving requirements led to the creation of different models of and +protocols for identity management systems (IdMS). While stand-alone organi- +zations run a centralized Identity & Access Management (I&AM) system, many +organizations with collaboration, especially in academia, introduced Federated +Identity Management (FIM). FIM is an arrangement between multiple entities in +order to let users use the same identification data as in their home organization. +By FIM, users obtain access to the services provided by partners, called service +providers (SPs), within organizational trust boundaries called federations. The +often-used Security Assertion Markup Language (SAML) [11] is rather static, +whereas OAuth and OpenID Connect (OIDC) [14] provide a dynamic approach, +known for example from Google. Limitations of FIM led to different approaches, +arXiv:2301.00444v1 [cs.CR] 1 Jan 2023 + +like inter-federations (e.g. eduGAIN [5]), the use of the Domain Name System +(DNS) for discovery and trust, e.g., LIGHTest [13], different assurance frame- +works and components. In parallel, user-centric solutions were developed. User +Managed Access (UMA) [8], an OAuth-based standard, enables the user to con- +trol the authorization of data sharing and other protected resources. The user +of Self-Sovereign Identities (SSIs) is the ultimate owner of the identity. SSIs are +typically realized by decentralized networks, like distributed ledger technologies +(DLTs) [2]. Decentralized Identifiers (DIDs) [12] often make use of DLTs. +IdM is one crucial pillar of security frameworks. Several different models +and approaches are currently developed and run. Not all approaches fit into +one single model, making a categorization challenging. This paper contributes +the following improvements: The developed identity management cube (IMC) +categorizes different IdM approaches. The cube is broadened by a morphology +describing aspects of collaboration within the life cycle. Both categorizations +are applied to different protocols and applications. This helps to identify fitting +approaches and missing tools for interoperability. It also provides an overview +of important aspects during the life cycle, helping stakeholders. +This paper is organized as follows. We discuss related work in Section 2. In +Section 3, we present a new categorization of IdM and provide a brief classifica- +tion of current approaches. Additionally, we present a morphology in Section 4, +which is then mapped to the life cycle of identities and identity management. The +newly developed IMC and the morphology are applied to current approaches and +then discussed in Section 5. The paper is concluded in Section 6 by a summary +of the results achieved so far and an outlook to ongoing work. +2 +Related Work +Yuan Cao and Lin Yang [16] identify three core components for IdM: user, ser- +vice provider (SP), and identity provider (IdP). The authors further describe +the three models isolated, centralized, and federated. According to them, the +IdM paradigms can be classified into network-centric paradigm, service-centric +paradigm, and user-centric paradigm. Sovrin [15] sees SSI as next step after +isolated, centralized, federated, and user-centric IdM models. In other papers, +either the models isolated, centralized, federated, and user-centric or centralized, +federated, and decentralized are used. +Boujezza et al. [1] describe a taxonomy for Internet of Things (IoT) by adapt- +ing the paradigms and requirements. The authors classify user model, service +provider model, and hybrid model, combining user and SP, and further submod- +els. In contrast, Pal et al. [10] relate IoT identities to things-centric identities. +Gao et al. [3] describe an IdM model for big data based on authorization, au- +thentication, identification, and audit modules. Habiba et al. [7] use the IdM +requirements taxonomy to classify cloud IdMS. Further approaches have been +developed, leading to different directions, which we integrate into our model. + +3 +Identity Management Models +The main functionalities of IdM are identification, authentication, and autho- +rization. In most cases, a password is provided for authentication, which fulfils +a required complexity or entropy. Second factor, multi-factor, and anonymous +are also possible. The authorization is based on policies, which describe whether +the user is allowed to access a certain functionality or data. With collabora- +tions, the information about the user is stored at the IdP. The user wants to +access a service of the entity SP. Minor entities are trusted third parties (TTPs), +attribute authorities (AAs), having additional information about the user, and +federation operators, if IdPs and SPs form trust boundaries. As new require- +ments are evolving, different approaches for IdM have been developed and will +be emerging in the future. The existing IdM models do not work for several use +cases. Therefore, new models are developed and applied in the following. +3.1 +Analysis of Identity Management Models +In order to distinguish different IdM approaches, models have been established. +These models were updated for user-centric models and partly for SSI. As de- +scribed in Section 2, the following IdM models are mostly used. +Isolated: I&AM per service. +Centralized: Network-centric. I&AM per entity, e.g., with single sign-on (SSO). +Federated: Application-centric. I&AM per federation, which is a set of IdPs +and SPs. Possible protocols are, e.g., OIDC and SAML. +Decentralized: User-centric. I&AM, where the user is in control. Used for FIM +in many cases. Possible approaches are UMA and SSI. Decentralized is partly +divided into user-centric and SSI. +The models are seen as evolution with almost no intersection, displayed in +Figure 1a. The models describe the topology and the source of truth, i.e., the +user or another entity. Approaches can fit into two models at the same time, see +Figure 1b, e.g., if the IdM is user-centric but the SPs form a federation. In this +case, SSI respectively UMA belong to two models. +3.2 +The Identity Management Cube (IMC) +In order to distinguish the approaches, we use the following dimensions. +Topology: Topology of the IdM approach. +Type of User: Type of user, using the approach. +Type of Service: Type of service featured by the approach. +The topology is orthogonal to user-centric and can be used as one category. +Based on existing approaches, the topology can be described as follows. Isolated +is left out of the category as it disappears due to the management overhead. + +(a) Evolution of Identity Management +Models +(b) Orthogonality of Identity Manage- +ment Models +Fig. 1: Identity Management Models +Centralized: I&AM per entity. +With TTP: I&AM with several entities, where at least one TTP is involved. +This applies to many cases of FIM and is, therefore, similar to federated. +Without TTP: I&AM with several entities, where no TTP is involved. As +it describes a distributed, completely decentralized structure, it addresses +different approaches. Most cases of SSI belong to this category. +User-centric describes two things: a human user and user as source of truth. +Other user types are computers, like servers, and IoT devices. Therefore, the +second category is type of user. The human user is further divided into user- +centric and provider-centric, describing attribute handling. +User: Divided into user-centric and provider-centric. This includes cases of +UMA, SSI, but also SAML and OIDC. +Computer: Machine to machine (M2M) communication, for example. +IoT Device: IoT devices usually have less computing power, which restricts +computationally intensive cryptographic operations. +Although an increasing number of web services are used, like Office 365, +several services are non-web-based. In order to distinguish the type of service, +the following characteristics are set. +Non-Web Service: M2M communication, but also local services. +Background Web Service: Services, which are need for interactive web ser- +vices, like localization of the user’s home organisation. +Interactive Web Service: Services the end-user uses. +As a result, the new model comprises three categories, topology, type of +user, and type of service, displayed as axes. In reference to the Life cycle, Aspect, +Layer (LAL) Brick [4], the categories result in a cube. In Figure 2, the developed +IdM cube including the labels of the different axes is shown. User-centric and +provider-centric are thus left out for clarity reasons. + +isolated +centralized +federated +user-centricisolated +centralized +federated +user-centricFig. 2: Identity Management Cube +3.3 +IMC Applied to Current Approaches +In order to depict the IMC, different IdM approaches are classified by the cate- +gories described above. As examples, centralized IdM with SAML federations in +research and education (R&E), OIDC in the web, UMA for private users, and +SSI as new approach are chosen. In addition, IdM for servers, IoT, and with +Active Directory (AD) are explained. +SAML is used in R&E to let users access web services at research partners. +It is based on lightweight directory access protocol (LDAP), databases, or even +AD with the add-on federation. The entities form a federation, which relies on +contracts with the federation operator. As a result, it has the following charac- +teristics, as shown in Figure 3a. +Topology: With federation tools as TTPs. +Type of User: User are humans, but the type is provider-centric. +Type of Service: Interactive web services for end users. +OIDC is used in web as well, but is a more dynamic protocol without a TTP, +based on OAuth. UMA is also developed on top of OAuth, but more user-centric. +This can be seen in the characteristics, shown in Figure 3b. +Topology: Using Webfinger technology is without a TTP, but can be central- +ized in some use cases. +Type of User: Human end user in most cases, which can be either provider- +centric (OIDC) or user-centric (UMA). +Type of Service: Typically interactive web for end users, but others types are +also possible. +SSI is seen as the new step in evolution of IdM, as the user is in control of +everything. The concept is without a TTP, but it evolves to a topology with a +TTP for scalability and performance reasons. Most approaches concentrate on +interactive web services, though the concept could be applied to other services +as well. SSI, therefore, has the following characteristics, displayed in Figure 3c. + +user +computer +loTdevice +!TTP +centralized +non-web +background +interactiveTopology: Originally, SSI is without a TTP, but is evolving to centralized ser- +vices. +Type of User: SSI focuses on the user, therefore, user-centric. +Type of Service: Interactive web services for end users. +Besides web application, servers are run at the backend, which are normally ac- +cess through keys. The public key is stored at the server, while the administrator +is in possession of the private key. So, the service is non-web and it is typically +either centralized or with a TTP. As a result, identity management for servers +can be described as following, shown in Figure 3d. +Topology: Either centralized, with a centralized IdM, or with a TTP. +Type of User: Both, computer in M2M or human users are possible. +Type of Service: The services are typically non-web. +Centralized IdM with AD is used in companies to enable employees to login at +their computer, provision folders and shares, but also to access web services with +single sign-on (SSO). It has the following characteristics, shown in Figure 3e. +Topology: The AD itself is centralized. +Type of User: The human user is in focus, but the IdM is provider-centric. +Additionally, Windows computer can be a user. +Type of Service: All types of services are possible, as it relies on Windows. +IoT devices often communicate with Constrained Application Protocol (CoAP) +instead of Hypertext Transfer Protocol (HTTP). The devices, which either lack +a browser to perform user-agent based authorization or are input constrained, +cannot make use of typical web protocols, like OAuth or SAML. One option is, +e.g., to utilize shared keys, another is ACE-OAuth. ACE-OAuth maps OAuth +methods to Authentication and Authorization for Constrained Environments +(ACE). The characteristics are shown in Figure 3f. +Topology: IoT networks are typically centralized managed, which can be with +or without a TTP. +Type of User: The type is IoT device. +Type of Service: IoT devices are mainly background services. +The selected approaches can be merged in one IMC. The colors are used as in +the figure above: AD dark green, SAML yellow, OIDC dark blue, SSI light blue, +servers orange, IoT light green. The cube illustrates that many approaches are +used for interactive web and human users, while the protocols themselves could +be used for other user cases as well. The figure at the same time visualizes the +differences between the approaches. While AD is focused on centralized topology, +SAML typically uses a TTP, while OIDC, UMA, and SSI tend to work without +TTP. The most common type of service are used in Figure 4a, while Figure 4b +adds also unusual use cases. Both figures show that the selected approaches do +not cover all aspects of the IMC. SSI with a centralized party would partly fulfil +the application of SAML, as it would double to with a TTP from the later figure. + +(a) IMC for SAML +(b) IMC for OIDC +(c) IMC for SSI +(d) IMC for Server +(e) IMC for AD +(f) IMC for IoT +Fig. 3: Identity Management Cubes Applied to Different Use Cases + +user +computerloTdevice +!TTP +TTP +centralized +non-web +background +interactiveuser +computer +loTdevice +!TTP +TTP +centralized +non-web +background +interactiveuser +computerloTdevice +!TTP +TTP +centralized +non-web +background +interactiveuser +computer +loT device +!TTP +TTP +centralized +non-web +background +interactiveuser +computer +loT device +!TTP +TTP +centralized +non-web +background +interactiveuser +computer +loTdevice +/dli +TTP +centralized +non-web +background +interactiveEspecially these shared single cubes illustrate that interoperability between the +approaches should be easily reached, while combining different approaches ar- +ranged in different cubes probably needs more effort and tools. Additionally, one +can either have user-centric or service-centric. Most approaches cannot provide +both, as trust into the attributes is missing. +(a) Merged IMC +(b) Merged IMC Including Unusual Use +Cases +Fig. 4: Merged Identity Management Cubes +4 +Morphology of Identity Management +In order to determine the degree of fulfilment, a uniform format is needed de- +scribing the approaches in more detail. Therefore, a characteristic similarly to a +morphology is established. The morphology focusses on organizational aspects, +while the IMC categorizes the technology. The authors of [6] describe the char- +acteristics of Inter-FIM based on a morphology. As the characteristics need to +comprise all IdM approaches and therefore relates to the IMC, the morphology +is extended for the needs of universal IdM. In a next step, the morphology is +mapped to the life-cycle, in order to clarify when which decision is taken. Last +but not least, approaches are characterized by the morphology. +4.1 +Design of the Morphology +The morphology describes the characteristics of the cooperation. [6] uses cooper- +ation structure, members, group structure, federation dimension, organizational +dimension, duration, sort of collaboration, coordination, establishment, circle of +trust, degree of commitment, and trust relationship. As this approach concen- +trates on Inter-FIM, the following characteristics can be left out or need changed. + +user +computer +loTdevice +dlli +TTP +centralized +non-web +background +interactiveuser +computer +loTdevice +!TTP/ +TTP +centralized +non-web +background +interactiveStructure of Cooperation: The structure described topology and coopera- +tion customized for federations. The topology is described by the cube, while +different aspects of the cooperation are part of the morphology. +Cooperation: Instead of FIM, Projects, and Communities, this characteristic +is now described in “Reason for Joining” as well as “Order”. +Formalization Differentiates between “limited contract” and “cooperation agree- +ments”, in order to describe the distinction of contracts. +Dynamic of Joining: Broader scope with “stable” and “unstable”. +To describe different organizational aspects, other characteristics need to be +added. Several characteristics relate to the IdM architecture. +Reason for Joining: In order to differentiate between private usage and busi- +ness reasons, this characteristic was included. +Connectivity: Describes the interaction between involved parties, which might +have consequences for the architecture. +Direction of Cooperation: Broadens the scope. +Administration: Degree of automation, which relates to the architecture. +Cooperation Structure: Either “hierarchical” or “heterarchical”. +Level of Trust: Trust between involved parties. +Identities: Included as it has implications for the architecture. +4.2 +Identity Management Morphology +This results in a morphology, which includes more and broader characteristics. +The morphology has the following categorize, as shown in Table 1. +Initiation: Initiation of the cooperation. +Cooperation: Settings of the cooperation. +Coordination: Settings of the coordination. +Trust: Trust between participating entities. +Identities: Settings of the identities. +Initiation comprises of reason for joining and dynamic of joining. The reason +can be “personal”, “social”, like in social media, “by law” or “economic”. Eco- +nomic reasons can further be split into “time”, “risk”, “earnings”, “competence”, +“costs”, “pressure”, and “protection”. Another distinction could be “planned”, +if necessary, and “spontaneously event-driven”. The dynamic is either “stable” +or “unstable”, i.e., it is either predictable or not. +The cooperation itself is described by degree of integration, connectivity, pro- +fessional limits, factual limits, direction of cooperation, order, locality, organiza- +tional, and formalization. Both, the degree of integration and the connectivity +between partners, are part of the networking between partners. The degree of +integration can either be “autonomous”, “coordinated” or “integrated”. This +means that either the partners work autonomous, coordinated towards a goal. +Integrated can be a fusion of organizations. The connectivity has two steps: “low” +and “high”. It partly relates to integration. The next category are both limits, + +Table 1: Morphology for Identity Management in Detail +Initiation +Reason for Joining +personal +social +economic +law +Dynamic of Joining +stable +unstable +Cooperation +Degree of Integra- +tion +autonomous +coordinated +integrated +Connectivity +low +high +Professional Limits +user +R&D +department +value chain +Factual Limits +short +medium +long +permanent +Direction of Coop- +eration +vertical +horizontal +diagonal +Order +strategy +project +R&E +region +Locality +local +regional +national +international +Organizational +micro +meso +macro +Formalization +arrangement +limited +con- +tract +cooperation +agreement +capital +inter- +weaving +Coordination +Administration +manual +supported +automated +Number of Partici- +pants +bilateral +simple +complex +Group Structure +open +with +limita- +tions +closed +Cooperation Struc- +ture +hierarchical +heterarchical +Sort +of +Coordina- +tion +implicit +explicit +Trust +Directness +direct +transitive +Circle of Trust +static +dynamic +virtual +Level of Trust +zero +low +medium +high +Identities +Transparency +low +medium +high +Controllability +low +medium +high +Identification +internal +external +combination +Authentication +Method +anonymous +simple +2FA +MFA +Authentication Or- +ganization +internal +external +combination +Authorization +internal +external +combination + +professional and factual. Professional limits describe which organization part is +involved in the cooperation. It can be “research”, a “department”, the complete +“value chain”, or just one or more “users”. Factual limits are described by “per- +manent” or “restricted”. Restricted can further be split into “short”, “medium”, +and “long”. The direction of cooperation depicts how close both economic levels +are related. “Horizontal cooperation” describes the cooperation of companies of +the same business or same level of the value chain, while “vertical cooperation” +is a cooperation between organizations of different economic levels, like retail +company and production company. A cooperation is “diagonal”, if all involved +companies are neither on the same economic level nor business, e.g., travel com- +pany and food company. The order characterizes the reason for the cooperation, +which is “strategic”, a “project”, “R&E”, or based on the “region”. Both, the +locality and the organizational are dimensions of the cooperation. The locality +of the cooperation is either “local”, “regional”, “national”, or “international”. +A national federation are the R&E federations, like SWITCHaai in Switzerland. +eduGAIN is the international umbrella federation for the national pendants. +The organizational dimension describes the viewing plane of the coopera- +tion. Terminology from economics is used. “Micro” plane consists of one sin- +gle entity, while the “meso” plane comprises of several organizations, e.g., in a +federation. The “macro” plane shows the cooperation of cooperation, e.g., an +inter-federation. The formalization classifies the kind of formality between the +entities. While an “arrangement” can be oral or somehow written, a “contract” +is divided into limited length and cooperation. The last step is a “capital” in- +terweaving of the involved entities. An example for an arrangement is the usage +of social media for end users, while contracts are typical for projects. The for- +malization also describes the binding intensity, which is the degree by with the +involved entities give up their autonomy. +The coordination explains the management of the cooperation, which con- +sists of administration, number of participants, group structure, order, and sort +of coordination. The number of participants is related to the group structure. +Open cooperation do not have a firm number of participants. Closed cooperation +allow simple as well as bilateral structures. The coordination further relates to +trust. The administration can be “manual”, “supported” or “automated”. The +number of participants is strongly related to cooperation. The participating enti- +ties can either have a “bilateral” agreement, the cooperation can have a “simple” +structure, or it can be “complex”. While bilateral cooperation still work with +duplicated user bases, this is not possible with more entities involved. Simple +networks can be realized with security assertion markup language (SAML) feder- +ations, while complex structures are also more complex for technical realization. +OpenID Connect (OIDC) can be used for it. The group structure is either “open”, +“with limitations” or “closed”. OIDC is typically open, while SAML federations +have limitations in R&E or are closed in industry. The cooperation structure is +“hierarchical” or “heterarchical”, when all partners are more or less of the same +level. The sort of coordination has two possible values: “implicit” or “explicit”. + +With an explicit coordination, the integration of an institutional coordination +instance is supported. An implicit coordination needs a local coordination. +Trust between entities is the result of several different factors, like recom- +mendation or past experience. Within the morphology, only the basics for the +cooperation are described, which includes directness, dynamics, and the average +level of trust. The circle of trust (CoT) relates to the sort of cooperation. If +the group structure is limited, then the CoT can be static. If the number of +participants is complex, is the CoT virtual as not all the information about all +participants cannot be fully known. Direct trust implicates static or dynamic +CoT. Directness describes how the trust between two entities is derived. The +trust is either “direct” or “transitive / indirect”, via another entity. The dynam- +ics characterize the trust over time, which is either “static” or “dynamic”. Last +but not least, the level of trust can be “zero”, “low”, “medium”, or “high”. +As final category, identities are classified. Identities especially describe factors +of trust and also user-centric features. This includes transparency, controllability, +identification, authentication, consisting of methods and organizational factors, +as well as authorization. The transparency is either “low”, “medium” or “high”. +The same characteristics can be applied to controllability. The identification, +authentication, and authorization can be done “internally”, “externally”, or in +a “combination” of different entities. The authentication methods describe the +sort of credentials used, which is either “anonymous”, “simple” (like a password), +“second-factor authentication (2FA)” or “multi-factor authentication (MFA)”. +In order to reduce the complexity of the morphology, suited characteristics can +be left out. In the next step, special characteristics for the use case, like topology +of federation can be added. This depends on the specific use case. +4.3 +Morphology Mapped to Life Cycle +The morphology can be mapped to the life cycle of IdM, helping starting cooper- +ation to identify their framework. The life cycle is similar to the Deming Cycle [9], +which has the phases plan, do, check, act. The Deming Cycle is an iterative four- +step management method used in business to control improvements of processes +and products. It can be applied to service management, security management, +and many other, like identity management. The life cycle of IdM has the phases +initiation, agreement, cooperation, reconsideration and improvement, and ter- +mination. Reconsideration can either lead to improvement or termination. The +phases of the IdM life cycle have the following characteristics. +Initiation: A purpose leads to the initiation of the cooperation. +Agreement: After discussions, an agreement is signed, describing the frame- +work of the cooperation. IdM should be a part of the agreement. Otherwise, +the parties need to agree on IT aspects outside of the agreement. +Cooperation: The cooperation is starting. In many cases, the cooperation is +starting slowly, setting everything in place. Then there is a hype of coopera- +tion, where everything is running and the original purpose is hopefully met. +In IT, the start requires work, setting up the infrastructure. + +Reconsideration: It describes if the cooperation is proceeded and if changes +need to be made. The same appears for IdM. +Improvement: The changes lead to improvements, which are implemented. +Termination: If the purpose is met or other reasons lead to the end of the +cooperation, the IdM is also terminated for the project. +The morphology, described in the previous section, can be mapped to the life +cycle. This helps to gain a better picture of the required decisions, as shown in +Figure 5a. The initiation phase comprises both characteristics of the morphol- +ogy of initiation, which means “Reason of Joining” and “Dynamics of Joining”. +Cooperation and coordination, have characteristics in “Agreement” and “Coop- +eration”. This is the case as some characteristics are decided at the agreement, +while others have more impact on the cooperation. The agreement thereby fea- +tures: “Formalization”, “Limitations”, “Direction of Cooperation”, “Coopera- +tion Structure”, “Dimensions”, “Number of Participants”, and “Group Struc- +ture”. The cooperation as a result includes “Trust”, “Identity”, “Degree of Inte- +gration”, “Connectivity”, “Administration”, and “Sort of Cooperation”. During +reconsideration every aspect is re-evaluated. Some aspects are enhanced during +improvement. The life cycle is terminated, if the cooperation ends. +The IdM life cycle includes the user life cycle, because users change through- +out a project or life cycle. During a project, users leave, while others join. This +is also the case for IdM in organizations. In the end, every user account needs +to be closed. The life cycle of the user includes the following phases. +Request: The user requests an account at an IdMS. +Provisioning: The account is provisioned (attributes, roles, and permissions). +Identification: The user identifies himself. +Authentication and Authorization: First authentication, then authorization. +Self-Service: The user can access the self-service. +De-Provisioning: In the end, the user account is de-provisioned. +The life cycle of the user has only few interactions with the morphology: +trust into the service provider during request, then identification, authentication, +authorization themselves as well as transparency in the self-service phase. This +is also visualized in Figure 5b. +As a result, IdM models describe the approaches in general, while the mor- +phology details aspects of the cooperation. The mapping of morphology with the +life cycles explains the order of the actions, which need to be taken. This can +guide projects and organizations to identity management processes. Neverthe- +less, a decision matrix for choosing the best fitting approach is missing, although +the cube gives a first hint. Additionally, interfaces to already established pro- +cesses, like service management and security management, are needed. +4.4 +Morphology Applied to Current Approaches +Following, the morphology is applied to centralized IdM with AD and the differ- +ences for SAML, OIDC, SSI, server, and IoT devices are described. Centralized + +(a) +Morphology +mapped +to +Identity +Management System Life Cycle +(b) Morphology mapped to User Life Cy- +cle +Fig. 5: Morphology mapped to Life Cycles +IdM with AD is typically used in companies. Depending on whether they have +cooperation, several branches or not, the complexity is different. Also depending +on the point of view, e.g., user or company, different properties can be coloured. +Let us assume the company in this example just uses AD for its users, while they +have other methods for cooperation. Therefore, the following morphology can be +formed, see Table 2. The reason for joining is economic, while the dynamic is +stable. AD was introduced at some point in time. When regarding the coop- +eration, the cooperation within the company is considered. The integration is +therefore integrated. The connectivity is high as all participants work together. +AD was introduced for the complete value chain of the company. It should be a +permanent solution, though technologies and decisions change. The direction of +cooperation cannot be described by the categorization. The order was based on +a strategy, while the company is local. The organizational factor is micro. The +cooperation within the company depends on contracts with its employees. The +administration is hopefully supported, while the number of participants can be +described with bilateral. The group structure is at least currently closed, while +the coordination is hierarchical and explicit. The trust is direct, rather static, +with medium trust, as all employees needed to submit papers. The identities are +managed within the company, with simple and second factors. +The entities of SAML in R&E form federations, which are spread over regions, +countries, and the world. The entities have contracts with federation operators, +which have contracts with the inter-federation operator. The coordination be- +tween the entities is rather low. As the identities are managed by the home +organization, trust is lower. Therefore, the following morphology can be seen. +Initiation: Individuals join for R&E, while companies have economic reasons. +The dynamic is rather stable, as entities have to sign on contracts. +Cooperation: The cooperation is autonomous, only little coordinated by the +federation and inter-federation operators. The connectivity between the en- +tities is low, as there are many different services within a federation and only + +Formalisation +Limitations +Initiation +Direction of Cooperation +Agreement +Phase +Order Dimensions +Number of Participants +Initiation +Group Structure +★ +Trust +Improve- +Identity +Cooperation +ment +Degree of +Integration +Connectivity +Administration +Sort of +Reconside +Terminationk +Coordination +rationldentification +Request +Provisioning +Self-Service +Permissions +Transparency +De- +AuthNZ +ProvisioningTable 2: Morphology for Centralized Identity Management with AD +Initiation +Reason for Joining +personal +social +economic +law +Dynamic of Joining +stable +unstable +Cooperation +Degree of Integra- +tion +autonomous +coordinated +integrated +Connectivity +low +high +Professional Limits user +R&D +department +value chain +Factual Limits +short +medium +long +permanent +Direction of Coop- +eration +vertical +horizontal +diagonal +Order +strategy +project +R&E +region +Locality +local +regional +national +international +Organizational +micro +meso +macro +Formalization +arrangement +limited +con- +tract +cooperation +agreement +capital +inter- +weaving +Coordination +Administration +manual +supported +automated +Number of Partici- +pants +bilateral +simple +complex +Group Structure +open +with +limita- +tions +closed +Cooperation Struc- +ture +hierarchical +heterarchical +Sort +of +Coordina- +tion +implicit +explicit +Trust +Directness +direct +transitive +Circle of Trust +static +dynamic +virtual +Level of Trust +zero +low +medium +high +Identities +Transparency +low +medium +high +Controllability +low +medium +high +Identification +internal +external +combination +Authentication +Method +anonymous +simple +2FA +MFA +Authentication Or- +ganization +internal +external +combination +Authorization +internal +external +combination +a small percentage of users will use the specific service of a service provider. +Mostly, the entities have contact with the federation operators. The cooper- +ation is limited to research, while the time depends on several reasons. The +cooperation can be vertical as well as horizontal. The locality is national or +international in most cases. Also regional federations are established. The +organization form is either meso or macro, depending on the type of feder- + +ation. Federations are formalized by contracts with the federation operator +and partly arrangements between entities. +Coordination: The administration is supported with manual steps needed. As +contracts need to be signed, the number of participants is simple and the +group structure is with limitations. The order is more heterarchical than +hierarchical, while the coordination is more implicit than explicit. +Trust: Trust is transitive via federation or inter-federation operator. With a +static number of participants, the circle of trust is also static with little +dynamics. The level of trust is low or medium, depending on separate means. +Identities: Since communities with additional attribute authorities were formed +and other means of identification are in use, authorization and identification +are either internal or a combination, while authentication is internal. Trans- +parency and controllability are rather low as a result of the structure. +IdM with OIDC distinguishes from SAML as the protocol is dynamic and +the widely known use case is web authentication. For OIDC, the initiation can +have several reasons, therefore, the dynamic is unstable. The cooperation is +loose, which is true for the coordination as well. Trust is rather low, but can be +stepped up with a second factor. The different constellations also have impact +on the identities. SSI is different as the user is in control of the attributes, which +then impacts trust and identities. If a company hosts their servers in-house, then +the cooperation is within the company and maybe with other offices. IoT devices +can be used at home as well as at organizations. The trust into the devices is +typically low, as others might manipulate the device without notice. +5 +Discussion +We characterized IdM approaches in two ways: the IMC describes the technical +aspects, followed the morphology for organizational aspects. In order to compile +an overview of IdM approaches, we noticed intersections between existing IdM +models. These intersections helped us to identify categories, which are needed to +differentiate IdM approaches. The three categories topology, type of user, and +type of service are arranged in a cube, the IMC. IMC clarifies the characteristics +type of user and topology. Additionally, the perspective is made clear, i.e., user- +centric or provider-centric. While an approach could belong to two models used +beforehand, it can be clearly classified by the IMC. With the flexibility of the +three categories in mind, future approaches should be able to be characterized. In +the next step, we applied different IdM approaches to the cube. These approaches +were typical web services for end users, but also servers and IoT, resulting in a +colourful IMC. Some use cases are more typical than others. Besides this fact, +the application was straight forward and showed us similarities and differences +between the approaches. These findings might indicate a possible combination of +approaches. It is noticeable that trust and user-centric are not featured together +in the shown examples. The IMC can, therefore, help to combine different IdM +approaches and explore missing tools. + +In a next step, a morphology for IdM was developed. The morphology de- +scribes different aspects of a cooperation. In this case, the categories initiation, +cooperation, coordination, trust, and identities with their categorizations need +to be regarded. The relationship between morphology and the different life cycles +were shown in a next step. For guidance, the morphology can be used to speed +up the implementation and evaluation in a later step. The morphology was then +mapped to different approaches. A certain variance is seen, which depends on +the actual implementation. Nevertheless, organizational settings are made clear. +This does not include internal processes, which will be regarded in future work. +Both, the IMC and the morphology, do not describe IdM in all aspects, but +help to categorize different approaches, use cases, and their implementation. The +categorization helps within the life cycle of IdM to mix different approaches, see +missing tools, and to regard all relevant aspects. +6 +Conclusion and Future Work +Identities are everywhere nowadays. With the growing number of internet users +and accounts, more servers are used. With new opportunities, also new use cases +come into sight. The IdM model was developed before the hype of blockchain. +New approaches were established since then. +In this paper, we introduced a broad classification of IdM. The existing IdM +modes were extended to an IMC with three axes. Topology, type of user, and +type of environment describe the IdM approach in more details while still being +vague about the actual protocols. The IMC was applied to different approaches, +showing silos as well as approaches, which should be comparably easy to inter- +operate with additional tools. This showed that many aspects rely on the actual +implementation within the organization. Also, it visualizes a trade-off between +user control and trust into attributes. The IMC was extended by a morphology +of IdM, which describes the characteristics of cooperation. This morphology was +mapped to the life cycle of users and IdM in a further step. The result of the +mapping can help to distinguish relevant questions during a cooperation. Both +methods, the IMC and the IdM morphology, combined provide a comprehensive +characterization of IdM approaches. This helps to choose suitable approaches for +an organization or cooperation. Furthermore, needed tools for interoperability +can be explored. An integration into processes and a guide to choose the best +fitting IdM approach were left out and will be further work. The methods also +reveal that interesting features for a holistic IdM have not been designed yet. +In order to create one holistic IdM framework, integrating different IdM +approaches, an architecture is being developed. This architecture is extended +by service models, visualizing needed processes. As a another step, processes +interacting with already established management processes are investigated. To +decide for the best fitting IdM approach, a decision matrix is created, all helping +to ease the use and improve the quality of IdM. + +References +1. 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In: 2010 IEEE +International Conference on Information Theory and Information Security. pp. +287–293 (Dec 2010). https://doi.org/10.1109/ICITIS.2010.5689468 + diff --git a/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/load_file.txt b/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78700960a84e89f6b024d8f5fc876a3b25d9a6cd --- /dev/null +++ b/ldAyT4oBgHgl3EQfk_iC/content/tmp_files/load_file.txt @@ -0,0 +1,903 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf,len=902 +page_content='IMC: A Classification of Identity Management Approaches Daniela P¨ohn1 and Wolfgang Hommel1 Bundeswehr Universit¨at M¨unchen, Research Institute CODE, Munich, Germany {daniela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='poehn,wolfgang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='hommel}@unibw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This paper presents a comprehensive classification of iden- tity management approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The classification makes use of three axes: topology, type of user, and type of environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The analysis of ex- isting approaches using the resulting identity management cube (IMC) highlights the trade-off between user control and trust in attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' A comparative analysis of IMC and established models identifies missing links between the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IMC is extended by a morphology of identity management, describing characteristics of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology is then mapped to the life cycle of users and identity man- agement in a further step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These classifications are practically underlined with current approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both methods combined provide a comprehen- sive characterization of identity management approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The methods help to choose suited approaches and implement needed tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Keywords: Security · Identity Management · Model · Taxonomy 1 Introduction Thousands of web applications around the world provide different services via the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These services require the user to present an identity for authentication, otherwise the user is not able to access them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' To manage different users with their identities, identity management (IdM) was introduced as a paradigm more than two decades ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It focuses on managing usernames, which are used as identifier assigned to users, some sort of credential, usually a password, and further information, like email address and postal address, called user attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Different evolving requirements led to the creation of different models of and protocols for identity management systems (IdMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' While stand-alone organi- zations run a centralized Identity & Access Management (I&AM) system, many organizations with collaboration, especially in academia, introduced Federated Identity Management (FIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' FIM is an arrangement between multiple entities in order to let users use the same identification data as in their home organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' By FIM, users obtain access to the services provided by partners, called service providers (SPs), within organizational trust boundaries called federations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The often-used Security Assertion Markup Language (SAML) [11] is rather static, whereas OAuth and OpenID Connect (OIDC) [14] provide a dynamic approach, known for example from Google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Limitations of FIM led to different approaches, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='00444v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='CR] 1 Jan 2023 like inter-federations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' eduGAIN [5]), the use of the Domain Name System (DNS) for discovery and trust, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', LIGHTest [13], different assurance frame- works and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In parallel, user-centric solutions were developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' User Managed Access (UMA) [8], an OAuth-based standard, enables the user to con- trol the authorization of data sharing and other protected resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The user of Self-Sovereign Identities (SSIs) is the ultimate owner of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SSIs are typically realized by decentralized networks, like distributed ledger technologies (DLTs) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Decentralized Identifiers (DIDs) [12] often make use of DLTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IdM is one crucial pillar of security frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Several different models and approaches are currently developed and run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Not all approaches fit into one single model, making a categorization challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This paper contributes the following improvements: The developed identity management cube (IMC) categorizes different IdM approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cube is broadened by a morphology describing aspects of collaboration within the life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both categorizations are applied to different protocols and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This helps to identify fitting approaches and missing tools for interoperability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It also provides an overview of important aspects during the life cycle, helping stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' We discuss related work in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In Section 3, we present a new categorization of IdM and provide a brief classifica- tion of current approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Additionally, we present a morphology in Section 4, which is then mapped to the life cycle of identities and identity management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The newly developed IMC and the morphology are applied to current approaches and then discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The paper is concluded in Section 6 by a summary of the results achieved so far and an outlook to ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 2 Related Work Yuan Cao and Lin Yang [16] identify three core components for IdM: user, ser- vice provider (SP), and identity provider (IdP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The authors further describe the three models isolated, centralized, and federated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' According to them, the IdM paradigms can be classified into network-centric paradigm, service-centric paradigm, and user-centric paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Sovrin [15] sees SSI as next step after isolated, centralized, federated, and user-centric IdM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In other papers, either the models isolated, centralized, federated, and user-centric or centralized, federated, and decentralized are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Boujezza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' [1] describe a taxonomy for Internet of Things (IoT) by adapt- ing the paradigms and requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The authors classify user model, service provider model, and hybrid model, combining user and SP, and further submod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In contrast, Pal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' [10] relate IoT identities to things-centric identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' [3] describe an IdM model for big data based on authorization, au- thentication, identification, and audit modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Habiba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' [7] use the IdM requirements taxonomy to classify cloud IdMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Further approaches have been developed, leading to different directions, which we integrate into our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 3 Identity Management Models The main functionalities of IdM are identification, authentication, and autho- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In most cases, a password is provided for authentication, which fulfils a required complexity or entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Second factor, multi-factor, and anonymous are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The authorization is based on policies, which describe whether the user is allowed to access a certain functionality or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With collabora- tions, the information about the user is stored at the IdP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The user wants to access a service of the entity SP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Minor entities are trusted third parties (TTPs), attribute authorities (AAs), having additional information about the user, and federation operators, if IdPs and SPs form trust boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As new require- ments are evolving, different approaches for IdM have been developed and will be emerging in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The existing IdM models do not work for several use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Therefore, new models are developed and applied in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='1 Analysis of Identity Management Models In order to distinguish different IdM approaches, models have been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These models were updated for user-centric models and partly for SSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As de- scribed in Section 2, the following IdM models are mostly used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Isolated: I&AM per service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Centralized: Network-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' I&AM per entity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', with single sign-on (SSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Federated: Application-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' I&AM per federation, which is a set of IdPs and SPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Possible protocols are, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', OIDC and SAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Decentralized: User-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' I&AM, where the user is in control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Used for FIM in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Possible approaches are UMA and SSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Decentralized is partly divided into user-centric and SSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The models are seen as evolution with almost no intersection, displayed in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The models describe the topology and the source of truth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', the user or another entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Approaches can fit into two models at the same time, see Figure 1b, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', if the IdM is user-centric but the SPs form a federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In this case, SSI respectively UMA belong to two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='2 The Identity Management Cube (IMC) In order to distinguish the approaches, we use the following dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: Topology of the IdM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: Type of user, using the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: Type of service featured by the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The topology is orthogonal to user-centric and can be used as one category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Based on existing approaches, the topology can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Isolated is left out of the category as it disappears due to the management overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' (a) Evolution of Identity Management Models (b) Orthogonality of Identity Manage- ment Models Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 1: Identity Management Models Centralized: I&AM per entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With TTP: I&AM with several entities, where at least one TTP is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This applies to many cases of FIM and is, therefore, similar to federated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Without TTP: I&AM with several entities, where no TTP is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As it describes a distributed, completely decentralized structure, it addresses different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Most cases of SSI belong to this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' User-centric describes two things: a human user and user as source of truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Other user types are computers, like servers, and IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Therefore, the second category is type of user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The human user is further divided into user- centric and provider-centric, describing attribute handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' User: Divided into user-centric and provider-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This includes cases of UMA, SSI, but also SAML and OIDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Computer: Machine to machine (M2M) communication, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IoT Device: IoT devices usually have less computing power, which restricts computationally intensive cryptographic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Although an increasing number of web services are used, like Office 365, several services are non-web-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In order to distinguish the type of service, the following characteristics are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Non-Web Service: M2M communication, but also local services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Background Web Service: Services, which are need for interactive web ser- vices, like localization of the user’s home organisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Interactive Web Service: Services the end-user uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As a result, the new model comprises three categories, topology, type of user, and type of service, displayed as axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In reference to the Life cycle, Aspect, Layer (LAL) Brick [4], the categories result in a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In Figure 2, the developed IdM cube including the labels of the different axes is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' User-centric and provider-centric are thus left out for clarity reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' isolated centralized federated user-centricisolated centralized federated user-centricFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 2: Identity Management Cube 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='3 IMC Applied to Current Approaches In order to depict the IMC, different IdM approaches are classified by the cate- gories described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As examples, centralized IdM with SAML federations in research and education (R&E), OIDC in the web, UMA for private users, and SSI as new approach are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In addition, IdM for servers, IoT, and with Active Directory (AD) are explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SAML is used in R&E to let users access web services at research partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It is based on lightweight directory access protocol (LDAP), databases, or even AD with the add-on federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The entities form a federation, which relies on contracts with the federation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As a result, it has the following charac- teristics, as shown in Figure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: With federation tools as TTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: User are humans, but the type is provider-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: Interactive web services for end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' OIDC is used in web as well, but is a more dynamic protocol without a TTP, based on OAuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' UMA is also developed on top of OAuth, but more user-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This can be seen in the characteristics, shown in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: Using Webfinger technology is without a TTP, but can be central- ized in some use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: Human end user in most cases, which can be either provider- centric (OIDC) or user-centric (UMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: Typically interactive web for end users, but others types are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SSI is seen as the new step in evolution of IdM, as the user is in control of everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The concept is without a TTP, but it evolves to a topology with a TTP for scalability and performance reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Most approaches concentrate on interactive web services, though the concept could be applied to other services as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SSI, therefore, has the following characteristics, displayed in Figure 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' user computer loTdevice !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP centralized non-web background interactiveTopology: Originally, SSI is without a TTP, but is evolving to centralized ser- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: SSI focuses on the user, therefore, user-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: Interactive web services for end users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Besides web application, servers are run at the backend, which are normally ac- cess through keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The public key is stored at the server, while the administrator is in possession of the private key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' So, the service is non-web and it is typically either centralized or with a TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As a result, identity management for servers can be described as following, shown in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: Either centralized, with a centralized IdM, or with a TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: Both, computer in M2M or human users are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: The services are typically non-web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Centralized IdM with AD is used in companies to enable employees to login at their computer, provision folders and shares, but also to access web services with single sign-on (SSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It has the following characteristics, shown in Figure 3e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: The AD itself is centralized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: The human user is in focus, but the IdM is provider-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Additionally, Windows computer can be a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: All types of services are possible, as it relies on Windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IoT devices often communicate with Constrained Application Protocol (CoAP) instead of Hypertext Transfer Protocol (HTTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The devices, which either lack a browser to perform user-agent based authorization or are input constrained, cannot make use of typical web protocols, like OAuth or SAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' One option is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', to utilize shared keys, another is ACE-OAuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' ACE-OAuth maps OAuth methods to Authentication and Authorization for Constrained Environments (ACE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The characteristics are shown in Figure 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology: IoT networks are typically centralized managed, which can be with or without a TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of User: The type is IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Type of Service: IoT devices are mainly background services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The selected approaches can be merged in one IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The colors are used as in the figure above: AD dark green, SAML yellow, OIDC dark blue, SSI light blue, servers orange, IoT light green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cube illustrates that many approaches are used for interactive web and human users, while the protocols themselves could be used for other user cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The figure at the same time visualizes the differences between the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' While AD is focused on centralized topology, SAML typically uses a TTP, while OIDC, UMA, and SSI tend to work without TTP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The most common type of service are used in Figure 4a, while Figure 4b adds also unusual use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both figures show that the selected approaches do not cover all aspects of the IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SSI with a centralized party would partly fulfil the application of SAML, as it would double to with a TTP from the later figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' (a) IMC for SAML (b) IMC for OIDC (c) IMC for SSI (d) IMC for Server (e) IMC for AD (f) IMC for IoT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 3: Identity Management Cubes Applied to Different Use Cases user computerloTdevice !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP TTP centralized non-web background interactiveuser computer loTdevice !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP TTP centralized non-web background interactiveuser computerloTdevice !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP TTP centralized non-web background interactiveuser computer loT device !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP TTP centralized non-web background interactiveuser computer loT device !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP TTP centralized non-web background interactiveuser computer loTdevice /dli TTP centralized non-web background interactiveEspecially these shared single cubes illustrate that interoperability between the approaches should be easily reached, while combining different approaches ar- ranged in different cubes probably needs more effort and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Additionally, one can either have user-centric or service-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Most approaches cannot provide both, as trust into the attributes is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' (a) Merged IMC (b) Merged IMC Including Unusual Use Cases Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 4: Merged Identity Management Cubes 4 Morphology of Identity Management In order to determine the degree of fulfilment, a uniform format is needed de- scribing the approaches in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Therefore, a characteristic similarly to a morphology is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology focusses on organizational aspects, while the IMC categorizes the technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The authors of [6] describe the char- acteristics of Inter-FIM based on a morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As the characteristics need to comprise all IdM approaches and therefore relates to the IMC, the morphology is extended for the needs of universal IdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In a next step, the morphology is mapped to the life-cycle, in order to clarify when which decision is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Last but not least, approaches are characterized by the morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='1 Design of the Morphology The morphology describes the characteristics of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' [6] uses cooper- ation structure, members, group structure, federation dimension, organizational dimension, duration, sort of collaboration, coordination, establishment, circle of trust, degree of commitment, and trust relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As this approach concen- trates on Inter-FIM, the following characteristics can be left out or need changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' user computer loTdevice dlli TTP centralized non-web background interactiveuser computer loTdevice !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='TTP/ TTP centralized non-web background interactiveStructure of Cooperation: The structure described topology and coopera- tion customized for federations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The topology is described by the cube, while different aspects of the cooperation are part of the morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation: Instead of FIM, Projects, and Communities, this characteristic is now described in “Reason for Joining” as well as “Order”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Formalization Differentiates between “limited contract” and “cooperation agree- ments”, in order to describe the distinction of contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Dynamic of Joining: Broader scope with “stable” and “unstable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' To describe different organizational aspects, other characteristics need to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Several characteristics relate to the IdM architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Reason for Joining: In order to differentiate between private usage and busi- ness reasons, this characteristic was included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Connectivity: Describes the interaction between involved parties, which might have consequences for the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Direction of Cooperation: Broadens the scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Administration: Degree of automation, which relates to the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation Structure: Either “hierarchical” or “heterarchical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Level of Trust: Trust between involved parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Identities: Included as it has implications for the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='2 Identity Management Morphology This results in a morphology, which includes more and broader characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology has the following categorize, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Initiation: Initiation of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation: Settings of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Coordination: Settings of the coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Trust: Trust between participating entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Identities: Settings of the identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Initiation comprises of reason for joining and dynamic of joining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The reason can be “personal”, “social”, like in social media, “by law” or “economic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Eco- nomic reasons can further be split into “time”, “risk”, “earnings”, “competence”, “costs”, “pressure”, and “protection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Another distinction could be “planned”, if necessary, and “spontaneously event-driven”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The dynamic is either “stable” or “unstable”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', it is either predictable or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation itself is described by degree of integration, connectivity, pro- fessional limits, factual limits, direction of cooperation, order, locality, organiza- tional, and formalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both, the degree of integration and the connectivity between partners, are part of the networking between partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The degree of integration can either be “autonomous”, “coordinated” or “integrated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This means that either the partners work autonomous, coordinated towards a goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Integrated can be a fusion of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The connectivity has two steps: “low” and “high”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It partly relates to integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The next category are both limits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Table 1: Morphology for Identity Management in Detail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Initiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Reason for Joining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='personal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='social ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='economic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='law ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Dynamic of Joining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='stable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='unstable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Cooperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Degree of Integra- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='tion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='autonomous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='coordinated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='integrated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Professional Limits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='R&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='department ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='value chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It can be “research”, a “department”, the complete “value chain”, or just one or more “users”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Factual limits are described by “per- manent” or “restricted”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Restricted can further be split into “short”, “medium”, and “long”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The direction of cooperation depicts how close both economic levels are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' “Horizontal cooperation” describes the cooperation of companies of the same business or same level of the value chain, while “vertical cooperation” is a cooperation between organizations of different economic levels, like retail company and production company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' A cooperation is “diagonal”, if all involved companies are neither on the same economic level nor business, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', travel com- pany and food company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The order characterizes the reason for the cooperation, which is “strategic”, a “project”, “R&E”, or based on the “region”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both, the locality and the organizational are dimensions of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The locality of the cooperation is either “local”, “regional”, “national”, or “international”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' A national federation are the R&E federations, like SWITCHaai in Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' eduGAIN is the international umbrella federation for the national pendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The organizational dimension describes the viewing plane of the coopera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Terminology from economics is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' “Micro” plane consists of one sin- gle entity, while the “meso” plane comprises of several organizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', in a federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The “macro” plane shows the cooperation of cooperation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', an inter-federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The formalization classifies the kind of formality between the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' While an “arrangement” can be oral or somehow written, a “contract” is divided into limited length and cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The last step is a “capital” in- terweaving of the involved entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' An example for an arrangement is the usage of social media for end users, while contracts are typical for projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The for- malization also describes the binding intensity, which is the degree by with the involved entities give up their autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The coordination explains the management of the cooperation, which con- sists of administration, number of participants, group structure, order, and sort of coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The number of participants is related to the group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Open cooperation do not have a firm number of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Closed cooperation allow simple as well as bilateral structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The coordination further relates to trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The administration can be “manual”, “supported” or “automated”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The number of participants is strongly related to cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The participating enti- ties can either have a “bilateral” agreement, the cooperation can have a “simple” structure, or it can be “complex”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' While bilateral cooperation still work with duplicated user bases, this is not possible with more entities involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Simple networks can be realized with security assertion markup language (SAML) feder- ations, while complex structures are also more complex for technical realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' OpenID Connect (OIDC) can be used for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The group structure is either “open”, “with limitations” or “closed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' OIDC is typically open, while SAML federations have limitations in R&E or are closed in industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation structure is “hierarchical” or “heterarchical”, when all partners are more or less of the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The sort of coordination has two possible values: “implicit” or “explicit”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With an explicit coordination, the integration of an institutional coordination instance is supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' An implicit coordination needs a local coordination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Trust between entities is the result of several different factors, like recom- mendation or past experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Within the morphology, only the basics for the cooperation are described, which includes directness, dynamics, and the average level of trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The circle of trust (CoT) relates to the sort of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' If the group structure is limited, then the CoT can be static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' If the number of participants is complex, is the CoT virtual as not all the information about all participants cannot be fully known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Direct trust implicates static or dynamic CoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Directness describes how the trust between two entities is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The trust is either “direct” or “transitive / indirect”, via another entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The dynam- ics characterize the trust over time, which is either “static” or “dynamic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Last but not least, the level of trust can be “zero”, “low”, “medium”, or “high”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As final category, identities are classified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Identities especially describe factors of trust and also user-centric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This includes transparency, controllability, identification, authentication, consisting of methods and organizational factors, as well as authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The transparency is either “low”, “medium” or “high”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The same characteristics can be applied to controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The identification, authentication, and authorization can be done “internally”, “externally”, or in a “combination” of different entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The authentication methods describe the sort of credentials used, which is either “anonymous”, “simple” (like a password), “second-factor authentication (2FA)” or “multi-factor authentication (MFA)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In order to reduce the complexity of the morphology, suited characteristics can be left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In the next step, special characteristics for the use case, like topology of federation can be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This depends on the specific use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='3 Morphology Mapped to Life Cycle The morphology can be mapped to the life cycle of IdM, helping starting cooper- ation to identify their framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The life cycle is similar to the Deming Cycle [9], which has the phases plan, do, check, act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The Deming Cycle is an iterative four- step management method used in business to control improvements of processes and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It can be applied to service management, security management, and many other, like identity management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The life cycle of IdM has the phases initiation, agreement, cooperation, reconsideration and improvement, and ter- mination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Reconsideration can either lead to improvement or termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The phases of the IdM life cycle have the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Initiation: A purpose leads to the initiation of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Agreement: After discussions, an agreement is signed, describing the frame- work of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IdM should be a part of the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Otherwise, the parties need to agree on IT aspects outside of the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation: The cooperation is starting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In many cases, the cooperation is starting slowly, setting everything in place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Then there is a hype of coopera- tion, where everything is running and the original purpose is hopefully met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In IT, the start requires work, setting up the infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Reconsideration: It describes if the cooperation is proceeded and if changes need to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The same appears for IdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Improvement: The changes lead to improvements, which are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Termination: If the purpose is met or other reasons lead to the end of the cooperation, the IdM is also terminated for the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology, described in the previous section, can be mapped to the life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This helps to gain a better picture of the required decisions, as shown in Figure 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The initiation phase comprises both characteristics of the morphol- ogy of initiation, which means “Reason of Joining” and “Dynamics of Joining”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation and coordination, have characteristics in “Agreement” and “Coop- eration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This is the case as some characteristics are decided at the agreement, while others have more impact on the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The agreement thereby fea- tures: “Formalization”, “Limitations”, “Direction of Cooperation”, “Coopera- tion Structure”, “Dimensions”, “Number of Participants”, and “Group Struc- ture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation as a result includes “Trust”, “Identity”, “Degree of Inte- gration”, “Connectivity”, “Administration”, and “Sort of Cooperation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' During reconsideration every aspect is re-evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Some aspects are enhanced during improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The life cycle is terminated, if the cooperation ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IdM life cycle includes the user life cycle, because users change through- out a project or life cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' During a project, users leave, while others join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This is also the case for IdM in organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In the end, every user account needs to be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The life cycle of the user includes the following phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Request: The user requests an account at an IdMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Provisioning: The account is provisioned (attributes, roles, and permissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Identification: The user identifies himself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Authentication and Authorization: First authentication, then authorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Self-Service: The user can access the self-service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' De-Provisioning: In the end, the user account is de-provisioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The life cycle of the user has only few interactions with the morphology: trust into the service provider during request, then identification, authentication, authorization themselves as well as transparency in the self-service phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This is also visualized in Figure 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As a result, IdM models describe the approaches in general, while the mor- phology details aspects of the cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The mapping of morphology with the life cycles explains the order of the actions, which need to be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This can guide projects and organizations to identity management processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Neverthe- less, a decision matrix for choosing the best fitting approach is missing, although the cube gives a first hint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Additionally, interfaces to already established pro- cesses, like service management and security management, are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='4 Morphology Applied to Current Approaches Following, the morphology is applied to centralized IdM with AD and the differ- ences for SAML, OIDC, SSI, server, and IoT devices are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Centralized (a) Morphology mapped to Identity Management System Life Cycle (b) Morphology mapped to User Life Cy- cle Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 5: Morphology mapped to Life Cycles IdM with AD is typically used in companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Depending on whether they have cooperation, several branches or not, the complexity is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Also depending on the point of view, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', user or company, different properties can be coloured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Let us assume the company in this example just uses AD for its users, while they have other methods for cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Therefore, the following morphology can be formed, see Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The reason for joining is economic, while the dynamic is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' AD was introduced at some point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' When regarding the coop- eration, the cooperation within the company is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The integration is therefore integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The connectivity is high as all participants work together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' AD was introduced for the complete value chain of the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It should be a permanent solution, though technologies and decisions change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The direction of cooperation cannot be described by the categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The order was based on a strategy, while the company is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The organizational factor is micro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation within the company depends on contracts with its employees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The administration is hopefully supported, while the number of participants can be described with bilateral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The group structure is at least currently closed, while the coordination is hierarchical and explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The trust is direct, rather static, with medium trust, as all employees needed to submit papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The identities are managed within the company, with simple and second factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The entities of SAML in R&E form federations, which are spread over regions, countries, and the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The entities have contracts with federation operators, which have contracts with the inter-federation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The coordination be- tween the entities is rather low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As the identities are managed by the home organization, trust is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Therefore, the following morphology can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Initiation: Individuals join for R&E, while companies have economic reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The dynamic is rather stable, as entities have to sign on contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Cooperation: The cooperation is autonomous, only little coordinated by the federation and inter-federation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The connectivity between the en- tities is low,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' as there are many different services within a federation and only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Formalisation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Limitations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Initiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Direction of Cooperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Agreement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Phase ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Order Dimensions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Number of Participants ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Initiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Group Structure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='★ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Trust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Improve- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Identity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Cooperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='ment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Degree of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Integration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Administration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Sort of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Reconside ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Terminationk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Coordination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='rationldentification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Request ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Provisioning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Self-Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Permissions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Transparency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='De- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='AuthNZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='ProvisioningTable 2: Morphology for Centralized Identity Management with AD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Initiation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Reason for Joining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='personal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='social ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='economic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='law ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Dynamic of Joining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='stable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='unstable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Cooperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Degree of Integra- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='tion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='autonomous ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='coordinated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='integrated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Connectivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='low ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Professional Limits user ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='R&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='department ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='value chain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Factual Limits ' metadata={'source': 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+page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Coordina- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='tion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='implicit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='explicit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Trust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Directness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='transitive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Circle of Trust ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='static ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='dynamic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='virtual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Level of Trust ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='2FA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='MFA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Authentication Or- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='ganization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='internal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='external ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='combination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='Authorization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='internal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='external ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='combination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='a small percentage of users will use the specific service of a service provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Mostly, the entities have contact with the federation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooper- ation is limited to research, while the time depends on several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation can be vertical as well as horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The locality is national or international in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Also regional federations are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The organization form is either meso or macro, depending on the type of feder- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Federations are formalized by contracts with the federation operator and partly arrangements between entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Coordination: The administration is supported with manual steps needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As contracts need to be signed, the number of participants is simple and the group structure is with limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The order is more heterarchical than hierarchical, while the coordination is more implicit than explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Trust: Trust is transitive via federation or inter-federation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With a static number of participants, the circle of trust is also static with little dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The level of trust is low or medium, depending on separate means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Identities: Since communities with additional attribute authorities were formed and other means of identification are in use, authorization and identification are either internal or a combination, while authentication is internal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Trans- parency and controllability are rather low as a result of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IdM with OIDC distinguishes from SAML as the protocol is dynamic and the widely known use case is web authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' For OIDC, the initiation can have several reasons, therefore, the dynamic is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The cooperation is loose, which is true for the coordination as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Trust is rather low, but can be stepped up with a second factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The different constellations also have impact on the identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' SSI is different as the user is in control of the attributes, which then impacts trust and identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' If a company hosts their servers in-house, then the cooperation is within the company and maybe with other offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IoT devices can be used at home as well as at organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The trust into the devices is typically low, as others might manipulate the device without notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 5 Discussion We characterized IdM approaches in two ways: the IMC describes the technical aspects, followed the morphology for organizational aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In order to compile an overview of IdM approaches, we noticed intersections between existing IdM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These intersections helped us to identify categories, which are needed to differentiate IdM approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The three categories topology, type of user, and type of service are arranged in a cube, the IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' IMC clarifies the characteristics type of user and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Additionally, the perspective is made clear, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=', user- centric or provider-centric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' While an approach could belong to two models used beforehand, it can be clearly classified by the IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With the flexibility of the three categories in mind, future approaches should be able to be characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In the next step, we applied different IdM approaches to the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These approaches were typical web services for end users, but also servers and IoT, resulting in a colourful IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Some use cases are more typical than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Besides this fact, the application was straight forward and showed us similarities and differences between the approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' These findings might indicate a possible combination of approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' It is noticeable that trust and user-centric are not featured together in the shown examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IMC can, therefore, help to combine different IdM approaches and explore missing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In a next step, a morphology for IdM was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology de- scribes different aspects of a cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In this case, the categories initiation, cooperation, coordination, trust, and identities with their categorizations need to be regarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The relationship between morphology and the different life cycles were shown in a next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' For guidance, the morphology can be used to speed up the implementation and evaluation in a later step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The morphology was then mapped to different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' A certain variance is seen, which depends on the actual implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Nevertheless, organizational settings are made clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This does not include internal processes, which will be regarded in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both, the IMC and the morphology, do not describe IdM in all aspects, but help to categorize different approaches, use cases, and their implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The categorization helps within the life cycle of IdM to mix different approaches, see missing tools, and to regard all relevant aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 6 Conclusion and Future Work Identities are everywhere nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With the growing number of internet users and accounts, more servers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' With new opportunities, also new use cases come into sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IdM model was developed before the hype of blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' New approaches were established since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In this paper, we introduced a broad classification of IdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The existing IdM modes were extended to an IMC with three axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Topology, type of user, and type of environment describe the IdM approach in more details while still being vague about the actual protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IMC was applied to different approaches, showing silos as well as approaches, which should be comparably easy to inter- operate with additional tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This showed that many aspects rely on the actual implementation within the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Also, it visualizes a trade-off between user control and trust into attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The IMC was extended by a morphology of IdM, which describes the characteristics of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This morphology was mapped to the life cycle of users and IdM in a further step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The result of the mapping can help to distinguish relevant questions during a cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Both methods, the IMC and the IdM morphology, combined provide a comprehensive characterization of IdM approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This helps to choose suitable approaches for an organization or cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Furthermore, needed tools for interoperability can be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' An integration into processes and a guide to choose the best fitting IdM approach were left out and will be further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' The methods also reveal that interesting features for a holistic IdM have not been designed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In order to create one holistic IdM framework, integrating different IdM approaches, an architecture is being developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' This architecture is extended by service models, visualizing needed processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' As a another step, processes interacting with already established management processes are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' To decide for the best fitting IdM approach, a decision matrix is created, all helping to ease the use and improve the quality of IdM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 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+page_content=': The Inevitable Rise of Self-Sovereign Identity (2017), [Online, January 4, 2023] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' Yuan Cao, Lin Yang: A survey of Identity Management technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' In: 2010 IEEE International Conference on Information Theory and Information Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' 287–293 (Dec 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf'} +page_content=' https://doi.' metadata={'source': 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We propose a new family of preconditioners generated +by symmetric polynomials. They provide first-order optimization methods with a +provable improvement of the condition number, cutting the gaps between highest +eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic +interpretation of this preconditioning in terms of coordinate volume sampling and +compare it with other classical approaches, including the Chebyshev polynomials. +We show how to incorporate a polynomial preconditioning into the Gradient and +Fast Gradient Methods and establish the corresponding global complexity bounds. +Finally, we propose a simple adaptive search procedure that automatically chooses the +best possible polynomial preconditioning for the Gradient Method, minimizing the +objective along a low-dimensional Krylov subspace. Numerical experiments confirm +the efficiency of our preconditioning strategies for solving various machine learning +problems. +Keywords: Convex optimization, Preconditioning, Gradient methods, Accelerated methods, +Symmetric polynomials +1 +Introduction +Motivation. +Preconditioning is an important tool for improving the performance of +numerical algorithms. The classical example is the preconditioned Conjugate Gradient +Method [9] for solving a system of linear equations. It proposes to modify the initial system +in a way to improve its eigenvalue distribution and thus to accelerate the convergence +of the method. The question of choosing the right preconditioner heavily depends on +the problem structure, and there exist many problem-specific recommendations which +provide us with a good trade-off between computational cost and the spectrum properties +of the new system. Some notable examples include Jacobi or the diagonal preconditioners, +symmetric successive over-relaxation, the incomplete Cholesky factorization [6], Laplacian +preconditioning for graph problems [28, 29], preconditioners for discretizations of system +of partial differential equations [15]. +Another important class of numerical algorithms are the second-order methods or +Newton’s Method (see, e.g. [21]), that aims to solve difficult ill-conditioned problems by +using local curvature information (the Hessian matrix) as a preconditioner at every step. +∗The work of the first author was supported by the Swiss State Secretariat for Education, Research and +Innovation (SERI) under contract number 22.00133. The work of the second author received funding from +the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation +programme (grant agreement No. 788368) +†EPFL, Switzerland. E-mail: nikita.doikov@epfl.ch. +‡Catholic University of Louvain (UCLouvain), Belgium. E-mail: anton.rodomanov@uclouvain.be. +1 +arXiv:2301.13194v1 [math.OC] 30 Jan 2023 + +However, being a powerful algorithm, each iteration of Newton’s Method is very expensive. +It requires to solve a system of linear equations with the Hessian matrix, and in case of +quadratic objective it is equivalent to solving the original problem. +In this paper, our goal is to solve a general nonlinear optimization problem with a +structured convex objective by the efficient first-order methods. Thus, in the case of +unconstrained minimization of a smooth function: minx f(x), the simplest method that +we study is as follows, for k ≥ 0: +xk+1 += +xk − αkP ∇f(xk), +(1.1) +where αk > 0 is a stepsize and P is a fixed preconditioning matrix. P := I corresponds to +the classical gradient descent. Another natural choice is P := B−1, where B is a curvature +matrix of our problem1, which is directly available for the algorithm. That resembles the +Newton-type direction, and the method with this preconditioner tends to converge much +faster in practice (see Figure 1.1). However, computing B−1 (or solving the corresponding +linear system with B) is a very expensive operation in the large scale setting. +0 +250 +500 +750 +1000 +Iteration +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Func. residual +a9a +0 +250 +500 +750 +1000 +Iteration +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +phishing +0 +250 +500 +750 +1000 +Iteration +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +breast-cancer +0 +250 +500 +750 +1000 +Iteration +10 +9 +10 +7 +10 +5 +10 +3 +10 +1 +Func. residual +ionosphere +0 +250 +500 +750 +1000 +Iteration +10 +4 +10 +3 +10 +2 +10 +1 +covtype +0 +2500 +5000 +Iteration +10 +8 +10 +6 +10 +4 +10 +2 +100 +mnist +P = I +P = B +1 +Figure 1.1: Training logistic regression with the standard gradient descent (P = I), and using the +inverse of the curvature matrix (P = B−1) as a preconditioner in (1.1). The latter method works +much faster, while it can be very expensive to compute B−1 for large scale problems. +Instead of using B−1, we propose a new family of Symmetric Polynomial Precondi- +tioners, that provably improve the spectrum of the objective. The first member of our +family is +P +:= +tr (B)I − B. +(1.2) +We prove that using preconditioner (1.2) within method (1.1), makes the condition +number insensitive to the gap between the top two eigenvalues of the curvature matrix. +Since it is quite common for real data to have a highly nonuniform spectrum with several +large gaps between the top eigenvalues (see Figure 1.2), our preconditioning can significantly +1See the definition of B in our Assumption 2.1 and the corresponding Examples 2.2, 2.3, 2.4 of different +problems. +2 + +improve the convergence of the first-order methods. At the same time, one step of the +form (1.1),(1.2) is still cheap to compute. It involves just the standard matrix operations +(trace and the matrix-vector product), without the need to solve linear systems with the +curvature matrix as in Newton’s Method. +1 +7 +13 +19 +25 +31 +i +104 +105 +i +a9a +1 +4 +7 +10 +13 +16 +i +102 +103 +phishing +1 +2 +3 +4 +5 +6 +7 +8 +9 +i +102 +103 + breast-cancer +1 +6 +11 +16 +21 +26 +i +102 +103 +i +ionosphere +1 +3 +5 +7 +9 +11 +13 +15 +i +105 +106 +covtype +1 +5 +9 +13 +17 +21 +25 +i +106 +107 +mnist +Figure 1.2: Leading eigenvalues (in the logarithmic scale) of the curvature matrix B, for several +typical datasets2. There are large gaps between the top eigenvalues. +This approach works for general structured nonlinear problems (not necessarily quadrat- +ics) and also for the problems with possible composite parts (e.g., constrained minimization +or non-smooth regularization). +Our new family of Symmetric Polynomial Preconditioners gradually interpolate between +the first preconditioner (1.2) and P ∝ B−1 as the other extreme case. We show that +increasing the order of the preconditioner, we are able to cut off several top eigenvalues +of the curvature matrix, without knowing the actual spectrum. We can incorporate +these preconditioners both into the Gradient Method, as well as into the accelerated Fast +Gradient Method [19], with a further provable improvement of the condition number. +Finally, we address the common question of choosing the best possible preconditioner. +We propose a new adaptive strategy for the basic nonlinear Gradient Method based on the +Krylov subspace minimization. In this approach, preconditioner P is defined as a general +polynomial of the curvature matrix B of a fixed (small) degree τ: +P +:= +a0I + a1B + . . . + aτBτ, +where the vector of coefficients a ∈ Rτ+1 is found by solving a certain linear system of +size τ + 1 in each iteration of the method. It has a plain interpretation of projecting the +direction B−1∇f(xk) onto an affine set Kτ +xk, which is the Krylov subspace: +Kτ +x +def += +span +� +∇f(x), B∇f(x), . . . , Bτ∇f(x) +� +. +(1.3) +2https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/ +3 + +In case of small τ, we can solve this linear system easily and obtain the best preconditioning +guarantee for our method, which is adaptive for each iteration. +Related Work. +It is widely known that the standard Conjugate Gradient Method is +optimal in the class of the first-order algorithms for unconstrained minimization of convex +quadratic functions [17]. The kth iteration of the Conjugate Gradients finds the full +minimum of the objective over the k-dimensional Krylov subspace, and thus it provably +solves the problem after k = n iterations, where n is the dimension of the problem. +Quadratic minimization is equivalent to solving a system of linear equations, therefore it +is often referred as the linear case. Polynomial preconditioning for solving large linear +systems has been extensively studied during the last several decades; see [4, 10, 13, 14, 26, +30] and references therein. See also Section 5.3 for the comparison of our preconditioning +strategies with the linear Conjugate Gradient Method. +The situtation with nonlinear problems is more difficult. Along with the basic Gra- +dient Method, the classical approaches include the Nonlinear Conjugate Gradients and +Quasi-Newton Methods (see, e.g. [22]), which typically demonstrate a decent practical +performance, while replicating the standard Conjugate Gradients in the linear case. How- +ever, these methods lack of having any good global complexity bounds, and thus in the +worst-case scenario they can actually perform even worse than the Gradient Method [8]. +At the same time, the Fast Gradient Method developed by [19] is optimal for the class +of nonlinear problems with a uniformly bounded eigenvalues of the Hessian [18]. This +assumption does not take into account the actual distribution of the spectrum. Hence, +it can not distinguish the problems with large gaps between the top eigenvalues, as in +Figure 1.2. +There have been several attempts to study more specific problem formulations, and so +to gain a provable advantage for the optimization algorithms by leveraging the spectrum +of the Hessian. +Thus, the quadratic minimization problems were studied under the +assumption of a particular probability distribution for the eigenvalues [2, 27], or assuming +a certain fixed spectral gap [7], revealing the advantages of employing the Heavy-ball +Method [23] in these cases. Another example is the Stochastic Spectral Descent [12], which +improves the condition number for quadratic problems if we know some of the eigenvectors. +In this work, we consider a refined smoothness characterization of the objective with +the curvature matrix B (Assumption 2.1). It is similar in spirit to that one used in +Stochastic Dual Newton Ascent [24]. An important particular instance of this class of +algorithms is the Randomized Coordinate Descent with Volume Sampling [25]. In the +latter method, it was proposed to select subsets of variables of certain size m proportionally +to the determinants of principal submatrices of B. While this approach was practically +implementable only for the subsets of size m = 1 or 2, it was shown that, in theory, the +method is insensitive to the large spectral gap between the top m − 1 eigenvalues. +Surprisingly, our new family of the Symmetric Polynomial Preconditioners can be +viewed as a deterministic version of the Volume Sampling technique (with m = τ + 1 +where τ is the degree of a preconditioning polynomial; preconditioner (1.2) corresponds to +τ = 1). Thus, we provide the Volume Sampling with a novel deterministic interpretation, +which also leads to new accelerated and composite optimization algorithms (see Section 4.3 +for a detailed comparison). +4 + +Preconditioner +Cond. number, β/α +Methods +Cost +Classic Gradient Method +P = I +λ1/λn +GM, +FGM +cheap +“Full Preconditioning” +P = B−1 +1 +GM, +FGM +expensive +Symmetric Polynomial +Preconditioning (ours) +P = Pτ (4.1) +λ1/λn · ξτ(λ) +GM, +FGM +cheap +for small τ +Krylov Subspace +Minimization (ours) +optimal poly. +λτ+1/λn +GM +cheap +for small τ +Table 1.1: +β +α for different preconditioning strategies, λ = λ(B). +Note that ξτ(λ) ≤ 1, and +ξτ(λ) → 0 in case of large spectral gaps, namely when +λ1 +λτ+1 → ∞ (see Section 4). For solving the +problem with ϵ-accuracy, GM needs k(ϵ) = O( β +α · 1 +ϵ ) and k(ϵ) = O( β +α · L +µ ·log 1 +ϵ ) iterations for convex +and strongly convex functions, respectively. FGM needs only +� +k(ϵ) iterations (Theorems 3.1 +and 3.2). +Contributions. +We propose several polynomial preconditioning strategies for first-order +methods for solving a general composite convex optimization problem, and prove their +better global complexity guarantees, specifically: +• We study the convergence of the basic Gradient Method (GM, Algorithm 3.1) +and the accelerated Fast Gradient Method (FGM, Algorithm 3.2) with a general +(arbitrarily fixed) preconditioning matrix. We introduce two condition numbers, that +are designated to the different parts of the objective (L/µ for nonlinearity and β/α +for the curvature matrix), and show that they serve as main complexity factors. +• We develop a new family of Symmetric Polynomial Preconditioners (Section 4). +Combining them with the preconditioned Gradient Methods, we establish a significant +improvement of the curvature condition number β/α in case of large gaps between +the top eigenvalues of the matrix (see Table 1.1). +• Then, we propose a new adaptive procedure based on the Krylov subspace mini- +mization (Algorithm 5.1) that achieves the best polynomial preconditioning. We +present the guarantees we can get, including cutting off the top eigenvalues directly +and by employing the Chebyshev polynomials, and compare this approach with the +Symmetric Polynomial Preconditioning. +• Numerical experiments are provided. +2 +Notation and Assumptions +We consider the following optimization problem given in the composite form: +F ⋆ += +min +x∈Rn +� +F(x) +def += +f(x) + ψ(x) +� +, +(2.1) +5 + +where f : Rn → R is a differentiable convex function which is the main part of the problem, +and ψ : Rn → R ∪ {+∞} is a proper closed convex function that can be nondifferentiable +but has a simple structure. For example, it can be an indicator of a convex set, or +ℓ1-regularizer. +Additionally, we fix some symmetric positive-definite matrix B ∈ Rn×n (notation +B = B⊤ ≻ 0). This matrix plays the key role in our characterization of the smoothness +properties of f. Namely, we assume the following (considering for simplicity two-times +differentiable functions): +Assumption 2.1. The Hessian of f is uniformly bounded, for some constants L ≥ +µ ≥ 0: +µB +⪯ +∇2f(x) +⪯ +LB, +∀x ∈ Rn. +(2.2) +Having fixed the matrix B, we define the corresponding induced norm by ∥x∥B +def += +⟨Bx, x⟩1/2, x ∈ Rn. Thus, matrix B is responsible for fixing the coordinate system in the +problem. Then, condition (2.2) can be rewritten in terms of the global lower and upper +bound on the first-order approximation of f [21]: +µ +2∥y − x∥2 +B +≤ +f(y) − f(x) − ⟨∇f(x), y − x⟩ +≤ +L +2 ∥y − x∥2 +B, +∀x, y ∈ Rn. +(2.3) +In what follows, we denote by λ = λ(B) ∈ Rn the vector of eigenvalues for the matrix +B, sorted in a nonincreasing order: λ1 ≥ λ2 ≥ . . . ≥ λn. +The classical example is B := I (identity matrix). Then, condition (2.2) implies that +the function f is (strongly) convex and has the Lipschitz continuous gradient. However, +by choosing a specific B, we tend to achieve a better granularity of the description of our +problem class and thus to improve the convergence properties of the methods. +Example 2.2. Let a ∈ Rn. Then, the quadratic function +f(x) += +1 +2⟨Bx, x⟩ − ⟨a, x⟩, +satisfies condition (2.2) with L = µ = 1. +We see that in this case, the so-called condition number L/µ is just 1, which means +that preconditioning the Gradient Method (1.1) with the matrix P := B−1 would give +an immediate convergence to the solution. However, inverting the matrix is prohibitively +expensive for large scale problems. Our aim is to find a suitable trade-off between improving +the condition number and the arithmetic cost of algorithm steps. Let us consider the +following important example which can be met in many practical applications. +Example 2.3. Let A ∈ Rm×n be a given data matrix, and b ∈ Rm be a given vector. +Denote, +f(x) += +g(Ax + b) +Then, the derivatives are as follows: ∇f(x) = A⊤∇g(Ax+b) and ∇2f(x) = A⊤∇2g(Ax+ +b)A. Hence, assuming: µIm ⪯ ∇2g(x) ⪯ LIm, ∀x, with some L ≥ µ ≥ 0, condition +(2.2) is satisfied 3 with +3Here, we assume that A⊤A ≻ 0 which is typically the case when m ≫ n. Otherwise, we can reduce +the dimensionality. +6 + +B +:= +A⊤A. +At the same time, for B := In (the standard Euclidean norm), the Lipschitz constant +increases by the factor λ1(A⊤A), which makes the problem extremely ill-conditioned. +A particular case of this example is separable optimization, or generalized linear models +[1], which covers the classical regression and classification models. +Example 2.4. Let +f(x) += +1 +m +m +� +i=1 +φ(⟨ai, x⟩), +x ∈ Rn, +where φ : R → R is a loss function satisfying: µ ≤ φ′′(t) ≤ L, ∀t ∈ R, with some L ≥ µ ≥ 0. +Then, forming the matrix A ∈ Rm×n whose rows are a⊤ +1 , . . . , a⊤ +m and setting B := A⊤A, +condition (2.2) holds. +3 +Preconditioned Gradient Methods +A natural intention would be to use the global upper bound (2.3) as a model for the smooth +part of the objective. However, the direct minimization of such upper model requires to +solve the linear system with the matrix B, which can computationally unfeasible for large +scale problems. +Instead, let us fix for our preconditioner some positive definite symmetric matrix P = +P ⊤ ≻ 0, which satisfies the following bound, for some α := α(P ) and β := β(P ) ≥ α > 0: +αB−1 +⪯ +P +⪯ +βB−1. +(3.1) +We are going to use this matrix instead of B−1 in our methods. For a fixed symmetric +positive definite matrix P and parameter M > 0, we denote the gradient step from a point +x ∈ dom ψ along a gradient direction g ∈ Rn by +GradStepM,P (x, g) +def += argmin +y∈dom ψ +� +⟨g, y⟩ + ψ(y) + M +2 ∥y − x∥2 +P −1 +� +. +This operation is well-defined since the objective function in the above minimization +problem is strongly convex. We assume that both ψ and P are reasonably simple so +that the corresponding gradient step can be efficiently computed. An important case is +ψ = 0 for which we have GradStepM,P (x, g) = x − 1 +M P g. The latter expression can be +efficiently computed whenever one can cheaply multiply the matrix P by any vector. +3.1 +Preconditioned Basic Gradient Method +First, we consider the basic first-order scheme shown in Algorithm 3.1 for solving the +composite problem (2.1). For simplicity, in this section, we only present a version of +this method with a fixed step size and assume that all necessary constants are known. +An adaptive version of Algorithm 3.1 which does not have these limitations and is more +efficient in practice can be found in Appendix C. +7 + +Algorithm 3.1 Preconditioned Basic Gradient Method +Input: x0 ∈ dom ψ, P = P ⊤ ≻ 0, M > 0. +for k = 0, 1, . . . do +Compute xk+1 = GradStepM,P +� +xk, ∇f(xk) +� +. +end for +For Algorithm 3.1, we can prove the following results. +Theorem 3.1. Consider Algorithm 3.1 with M = βL. Then, at each iteration k ≥ 1, +we have +F(xk) − F ⋆ +≤ +β +α +L∥x0 − x⋆∥2 +B +k +. +(3.2) +When µ > 0, the convergence is linear: for all k ≥ 1, +F(xk) − F ⋆ +≤ +� +1 − 1 +4 +α +β +µ +L +�k +[F(x0) − F ⋆]. +(3.3) +We see that one of the principal complexity factors in the above estimates is the +condition number β/α which depends on the choice of our preconditioner P (see (3.1)). +For the basic choice P = I, we have β/α = λ1/λn. However, as we show in the following +sections, it is possible to use more efficient (and still quite cheap) preconditioners which +improve this condition number. +3.2 +Preconditioned Fast Gradient Method +Now let us consider an accelerated scheme shown in Algorithm 3.2. This algorithm is one +of the standard variants of the Fast Gradient Method (FGM) known as the Method of +Similar Triangles (see, e.g., Section 6.1.3 in [21]) but adapted to our assumptions (2.2) +and (3.1). +Algorithm 3.2 Preconditioned Fast Gradient Method +Input: x0 ∈ dom ψ, P = P ⊤ ≻ 0, M > 0, ρ ≥ 0. +Set v0 = x0, A0 = 0. +for k = 0, 1, . . . do +Find ak+1 from eq. +Ma2 +k+1 +Ak+ak+1 = 1 + ρ(Ak + ak+1). +Set Ak+1 = Ak + ak+1, Hk = 1+ρAk+1 +ak+1 +. +Set θk = ak+1 +Ak+1 , ωk = +ρ +Hk , γk = ωk(1−θk) +1−ωkθk . +Set ˆvk = (1 − γk)vk + γkxk. +Set yk = (1 − θk)xk + θkˆvk. +Compute vk+1 = GradStepHk,P +�ˆvk, ∇f(yk) +� +. +Set xk+1 = (1 − θk)xk + θkvk+1. +end for +As in other versions of FGM, to properly handle strongly convex problems, Algo- +rithm 3.2 requires the knowledge of the strong convexity parameter α and µ (or, more +8 + +precisely, their product ρ = αµ). For non-strongly convex problems, we can always choose +α = µ = 0. See also Appendix C for a variant of Algorithm 3.2 which can automatically +adjust the constant M in iterations. +The convergence results for Algorithm 3.2 are as follows. +Theorem 3.2. Consider Algorithm 3.2 with M = βL and ρ = αµ. Then, at each +iteration k ≥ 1, we have +F(xk) − F ⋆ +≤ +2 β +α +L∥x0 − x⋆∥2 +B +k2 +. +(3.4) +When µ > 0, the convergence is linear: for all k ≥ 1, +F(xk) − F ⋆ +≤ +� +1 − +�α +β +µ +L +�k−1 β +α +L +2 ∥x0 − x⋆∥2 +B. +Comparing these estimates with those from Theorem 3.1, we see that the accelerated +scheme is much more efficient. For instance, to reach accuracy ϵ > 0 in terms of the +objective function in the non-strongly convex case, Algorithm 3.1 needs k(ϵ) = β +α +L∥x0−x⋆∥2 +B +ϵ +iterations, while for Algorithm 3.2 this number is only k2(ϵ) = +� +2k(ϵ). Similar conclusions +are valid in the strongly convex case. +Despite having much weaker dependency on the condition number β/α, Algorithm 3.2 +is still quite sensitive to it. Thus, the proper choice of the preconditioner P is important +for both our methods. +4 +Symmetric Polynomial Preconditioning +We would like to have a family of preconditioners Pτ for our problem indexed by some +parameter τ. Varying τ should provide us with a trade off between the spectral quality +of approximation (3.1) of the inverse matrix and the arithmetical cost of computing the +preconditioner. +Surprisingly, such a family of preconditioners can be built by using symmetric polyno- +mials, the classical objects of Algebra. We prove that our preconditioning improves the +condition number β +α of the problem, by automatically cutting off the large gaps between +the top eigenvalues. +4.1 +Definition and Basic Properties +We define the family of symmetric matrices {Pτ}0≤τ≤n−1 recursively. We start with +identity matrix: P0 +def += I. Then, +Pτ +def += +1 +τ +τ� +i=1 +(−1)i−1Pτ−iUi, +(4.1) +where Uτ +def += tr (Bτ)I − Bτ are the auxiliary matrices. It turns out that matrices (4.1) +serve as a good approximation of the inverse matrix: Pτ ≈ B−1, up to some multiplicative +9 + +constant, and the quality of such approximation is gradually improving when increasing +parameter τ. Let us look at several first members. Clearly, +P1 += +tr (B)I − B, +(4.2) +which is very easy to handle, by having computed the trace of the curvature matrix. Then, +multiplying P1 by any vector would require just one matrix-vector multiplication with our +original B. Further, +P2 += +1 +2tr (P1B)I − P1B += +1 +2 +� +[tr (B)]2 − tr (B2) +� +I − tr (B)B + B2, +(4.3) +thus its use would cost just two matrix-vector products with B, having evaluated4,5 the +numbers tr (B) and tr (B2). +It is clear that in general Pτ = pτ(B), where pτ is a polynomial of a fixed degree τ with +coefficients that can be found recursively from (4.1). Let us give a useful interpretation +for our family of preconditioners, that also explains their name. For a ∈ Rn−1, we denote +by σ0(a), . . . , σn−1(a) the elementary symmetric polynomials in n − 1 variables6. It is +known that every symmetric polynomial (that is invariant to any permutation of the +variables) can be represented as a weighed sum of elementary symmetric polynomials [5]. +We establish the following important characterization. +Lemma 4.1. Let B = QDiag (λ)Q⊤ be the spectral decomposition. Then, +Pτ = QDiag (στ(λ−1), . . . , στ(λ−n))Q⊤, +(4.4) +where λ−i ∈ Rn−1 is the vector that contains all elements of λ except λi. +In particular, we justify Pτ ≻ 0. For τ = n − 1, we get +Pn−1 +(4.1) += +det(B)B−1 +def += +Adj (B), +(4.5) +which gives us the true inverse matrix B−1 up to the constant factor det(B). +4Note that tr (B2) = �n +i=1 ∥B[:, i]∥2 +2, where B[:, i] ∈ Rn is the ith column of B. +5For general τ, we can also use a stochastic estimate of the trace: ξτ +def += n⟨Bτu, u⟩, where u ∈ Rn is +uniformly distributed on the unit sphere. It would give an unbiased estimate: E[ξτ] = nE[tr (u⊤Bτu)] = +ntr (E[uu⊤]Bτ) = tr (Bτ). +6That is στ(a) +def += � +1≤i1<... 0, we use the matrix P = pτ(B) as a preconditioner, where +pτ is a specifically constructed polynomial of degree τ such that P ≻ 0. +A natural question is how optimal is this choice of a polynomial? Indeed, the problem +of polynomial approximation has a long and rich history with an affirmative answer +provided by the classical Chebyshev polynomials [16] for the uniform approximation bound. +We present a new adaptive algorithm that automatically achieves the best polynomial +preconditioning. Then, we study what are the complexity guarantees that we can get with +this optimal approach. In this section, we focus on the non-composite case only, i.e. the +problem of unconstrained minimization of a smooth function: minx∈Rn f(x). +5.1 +Gradient Method with Krylov Preconditioning +We denote Pa +def += a0I + a1B + . . . + aτBτ, where vector a = (a0, . . . , aτ) ∈ Rτ+1 is a +parameter. In each iteration, it is found by solving the linear system: +a += +A−1 +τ gτ +∈ +Rτ+1, +(5.1) +where Aτ = Aτ(x) ∈ R(τ+1)×(τ+1) is the Gram matrix with the following structure +(0 ≤ i, j ≤ τ): +� +Aτ(x) +�(i,j) +def += +L · ⟨∇f(x), Bi+j+1∇f(x)⟩, +(5.2) +and gτ = gτ(x) ∈ Rτ+1 is defined by (0 ≤ i ≤ τ): +� +gτ(x) +�(i) +def += +⟨∇f(x), Bi∇f(x)⟩. +(5.3) +Note that this operation is exactly the projection of the direction 1 +LB−1∇f(xk) onto the +Krylov subspace (1.3): +xk+1 − xk +:= +argmin +h∈Kτxk +∥h + 1 +LB−1∇f(xk)∥2 +B. +Fortunately, for computing this projection we indeed do not need to invert the curvature +matrix B, but to solve only a small linear system (5.1) of size τ + 1. We are ready to +formulate our new adaptive method. +Algorithm 5.1 Gradient Method with Krylov Preconditioning +Initialization: x0 ∈ Rn, τ ≥ 0, L > 0. +for k = 0, 1, . . . do +Form matrix Aτ(xk) and vector gτ(xk) by (5.2), (5.3). +Compute ak = Aτ(xk)−1gτ(xk) ∈ Rτ+1. +Set xk+1 = xk − Pak∇f(xk). +end for +We prove the following optimality result. +13 + +Theorem 5.1. Let P ≻ 0 be any preconditioner that is a polynomial of degree τ of +the curvature matrix: +P += +pτ(B), +pτ ∈ R[s], +deg(pτ) = τ. +Then, for the iteration of Algorithm 5.1 we have the global rates (3.2),(3.4) with the +condition number that is attributed to P (3.1): β +α = β(P ) +α(P ). +Hence, our method automatically chooses the best possible preconditioning matrix +from the polynomial class. Let us understand what are the bounds for β +α that we can +achieve in this case. +5.2 +Bounds for the Condition Number +Let us assume that the top τ > 0 eigenvalues of B are all separated. Then, we can easily +cut them off with the following simple construction. Define +qτ(s) +def += +� +1 − s +λ1 +�� +1 − s +λ2 +� +· . . . · +� +1 − +s +λτ +� +. +(5.4) +Proposition 5.2. For any τ, taking P = pτ(B), where pτ(s) := 1+qτ(s)·(αs−1) +s +with qτ +defined by (5.4) and α = +2 +λτ+1+λn , the condition number is bounded by β +α ≤ λτ+1 +λn . +The worst case instance for the cutting strategy is when all the eigenvalues except one +share the same value: λ1 = λ2 = . . . = λn−1 > λn. Indeed, then the condition number +remains the same: +β +α = +λτ+1 +λn +≡ +λ1 +λn , while τ < n − 1. A better approach in such a +situation would be to find a bound from the uniform polynomial approximation for the +whole interval [λn, λ1], which is achieved with the Chebyshev polynomials [17, 23]. Then, +we can decrease the condition number exponentially by increasing the degree τ. +Proposition 5.3. For a fixed 0 < ϵ < 1, let τ := +�� +λ1 +λn ln 8 +ϵ +� +. Then, taking P = pτ(B), +where pτ(s) := 1−Qτ(s) +s +with Qτ is a normalized Chebyshev polynomial7of the first kind of +degree τ + 1, the condition number is bounded by β +α ≤ 1 + ϵ. +Clearly, we can combine this technique with the cutting strategy, getting the best of +two guarantees. +5.3 +Discussion +We see that in the case of unconstrained smooth minimization, it is possible to achieve the +guarantee of the best polynomial of a fixed degree τ, by computing a certain projection onto +the corresponding Krylov subspace. Namely, we can achieve β +α ≤ λτ+1 +λn +(Proposition 5.2), +which cuts off the top τ eigenvalues of the spectrum completely, if they are separated from +the others. At the same time, by using the Chebyshev polynomials, we can contract a part +of the spectrum uniformly, with an appropriate degree τ (Proposition 5.3). It remains +to be an open question where we can incorporate adaptive Krylov preconditioning into +7See Appendix B.8 for the precise definition. +14 + +the Fast Gradient Method, which would give us a further improvement of the condition +number. +Note that the linear Conjugate Gradient Method (CGM) as applied to a convex +quadratic function (thus L = µ = 1), has the same guarantee as in Theorem 5.1 with +τ := k, where k is the iteration number. I.e., in each iteration k, the method automatically +achieves the best polynomial approximation of degree k. As compared to CGM, it remains +to be the main advantage of Algorithm 5.1, that it is applicable to a wider class of nonlinear +problems. +Finally, for the methods with Symmetric Polynomial Preconditioning, we obtained +the bound: +β +α ≤ +λ1 +λn · ξτ(λ) (Theorem 4.2), where ξτ(λ) ≤ 1, and ξτ(λ) → 0 in the +case of large spectral gap: λ1/λτ+1 → ∞. This guarantee is similar to that for the +cutting guarantee of Algorithm 5.1. Moreover, we can apply our family of preconditioners +for the more general composite optimization problems. We also can incorporate the +Symmetric Polynomial Preconditioning into the Fast Gradient Method (Algorithm 3.2), +which significantly improves the product of the both condition numbers: L +µ · β +α by taking +the square root (Theorem 3.2). Therefore, in particular practical scenarios, any of these +strategies can be more preferable. +6 +Experiments +Huber Loss. +Let us present an illustrative experiment, with the regression model +(Example 2.4) with the Huber loss function: +φ(t) +:= +� +t2 +2µ, +if +|t| ≤ µ, +|t| − µ +2, +otherwise, +where µ := 0.1 is a parameter. The data is generated with a fixed distribution of eigenvalues: +λ1 > λ2 > λ3 = . . . λn = 1. Thus, we have two gaps between the leading eigenvalues. We +use the Gradient Method (Algorithm 3.1), with the adaptive search to fit the parameter +M. The results are shown in Figure 6.1. Using the preconditioner P1 helps the method to +deal with the large gap between λ1 and λ2, while P2 makes the method to be insensitive +to the gap between λ1 and λ3, as predicted by our theory. For example, increasing the +ratio λ1/λ2 by 10, the performance of the methods with P1 and P2 becomes ten times +faster than that of the classical gradient descent. +0 +200 +400 +600 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +Func. residual +Huber, +1 = 100, +2 = 10 +P0 +P1 +P2 +0 +2000 +4000 +6000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +Func. residual +Huber, +1 = 1000, +2 = 10 +P0 +P1 +P2 +0 +2000 +4000 +6000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +Func. residual +Huber, +1 = 1000, +2 = 800 +P0 +P1 +P2 +0 +20000 +40000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +Func. residual +Huber, +1 = 8000, +2 = 800 +P0 +P1 +P2 +Figure 6.1: Minimizing the Huber loss by Algorithm 3.1 with Symmetric Polynomial Preconditioning +(4.1). P0 corresponds to the classical gradient descent without preconditioning. +15 + +Logistic Regression. +We examine the training of logistic regression on real data. That +problem corresponds to Example 2.4 with the loss function φ(t) = log(1 + et). +In Figure 6.2, we see that the best convergence is achieved by the Fast Gradient +Method (FGM, Algorithm 3.2) with P2. Using Symmetric Polynomial Preconditioning +makes the methods to converge much better (two times faster for GM using P2 instead +of P0 ≡ I, and about 1.5 times faster for FGM). Among the versions of GM, the most +encouraging performance belongs to the Krylov preconditioning, which is consistent with +the theory. For all the methods tested, the arithmetic cost of every iteration remains to +be at the same level. See also Appendix A for the extra experiments. +0 +2000 +4000 +Iterations +10 +4 +10 +3 +10 +2 +10 +1 +100 +Func. residual +GM, mnist +GM, P0 +GM, P1 +GM, P2 +0 +2000 +4000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +Krylov, mnist +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +250 +500 +750 +1000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, mnist +FGM, P0 +FGM, P1 +FGM, P2 +0 +5000 +10000 +Matrix-vector products +10 +3 +10 +2 +10 +1 +100 +Func. residual +GM, mnist +GM, P0 +GM, P1 +GM, P2 +0 +5000 +10000 +Matrix-vector products +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +Krylov, mnist +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +2000 +4000 +6000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, mnist +FGM, P0 +FGM, P1 +FGM, P2 +Figure 6.2: Training logistic regression with Algorithm 3.1 (GM) and Algorithm 3.2 (FGM) +employing Symmetric Polynomial Preconditioning (4.1); and with Algorithm 5.1 (Krylov). +References +[1] +C. M. Bishop. Pattern recognition and machine learning, volume 4 of number 4. +Springer, 2006. +[2] +L. Cunha, G. Gidel, F. Pedregosa, D. Scieur, and C. Paquette. Only tails matter: +average-case universality and robustness in the convex regime. In International +Conference on Machine Learning, pages 4474–4491. PMLR, 2022. +[3] +A. Deshpande, L. Rademacher, S. S. Vempala, and G. Wang. Matrix approximation +and projective clustering via volume sampling. Theory of Computing, 2(12):225–247, +2006. +[4] +P. F. Dubois, A. Greenbaum, and G. H. Rodrigue. Approximating the inverse of a +matrix for use in iterative algorithms on vector processors. Computing, 22(3):257–268, +1979. +[5] +D. S. Dummit and R. M. Foote. Abstract algebra, volume 3. Wiley Hoboken, 2004. +[6] +G. H. Golub and C. F. Van Loan. Matrix computations. JHU press, 2013. +16 + +[7] +B. Goujaud, D. Scieur, A. Dieuleveut, A. B. Taylor, and F. Pedregosa. Super- +acceleration with cyclical step-sizes. 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Polynomial preconditioned gmres and gmres- +dr. SIAM Journal on Scientific Computing, 37(5):S407–S428, 2015. +[14] +J. A. Loe and R. B. Morgan. Toward efficient polynomial preconditioning for gmres. +Numerical Linear Algebra with Applications, 29(4):e2427, 2022. +[15] +K.-A. Mardal and R. Winther. Preconditioning discretizations of systems of partial +differential equations. Numerical Linear Algebra with Applications, 18(1):1–40, 2011. +[16] +J. C. Mason and D. C. Handscomb. Chebyshev polynomials. Chapman and Hall/CRC, +2002. +[17] +A. Nemirovski. Information-based complexity of convex programming. Lecture Notes, +834, 1995. +[18] +A. Nemirovski and D. Yudin. Problem complexity and method efficiency in opti- +mization, 1983. +[19] +Y. Nesterov. A method for solving the convex programming problem with convergence +rate O(1/kˆ2). In Dokl. akad. nauk Sssr, volume 269, pages 543–547, 1983. +[20] +Y. Nesterov. Gradient methods for minimizing composite functions. 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PMLR, 2020. +[28] +D. A. Spielman and S.-H. Teng. Nearly-linear time algorithms for graph partitioning, +graph sparsification, and solving linear systems. In Proceedings of the thirty-sixth +annual ACM symposium on Theory of computing, pages 81–90, 2004. +[29] +P. M. Vaidya. Solving linear equations with symmetric diagonally dominant matrices +by constructing good preconditioners. A talk based on this manuscript, 2(3.4):2–4, +1991. +[30] +M. Van Gijzen. A polynomial preconditioner for the gmres algorithm. Journal of +Computational and Applied Mathematics, 59(1):91–107, 1995. +[31] +N. K. Vishnoi. Lx=b. Foundations and Trends® in Theoretical Computer Science, +8(1–2):1–141, 2013. +18 + +A +Extra Experiments +Logistic Regression. +Let us present experimental results for our preconditioning +strategies, for the training of Logistic Regression with several real datasets. We investigate +both the number of iterations and the number of matrix-vector products (the most difficult +operation) required to reach a certain accuracy level in the functional residual. The results +are shown in Figure A.1. +0 +2000 +4000 +Iterations +10 +4 +10 +3 +10 +2 +10 +1 +Func. residual +GM, a9a +GM, P0 +GM, P1 +GM, P2 +0 +2000 +4000 +Iterations +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Krylov, a9a +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +2000 +4000 +Iterations +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +FGM, a9a +FGM, P0 +FGM, P1 +FGM, P2 +0 +5000 +10000 +Matrix-vector products +10 +4 +10 +3 +10 +2 +10 +1 +Func. residual +GM, a9a +GM, P0 +GM, P1 +GM, P2 +0 +5000 +10000 +Matrix-vector products +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Krylov, a9a +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +5000 +10000 +Matrix-vector products +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +FGM, a9a +FGM, P0 +FGM, P1 +FGM, P2 +0 +2000 +4000 +Iterations +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Func. residual +GM, phishing +GM, P0 +GM, P1 +GM, P2 +0 +2000 +4000 +Iterations +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +Krylov, phishing +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +2000 +4000 +Iterations +10 +7 +10 +5 +10 +3 +10 +1 +FGM, phishing +FGM, P0 +FGM, P1 +FGM, P2 +0 +5000 +10000 +Matrix-vector products +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Func. residual +GM, phishing +GM, P0 +GM, P1 +GM, P2 +0 +5000 +10000 +Matrix-vector products +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +Krylov, phishing +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +5000 +10000 +Matrix-vector products +10 +6 +10 +4 +10 +2 +100 +FGM, phishing +FGM, P0 +FGM, P1 +FGM, P2 +0 +200 +400 +600 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +Func. residual +GM, breast-cancer +GM, P0 +GM, P1 +GM, P2 +0 +50 +100 +150 +200 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +Krylov, breast-cancer +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +200 +400 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, breast-cancer +FGM, P0 +FGM, P1 +FGM, P2 +0 +1000 +2000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +Func. residual +GM, breast-cancer +GM, P0 +GM, P1 +GM, P2 +0 +500 +1000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +Krylov, breast-cancer +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +2000 +4000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, breast-cancer +FGM, P0 +FGM, P1 +FGM, P2 +0 +500 +1000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +Func. residual +GM, ionosphere +GM, P0 +GM, P1 +GM, P2 +0 +100 +200 +300 +400 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +Krylov, ionosphere +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +500 +1000 +Iterations +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, ionosphere +FGM, P0 +FGM, P1 +FGM, P2 +0 +2000 +4000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +Func. residual +GM, ionosphere +GM, P0 +GM, P1 +GM, P2 +0 +500 +1000 +1500 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +Krylov, ionosphere +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +5000 +Matrix-vector products +10 +8 +10 +6 +10 +4 +10 +2 +100 +FGM, ionosphere +FGM, P0 +FGM, P1 +FGM, P2 +0 +2000 +4000 +Iterations +10 +3 +10 +2 +10 +1 +Func. residual +GM, covtype +GM, P0 +GM, P1 +GM, P2 +0 +2000 +4000 +Iterations +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Krylov, covtype +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +2000 +4000 +Iterations +10 +7 +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +FGM, covtype +FGM, P0 +FGM, P1 +FGM, P2 +0 +5000 +10000 +Matrix-vector products +10 +3 +10 +2 +10 +1 +Func. residual +GM, covtype +GM, P0 +GM, P1 +GM, P2 +0 +5000 +10000 +Matrix-vector products +10 +4 +10 +3 +10 +2 +10 +1 +Krylov, covtype +Krylov, = 0 +Krylov, = 1 +Krylov, = 2 +Krylov, = 3 +0 +5000 +10000 +Matrix-vector products +10 +6 +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +FGM, covtype +FGM, P0 +FGM, P1 +FGM, P2 +Figure A.1: Training logistic regression with Algorithm 3.1 (GM) and Algorithm 3.2 (FGM) +employing Symmetric Polynomial Preconditioning (4.1); and with Algorithm 5.1 (Krylov). +We see that using Symmetric Polynomial Preconditioning (P1 and P2) significantly +accelerates both the Gradient Method (GM) and the Fast Gradient Method (GM), without +extra arithmetic efforts during each iteration. Using the Krylov preconditioning is more +costly, while it equips GM with the best possible iteration rates. +19 + +B +Proofs +B.1 +Proof of Theorem 3.1 +Let us consider one iteration of the method, for some k ≥ 0. By definition, xk+1 = +argmin +y∈dom ψ +� +Ωk(y) +� +, where +Ωk(y) +def += +f(xk) + ⟨∇f(xk), y − xk⟩ + M +2 ∥y − xk∥2 +P −1 + ψ(y) +is strongly convex with respect to P −1 norm with parameter M := βL. Thus, we have, +for any y ∈ dom ψ: +M +2 ∥y − xk∥2 +P −1 + F(y) ≥ Ωk(y) ≥ +Ωk(xk+1) + M +2 ∥y − xk+1∥2 +P −1 +≥ f(xk) + ⟨∇f(xk), xk+1 − xk⟩ + L +2 ∥xk+1 − xk∥2 +B + ψ(xk+1) + M +2 ∥y − xk+1∥2 +P −1 +(2.3) +≥ +F(xk+1) + M +2 ∥y − xk+1∥2 +P −1. +(B.1) +Hence, substituting y := x⋆ (solution to the problem), we establish the boundness for all +iterates: +∥xk+1 − x⋆∥P −1 +≤ +∥xk − x⋆∥P −1. +(B.2) +Further, let us take y := γkx⋆ + (1 − γk)xk, for some γk ∈ [0, 1]. We obtain +F(xk+1) +(B.1) +≤ +F(γkx⋆ + (1 − γk)xk) + γ2 +kM +2 ∥x⋆ − xk∥2 +P −1 +≤ +γkF ⋆ + (1 − γk)F(xk) + γ2 +kM +2 ∥x⋆ − xk∥2 +P −1. +(B.3) +Now, setting Ak +def += k · (k + 1), ak+1 +def += Ak+1 − Ak = 2(k + 1), and γk := ak+1 +Ak+1 = +2 +k+2, we +obtain +Ak+1 +� +F(xk+1) − F ⋆� +(B.3) +≤ +Ak +� +F(xk) − F ⋆� ++ +a2 +k+1 +Ak+1 · M +2 ∥x∗ − xk∥2 +P −1 +(B.2) +≤ +Ak +� +F(xk) − F ⋆� ++ +a2 +k+1 +Ak+1 · M +2 ∥x∗ − x0∥2 +P −1 +(3.1) +≤ +Ak +� +F(xk) − F ⋆� ++ +a2 +k+1 +Ak+1 · β +α · L +2 ∥x∗ − x0∥2 +B. +(B.4) +Telescoping this bound for the first k iterations, we get +F(xk) − F ⋆ +(B.4) +≤ +β +α · L +2 ∥x∗ − x0∥2 +B · +1 +Ak +k� +i=1 +a2 +i +Ai += +O +� +β +α · L +2k∥x∗ − x0∥2 +B +� +. +To prove the linear rate for the strongly convex case, we continue as follows +F(xk+1) +(B.3),(2.3) +≤ +γkF ⋆ + (1 − γk)F(xk) + γ2 +k · βL +αµ · +� +F(xk) − F ⋆� +. +20 + +Choosing γk := +αµ +2βL < 1, we get the exponential rate +F(xk+1) − F ⋆ +≤ +� +1 − αµ +4βL +�� +F(xk) − F ⋆� +, +which completes the proof. +B.2 +Proof of Theorem 3.2 +Let x ∈ dom ψ and k ≥ 0 be arbitrary. From (2.3), (3.1), and the fact that ρ = αµ, it +follows that +F(x) = f(x)+ψ(x) ≥ ℓk(x)+ρ +2∥x−yk∥2 +P −1, +ℓk(x) def += f(yk)+⟨∇f(yk), x−yk⟩+ψ(x). +Hence, +AkF(xk) + ak+1F(x) + 1 + ρAk +2 +∥x − vk∥2 +P −1 +≥ Akℓk(xk) + ak+1ℓk(x) + 1 + ρAk +2 +∥x − vk∥2 +P −1 + ρak+1 +2 +∥x − yk∥2 +P −1 +≥ Akℓk(xk) + ak+1ℓk(x) + 1 + ρAk+1 +2 +∥x − ˆvk∥2 +P −1 +def += ζk(x), +(B.5) +where the final inequality follows from the convexity of the squared norm and the fact +that, according to our definitions, +(1 + ρAk)vk + ρak+1yk +1 + ρAk+1 += (1 − ωk)vk + ωkyk = (1 − ωk)vk + ωk[(1 − θk)xk + θkˆvk] = ˆvk. +Note that ζk is a (1 + ρAk+1)-strongly convex function w.r.t. ∥·∥P −1, and vk+1 is precisely +its minimizer. Therefore, +ζk(x) ≥ ζk(vk+1) + 1 + ρAk+1 +2 +∥x − vk+1∥2 +P −1. +(B.6) +Since ℓk is a convex function, we have, by our definition of xk+1, +Akℓk(xk) + ak+1ℓk(vk+1) ≥ Ak+1ℓk(xk+1). +On the other hand, by the definition of xk+1 and yk, +xk+1 − yk = θk(vk+1 − ˆvk) = ak+1 +Ak+1 +(vk+1 − ˆvk). +Therefore, +ζk(vk+1) = Akℓk(xk) + ak+1ℓk(vk+1) + 1 + ρAk+1 +2 +∥vk+1 − ˆvk∥2 +P −1 +≥ Ak+1 +� +ℓk(xk+1) + Ak+1(1 + ρAk+1) +2a2 +k+1 +∥xk+1 − yk∥2 +P −1 +� +. +21 + +In view of our choice of ak+1, we have the following identity: +Ma2 +k+1 +Ak+1 += 1 + ρAk+1. +(B.7) +Combining this with the fact that M = βL and using (3.1) and (2.3), we get +ζk(vk+1) ≥ Ak+1 +� +ℓk(xk+1) + M +2 ∥xk+1 − yk∥2 +P −1 +� +≥ Ak+1 +� +ℓk(xk+1) + L +2 ∥xk+1 − yk∥2 +B +� += Ak+1 +� +f(yk) + ⟨∇f(yk), xk+1 − yk⟩ + L +2 ∥xk+1 − yk∥2 +B + ψ(xk+1) +� +≥ Ak+1F(xk+1). +Substituting the above bound into (B.6), and that one into (B.6), we thus obtain +AkF(xk)+ak+1F(x)+ 1 + ρAk +2 +∥x−vk∥2 +P −1 ≥ Ak+1F(xk+1)+ 1 + ρAk+1 +2 +∥x−vk+1∥2 +P −1. +This inequality is valid for any k ≥ 0. +Fixing an arbitrary k ≥ 1 and summing up the previous inequalities for all indices +k′ = 0, . . . , k − 1, we get +AkF(xk) ≤ AkF(x) + 1 + ρA0 +2 +∥x − v0∥2 +P −1 = AkF(x) + 1 +2∥x − x0∥2 +P −1. +Substituting further x = x⋆ (an optimal solution) and using (3.1), gives us the following +convergence rate estimate: +F(xk) − F ⋆ ≤ ∥x⋆ − x0∥2 +P −1 +2Ak +≤ ∥x⋆ − x0∥2 +B +2αAk +. +(B.8) +To complete the proof, it remains to use standard lower bounds on Ak (see [21]). +Specifically, dropping the second term from the right-hand side of (B.7) and rearranging, +we obtain, for any k ≥ 0, +� +Ak+1 +M +≤ ak+1 = Ak+1−Ak = ( +� +Ak+1− +� +Ak )( +� +Ak+1+ +� +Ak ) ≤ 2( +� +Ak+1− +� +Ak ) +� +Ak+1. +Cancelling +� +Ak+1 on both sides and using the fact that A0 = 0, we obtain, for any k ≥ 1, +� +Ak ≥ +k +2 +√ +M +. +Squaring both sides, substituting the resulting inequality into (B.8) and replacing M = βL, +we get (3.4). +When µ > 0, we can drop the first term from the right-hand side of (B.7). This gives +us +a2 +k+1 ≥ ρ +M A2 +k+1. +Hence, for any k ≥ 0, +Ak+1 − Ak = ak+1 ≥ qAk+1, +q def += +� ρ +M ≤ 1, +22 + +or, equivalently, +Ak+1 ≥ +Ak +1 − q. +Consequently, for any k ≥ 1, +Ak ≥ +A1 +(1 − q)k−1 ≥ +1 +M(1 − q)k−1 , +where the final inequality is due to (B.7) combined with the fact that A0 = 0. Substituting +this inequality into (B.8) and replacing M = βL, ρ = αµ, we get the second bound from +Theorem 3.2. +B.3 +Proof of Lemma 4.1 +Let us denote by uk(a) the k-th power sum of the variables: +uk(a) +def += +n−1 +� +i=1 +ak +i , +∀a ∈ Rn−1. +Then, the classical Newton-Girard identities (see, e.g. [11]) state the following relation +between the elementary symmetric polynomials: +στ(a) +≡ +1 +k +τ� +i=1 +(−1)i−1στ−i(a) · ui(a). +(B.9) +Note that for the matrix Uτ +def += tr (Bτ)I − Bτ, the following spectral decomposition holds: +Uτ += +QDiag +� n� +i=1 +λτ +i − λτ +1, +n� +i=1 +λτ +i − λτ +2, . . . , +n� +i=1 +λτ +i − λτ +n +� +Q⊤ += +QDiag +� +uτ(λ−1), uτ(λ−2), . . . , uτ(λ−n) +� +Q⊤. +(B.10) +Now, the identity that we need to prove is +Pτ += +QDiag +� +στ(λ−1), στ(λ−2), . . . , στ(λ−n) +� +Q⊤. +(B.11) +We justify (B.11) by induction. By definition, P0 +def += I and σ0(a) ≡ 1, therefore (B.11) +holds for τ = 0, which is our base. Let us fix τ ≥ 1 and assume that (B.11) is true for all +smaller indices. Then, +Pτ +def += +1 +τ +τ� +i=1 +(−1)i−1Pτ−iUi +(B.11),(B.10) += +QDiag +� τ� +i=1 +(−1)i−1στ−i(λ−1) · ui(λ−1), . . . , +τ� +i=1 +(−1)i−1στ−i(λ−n) · ui(λ−n) +� +Q⊤ +(B.9) += +QDiag +� +στ(λ−1), . . . , στ(λ−n) +� +Q⊤. +Hence, (B.11) is proven for all 0 ≤ τ ≤ n − 1. +23 + +B.4 +Proof of Theorem 4.2 +By Lemma 4.1, we have the following representation of our preconditioner: +Pτ += +QDiag (στ(λ−1), στ(λ−2), . . . , στ(λ−n)))Q⊤. +It is easy to see that, for the spectrum of the matrix +B1/2PτB1/2 += +QDiag +� +λ1 · στ(λ−1), λ2 · στ(λ−2), . . . , λn · στ(λ−n) +� +Q⊤, +it holds: +λ1 · στ(λ−1) +≥ +λ2 · στ(λ−2) +≥ +. . . +≥ +λn · στ(λ−n). +(B.12) +Indeed, without loss of generality, let us justify the first inequality: +λ1 · στ(λ−1) +≥ +λ2 · στ(λ−2) +Recall that +στ(a) +def += +� +1≤i1<... τ. Indeed, we can rewrite the left hand side sum as +L1 += +n−1 +� +k=τ+1 +λk · +� +� +2≤i1<... τ. +Finally, to prove the limit, let us divide the right hand side of +ξτ(λ) += +στ(λ−1) +στ(λ−n) +≤ +� +2≤i1<... 0), we continue as +f(xk+1) +(B.23),(2.3) +≤ +γkf⋆ + (1 − γk)f(xk) + γ2 +k +βL +αµ · +� +f(xk) − f⋆� +, +and choosing γk := +αµ +2βL we establish the exponential rate. +28 + +C +Adaptive Search +In this section, we briefly present adaptive versions of Algorithms 3.1 and 3.2 which do +not require the knowledge of the constant M = βL and can automatically “tune” it in +iterations yet preserving the original worst-case efficiency estimates. This is achieved by +using a standard “backtracking line search” which can be found, e.g., in [20]. +In what follows, for any x, y ∈ dom ψ, M > 0 and P = P ⊤ ≻ 0, we define the following +predicate: +QuadGrowthM,P (x, y): +f(y) ≤ f(x) + ⟨∇f(x), y − x⟩ + M +2 ∥y − x∥2 +P −1. +According to our assumptions (2.3) and (3.1), we know that this predicate is surely satisfied +for any pair of points once M ≥ βL. +The adaptive version of Algorithm 3.1 is presented in Algorithm C.1. This method +starts with a certain initial guess ˜ +M0 for the constant βL and then, at every iteration, +repeatedly increases the current guess in two times until the predicate becomes satisfied. +This process is guaranteed to terminate (when Mk becomes bigger or equal to βL, or even +sooner). After that, we accept the new point xk+1 and choose a new “optimistic” guess of +the constant M for the next iteration by halving the value of Mk that we have accepted +at the current iteration. +Algorithm C.1 Adaptive Preconditioned GM +Input: x0 ∈ dom ψ, P = P ⊤ ≻ 0, ˜ +M0 > 0. +for k = 0, 1, . . . do +Find smallest integer ik ≥ 0 such that +xk+1 = GradStepMk,P +� +xk, ∇f(xk) +� +, +Mk = 2ik ˜ +Mk +satisfies the predicate QuadGrowthMk,P (xk, xk+1). +Set ˜ +Mk+1 = Mk/2. +end for +We assume that the preconditioner P is sufficiently simple so that we can efficiently +check the predicate QuadGrowthMk,P (xk, xk+1). For example, if ψ = 0, then xk+1 = +xk − +1 +Mk P ∇f(xk) and Mk∥xk+1 − xk∥2 +P −1 = ⟨∇f(xk), xk − xk+1⟩ can be efficiently +computed. +For Algorithm C.1, we can prove exactly the same rates as in Theorem 3.1 (up to +absolute constants) provided that +˜ +M0 ≤ βL. +(C.1) +The proof is essentially the same as in Appendix B.1 with only two minor differences: 1) +inequality (B.1) is now guaranteed by our predicate; 2) instead of using M = βL in (B.4), +we should use the bound Mk ≤ 2βL which follows from (C.1) and the fact that any value +of M ≥ βL is always acceptable in the line search. Using a classical argument from [20], +it is not difficult to show that, on average, Algorithm C.1 makes only ∼ 2 gradient steps +at each iteration. +29 + +In contrast to an upper estimate of the constant βL, an initial guess satisfying (C.1) can +be easily generated. One simple recipe is to make a trial step x′ +1 = GradStepM′ +0,P +� +x0, ∇f(x0) +� +for an arbitrarily chosen M′ +0 > 0 and then compute +˜ +M0 = f(x′ +1) − f(x0) − ⟨∇f(x0), x′ +1 − x0⟩ +1 +2∥x′ +1 − x0∥2 +P −1 +. +Alternatively, we can find a suitable +˜ +M0 be choosing an arbitrary M′ +0 > 0 and then +repeatedly halving it until the predicate QuadGrowth(x0, x′ +1(M)) stops being satisfied +for x′ +1(M) = GradStepM,P (x0, ∇f(x0)). This auxiliary procedure either terminates in a +logarithmic number of steps, in which case we get a suitable ˜ +M0, or, otherwise, we quickly +find an approximate solution of our problem. +Similar technique can be applied for the Fast Gradient Method. Specifically, let us +introduce an auxiliary procedure shown in Algorithm C.2 for computing one iteration of +Algorithm 3.2 for a given value of M. Then, the adaptive FGM method can be constructed +as shown in Algorithm C.3. As in the basic method, we can show that the rates from +Theorem 3.2 still remain valid (up to absolute constants) for Algorithm C.3, provided +that ˜ +M0 satisfies (C.1). For generating the initial guess ˜ +M0, we can use exactly the same +techniques as before. +Algorithm C.2 (x+, v+, A+; y) = FastGradStepM,ρ,P (x, v, A) +Require: M > 0; ρ ≥ 0; P = P ⊤ ≻ 0; x, v ∈ dom ψ; A > 0. +Find a+ from eq. +Ma2 ++ +A+a+ = 1 + ρ(A + a+). +Set A+ = A + a+, H = 1+ρA+ +a+ +, θ = a+ +A+ , ω = ρ +H , γ = ω(1−θ) +1−ωθ . +Set ˆv = (1 − γ)v + γx, y = (1 − θ)x + θˆv. +Compute v+ = GradStepH,P +�ˆv, ∇f(y) +� +. +Set x+ = (1 − θ)x + θv+. +return (x+, v+, A+; y). +Algorithm C.3 Adaptive Preconditioned FGM +Input: x0 ∈ dom ψ, P = P ⊤ ≻ 0, ˜ +M0 > 0. +Set v0 = x0, A0 = 0. +for k = 0, 1, . . . do +Find smallest integer ik ≥ 0 such that +(xk+1, vk+1, Ak+1; yk) = FastGradStepMk,P +� +xk, vk, Ak +� +, +Mk = 2ik ˜ +Mk +satisfies the predicate QuadGrowthMk,P (yk, xk+1). +Set ˜ +Mk+1 = Mk/2. +end for +30 + diff --git a/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/2301.03694v1.pdf.txt b/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/2301.03694v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..397fd3c5ec6a9d9f68f7c9809adc1eb90c4b0dc6 --- /dev/null +++ b/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/2301.03694v1.pdf.txt @@ -0,0 +1,2617 @@ +Growth and Geometry Split in Light of the DES-Y3 Survey +Kunhao Zhong,1, ∗ Evan Saraivanov,1 Vivian Miranda,1, 2 Jiachuan Xu,3 Tim Eifler,3 and Elisabeth Krause3, 4 +1Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794, USA +2C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA +3Department of Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA +4Department of Physics, University of Arizona, 1118 E Fourth Str, Tucson, AZ, 85721-0065, USA +(Dated: January 11, 2023) +We test the smooth dark energy paradigm using Dark Energy Survey (DES) Year 1 and Year 3 weak lensing +and galaxy clustering data. Within the ΛCDM and 𝑤CDM model we separate the expansion and structure +growth history by splitting Ωm (and 𝑤) into two meta-parameters that allow for different evolution of growth and +geometry in the Universe. We consider three different combinations of priors on geometry from CMB, SNIa, +BAO, BBN that differ in constraining power but have been designed such that the growth information comes +solely from the DES weak lensing and galaxy clustering. For the DES-Y1 data we find no detectable tension +between growth and geometry meta-parameters in both the ΛCDM and 𝑤CDM parameter space. This statement +also holds for DES-Y3 cosmic shear and 3x2pt analyses. For the combination of DES-Y3 galaxy-galaxy lensing +and galaxy clustering (2x2pt) we measure a tension between our growth and geometry meta-parameters of +2.6𝜎 in the ΛCDM and 4.48𝜎 in the 𝑤CDM model space, respectively. We attribute this tension to residual +systematics in the DES-Y3 RedMagic galaxy sample rather than to new physics. We plan to investigate our +findings further using alternative lens samples in DES-Y3 and future weak lensing and galaxy clustering datasets. +I. +INTRODUCTION +Since the discovery of the accelerated expansion of our Uni- +verse [1, 2], the flat ΛCDM, which adopts a late-time Universe +dominated by the cosmological constant, has become the stan- +dard model of cosmology. From a fundamental physics view- +point, the origin of dark energy is still unknown. The cosmo- +logical constant modeled as vacuum energy is fine-tuned with +a value too small to any known quantum field theory [3]. Dy- +namical scalar fields, quintessence and k-essence, have been +proposed to solve the fine-tuning problem [4–8]. Modified +gravity is an alternative way to explain the Universe’s acceler- +ation without introducing a new component [9]. To date, none +of these proposed scenarios have been detected by observa- +tions. +With only six free parameters, the standard model of cos- +mology predicts the temperature and polarization anisotropy +statistics of the Cosmic Microwave Background (CMB) with +remarkable success. Additionally, imaging and spectroscopic +surveys show increasing power to constrain ΛCDM’s predic- +tions for the late-time evolution of large-scale structures (LSS); +current stage III LSS surveys include the Dark Energy Survey +(DES) [10–22], the Kilo-Degree Survey (KiDS) [23–27], the +Hyper Suprime-Cam Subaru Strategic Program (HSC) [28– +32], and the Baryon Oscillation Spectroscopic Survey (Boss +and eBOSS) [33–38]. +However, multiple tensions have arisen in the last few years +within the ΛCDM model, particularly between Planck mea- +surements of the Cosmic Microwave Background and data +from the late-time Universe. +The first tension involves the +value of the Hubble constant, 𝐻0 [39–42]. Local-Universe +𝐻0 estimates from type Ia supernova (SNIa), calibrated us- +ing Cepheid variable stars [43, 44], conflict with CMB pre- +∗ kunhao.zhong@stonybrook.edu +dictions [45, 46]. +Several studies show that this tension is +reaching a statistical significance of 5𝜎 [40–42]. +Hubble constant predictions from the Cosmic Microwave +Background are sensitive to changes in the late-time dark sec- +tor [47]. For example, cold dark matter models decaying to +relativistic species can affect the CMB predictions [48–50]. +These predictions are also sensitive to physics before recombi- +nation via the sound horizon. However, observations of SNIa +combined with Baryonic Acoustic Oscillations (BAO) show +that changes in the late-time Universe dark sector cannot solve +the 𝐻0 tension without creating additional problems [51–53]. +These constraints suggest that the new physics should come +from the time before recombination [54, 55]. +The Dark Energy Survey year one (DES-Y1) and year three +(DES-Y3) analysis conclude that the parameter 𝑆8 is in mild +tension with the ΛCDM model predicted by Planck CMB +data [20, 56, 57]. Multiple independent surveys have inde- +pendently discovered this discrepancy [24, 35, 58, 59]. The +projected one-dimensional 𝑆8 tension is not large; however, in- +vestigations of the multi-dimensional degeneracy directions in +ΛCDM parameter space offers a more complete picture [60]. +The generalizations of the late-time dark sector can reduce +this discrepancy, but the 𝑆8 tension generally increases with +statistical significance when an early-dark energy component +is added in the ΛCDM model [61–63]. +In this work, we split the matter density, Ωm, and the dark +energy equation of state, 𝑤, to test the consistency of smooth- +dark-energy between the background evolution and the late- +time scale-independent growth of structures [64–68]. Using +different data sets containing geometry or growth information, +we can verify such consistency in ΛCDM and 𝑤CDM models. +Parameter splitting has been extensively applied in multiple +contexts. For example, baryon density can be divided into +two parts with one only affecting ionization history [69], cold +matter density can be split into parts representing different +aspects of type Ia supernova [70], or the primordial inflationary +amplitude can be separated into one that affects the CMB and +arXiv:2301.03694v1 [astro-ph.CO] 9 Jan 2023 + +2 +another that only affects predictions from the effective field +theory of large-scale structure [71]. +This work is a follow-up investigation of two previous anal- +yses, one employing DES-Y1 data [72], and the other adopting +older weak lensing data from the Canada-France Hawaii Tele- +scope Lensing Survey [67]. +In this work, we employ the +new DES-Y3 3x2pt data, including different data combina- +tions that clarify some internal aspects of the galaxy-galaxy +lensing and galaxy clustering combination. The Kilo-Degree +Survey (KiDS) collaboration also analyzed their data with the +growth-geometry split type of parameters [27]. In addition to +weak lensing and galaxy clustering, redshift space distortion +(RSD) and clusters data are used to extract growth informa- +tion [73, 74]. See Sec. VI for the discussion of different split +methods. +The structure of the paper is as follows: In Sec. II, we ex- +plain the geometry-growth split and the 3x2pt combination of +two-point correlation functions. We summarize DES analy- +sis choices and the external data sets in Sec. IV, which also +contains a detailed description of our adopted pipeline and +the validation tests we performed based on synthetic ΛCDM +DES-Y1 and DES-Y3 data vectors. We present the results and +discussions in Sec. V, and conclusions, including an exposition +on planned follow-up improvements, in Sec. VI. +II. +THEORY AND METHODOLOGY +A. +Split Matter Power Spectrum +The linear matter power spectrum quantifies the inhomo- +geneity of matter distribution, and it can be written as the +product of the inflationary primordial spectrum, the transfer +function, and the growth function: +𝑃linear(𝑧, 𝑘) = 2𝜋2 +𝑘3 +4 +25 𝐴s +� +𝑘 +𝑘norm +�𝑛s−1 � 𝑘 +𝐻0 +�4 +× +× 𝑇2(𝑘) +� +𝐺(𝑧) +Ωm(1 + 𝑧) +�2 +(1) +The growth function, +𝐺(𝑧) = (1 + 𝑧)𝐷(𝑧) = (1 + 𝑧) 𝛿m(𝑧) +𝛿m(𝑧ini) , +(2) +describes the scale-independent time evolution of matter over- +density from initial conditions defined at redshift 𝑧ini = 1000. +In smooth dark energy cosmologies, the growth-factor evolu- +tion obeys the following ordinary differential equation: +𝐺′′ + +� +4 + 𝐻′ +𝐻 +� +𝐺′ + +� +3 + 𝐻′ +𝐻 − 3 +2Ωm(𝑧) +� +𝐺 = 0, +(3) +where the prime denotes derivative with respect to the log- +arithm of the scale factor, ln 𝑎. +The initial conditions are +𝐺ini = 1 and 𝐺′ +ini = −(3/5)(1 − 𝑤)ΩDE(𝑧ini) [65]. +Mod- +els that introduce clustering of dark energy break this scale- +independent relation between growth factor and dark energy +parameters [75, 76]. In this work, we confine our study to +the case of smooth dark energy with a constant equation of +state (𝑤CDM). Our results can be generalized, for example, by +considering instead principal component based 𝑤(𝑧) parame- +terizations [68, 77]. +We split the Ωm and 𝑤 parameters into geometry, +{Ωgeo +m , 𝑤geo}, and growth counterparts {Ωgrowth +m +, 𝑤growth}. The +growth parameters affect the late-time growth factor evolu- +tion via Eq. 3. The remaining parameters, {Ωb, 𝐻0, 𝐴s, 𝑛s, 𝜏}, +are not split. The split ΛCDM cosmology assumes 𝑤geo = +𝑤growth = −1. Since the linear power spectrum 𝑃linear(𝑧, 𝑘) ∝ +𝐺2(𝑧), we can define the split linear matter power spectrum to +be: +𝑃linear +split (𝑘, 𝑧) = +𝑃linear−camb +geo +(𝑘, 𝑧) +𝐺camb +geo (𝑧)2 +× 𝐺growth(𝑧)2 , +(4) +with +𝐺growth(𝑧) = 𝐺camb +geo (𝑧) × +�𝐺ODE +growth(𝑧) +𝐺ODE +geo (𝑧) +� +. +(5) +𝑃linear−camb +geo +and 𝐺camb +geo +are respectively the linear power spec- +trum and the growth factor both computed by the Boltzmann +code CAMB [78, 79] assuming the geometry parameters. +𝐺ODE +geo and 𝐺ODE +growth are solutions of the differential Eq. 3 given +geometry and growth parameters respectively. +Our slightly convoluted definition is analytically equivalent +to +𝑃linear +split (𝑘, 𝑧) = 𝑃linear−camb +geo +(𝑘, 𝑧) × +�𝐺ODE +growth(𝑧) +𝐺ODE +geo (𝑧) +�2 +. +(6) +Definition of Eq. 5 resolves the small numerical error between +the growth factor calculated by CAMB versus the solution from +Eq. 3 with no radiation and accurate background evolution of +massive neutrinos; the adopted multi-probe lensing pipeline +requires 𝐺growth(𝑧) itself when computing intrinsic alignment +contributions for cosmic shear and galaxy-galaxy lensing. +We follow the naming convention in the parameter split +literature. In our parameter split distance probes heavily con- +strain geometry parameters while growth parameters allow the +late-time growth factor to vary with extra degrees of freedom. +However Ωgeo +m +and 𝑤geo can also affect structure growth. +Specifically, the split matter power spectrum in our definition +is proportional to (Ωgeo +m )−2 (see Fig. 1). Additionally, early +universe physics that affect both background expansion and +structure formation are also modeled by Ωgeo +m . +The split +between geometry and growth is not uniquely defined, and +we defer to future work examining the impact of these choices. +The root mean variance within 8 Mpc/h is defined as: +𝜎2 +8 (𝑧) = +1 +2𝜋2 +∫ +d log 𝑘𝑊2(𝑘𝑅)𝑘3𝑃(𝑘, 𝑧), +(7) + +3 +k [h/Mpc] +101 +103 +P(k) [Mpc/h]3 +P(k), z = 0 +growth +m += 0.25 +growth +m += 0.35 +geo +m += 0.25 +geo +m += 0.35 +m = 0.3 +10 4 +10 3 +10 2 +10 1 +100 +k [h/Mpc] +0.75 +1.00 +1.25 +P(k)/P(k)fid +FIG. 1. Geometry and growth effects on the matter power spectrum +in the split-ΛCDM model. When changing one parameter, we keep +the remaining matter density at ΩXm = 0.3. The choice of employing +Ωgrowth +m +in the Euclid Emulator has the effect of roughly maintaining +the scale-independent amplitude shift induced by changes in Ωgrowth +m +on mildly non-linear scales. On 𝑘 ≳ 10−2 scales that are within +DES reach, changes in the shape parameter Γ = Ωgeo +m ℎ induced by +varying Ωgeo +m +are degenerate with the primordial power spectrum tilt, +𝑛s. This degeneracy motivates our choice of priors; CMB brings +external constraints on the inflationary and shape parameters, while +BAO and SNIa indirectly limits the shape parameter. +where 𝑊(𝑘𝑅) is a top-hat filter function in Fourier space with +radius 𝑅 = 8 Mpc/h. The split 𝜎split +8 +(𝑧) is then given by: +𝜎split +8 +(𝑧) = 𝜎geo−camb +8 +(𝑧) × +�𝐺ODE +growth(𝑧) +𝐺ODE +geo (𝑧) +� +. +(8) +The different behavior of 𝜎split +8 +(𝑧) versus 𝜎8(𝑧) with respect +to the change of growth and geometry parameters is shown in +Fig. 2. In the split ΛCDM case, the change of Ωgrowth +m +will give +a smaller change on 𝜎8(𝑧) compared with the unsplit case, +namely when changing Ωgrowth +m +and Ωgeo +m simultaneously. It is +the opposite in the split 𝑤CDM case where a change in 𝑤growth +gives a larger change on 𝜎8(𝑧). In both cases, the change is +larger at low redshift. +To account for non-linearities in the split power spec- +trum, we utilize the Euclid Emulator to compute the factor +𝐵(𝑘, 𝑧) ≡ 𝑃(𝑘, 𝑧)�𝑃linear(𝑘, 𝑧) [80]. In this work, we defined +the 𝐵(𝑘, 𝑧) as dependent on the growth parameters. We then +define the split matter power spectrum as +𝑃split(𝑘, 𝑧) = 𝑃linear +split (𝑘, 𝑧) × 𝐵growth(𝑘, 𝑧) . +(9) +The official DES-Y1 and DES-Y3 analyses adopt Halofit. +Figure 3 shows that Halofit and Euclid Emulator differ- +ences are within 5%. This disagreement doesn’t affect infer- +ences on the ΛCDM parameters as shown in the bottom panel +of Fig. 3. However, our definition has practical advantages. +0 +0.5 +1.0 +1.5 +z +0.4 +0.6 +0.8 +1.0 +8(z) +m = 0.3 +m = 0.35 +m = 0.25 +growth +m += 0.35 +growth +m += 0.25 +0 +0.5 +1.0 +1.5 +z +0.4 +0.6 +0.8 +1.0 +8(z) +w = +1 +w = +0.8 +w = +1.2 +wgrowth = +0.8 +wgrowth = +1.2 +FIG. 2. Both panels show 𝜎8 changes under the variations of the +unsplit and split Ωm and 𝑤 parameters. The solid lines show shifts +on the unsplit ΛCDM and 𝑤CDM models, and the dashed lines +change growth while keeping the geometry parameters at their fiducial +Ωgeo +m += 0.3 and 𝑤geo = -1. We can see that changes in the growth +parameter Ωgrowth +m +results in a minor shift in 𝜎8(𝑧), whereas 𝑤growth +gives a more considerable change compared with the unsplit model. +Massive neutrinos break the scale-independent evolution of +dark matter perturbations; neutrinos transition from relativis- +tic to non-relativistic behavior as the universe cools down. The +scale-dependant changes in the matter spectrum are absorbed +in 𝑃linear +geo (𝑘, 𝑧) calculated by the Boltzmann code. For this pa- +per, we only study fixed neutrino mass with � +𝜈 𝑚𝜈 = 0.06 eV. +III. +TWO-POINT CORRELATION FUNCTIONS +A. +Weak Lensing and Galaxy clustering +The dark matter distribution of the universe is traced +by two fields: i) the galaxy density field, and ii) the weak +lensing shear field. These fields generate three two-point cor- +relation functions (2PCF) as a function of angular separation 𝜃: + +4 +10 1 +100 +k [h/Mpc] +0.02 +0.02 +0.00 +0.04 +P(k)Halofit/P(k)Emulator +1 +z = 0 +z = 0.34 +z = 0.44 +0.30 +0.32 +0.34 +m +64 +66 +68 +70 +H0 +0.75 +0.80 +0.85 +8 +0.75 +0.80 +0.85 +8 +64 +66 +68 +70 +H0 +P1 + DES-Y1 ( ±) - Emul +P1 + DES-Y3 ( ±) - Emul +P1 + DES-Y1 ( ±) - Halofit +P1 + DES-Y3 ( ±) - Halofit +FIG. 3. Top panel: Fractional difference of non-linear power spec- +trum between Halofit and the Euclid Emulator at three different +redshifts; Fiducial parameter values are the same adopted in the syn- +thetic DES chains (see Sec. IV D). Bottom panel: posterior compari- +son between Halofit and the Euclid Emulator on P1 + DES-Y1/Y3 +cosmic shear combinations (P1 prior is defined in Sec. IV B). We +assumed the ΛCDM model in all four chains, and the DES-Y1/Y3 +data vector was synthetic with the same fiducial model adopted in +Sec. IV D. Both figures illustrate that within the prior ranges adopted +in this manuscript, the few percent differences between Halofit and +the Euclid Emulator do not impact our results. +Cosmic shear: +𝜉𝑖 𝑗 +± (𝜃): +the correlation, ⟨𝜅𝜅⟩, between +source galaxy shear in redshift bins 𝑖 and 𝑗. +Galaxy-Galaxy lensing 𝛾𝑖 𝑗 (𝜃): +the correlation, +⟨𝛿𝑔𝜅⟩, +between lens galaxy positions and source galaxy tangential +shear in redshift bins 𝑖 and 𝑗. +Galaxy Clustering 𝑤𝑖 𝑗 (𝜃): +the correlation, ⟨𝛿𝑔𝛿𝑔⟩, be- +tween lens galaxy position in redshift bins 𝑖 and 𝑗. +In combination, +these probes significantly increase the +information about the matter distribution and improve the +systematics mitigation. Throughout this paper, "3x2pt" refers +to the multi-probe analysis involving the combination of the +three 2PCF, and "2x2pt" refers to the multi-probe combination +of galaxy-galaxy lensing and galaxy clustering (𝛾𝑡 + 𝑤 𝜃). +Theory predictions and 2PCF are related by the angular +power spectra. In both DES-Y1 and DES-Y3, we calculate +the full non-Limber integral on large angles only in the galaxy +position auto power spectra, following Fang et al. [81]. Using +Limber approximation, the angular power spectra of tracer 𝐴 +at redshift bin 𝑖 and tracer 𝐵 at redshift bin 𝑗 is [82, 83]: +𝐶𝑖 𝑗 +𝐴𝐵(ℓ) = +∫ +𝑑𝑧 +𝑊𝑖 +𝐴(𝑧)𝑊 𝑗 +𝐵(𝑧) +𝜒2(𝑧) +𝑃split (𝑘, 𝑧) +��� +𝑘=(𝑙+1/2)/𝜒(𝑧), +(10) +where 𝜒 is the comoving radial distance. The weighting func- +tion of weak lensing shear 𝜅 and galaxy number density 𝛿𝑔 are +respectively [84] +𝑊𝑖 +𝜅 (𝜒) = +3𝐻2 +0Ωgeo +m +2c2 +𝜒 +𝑎(𝜒) +∫ +d𝜒′ 𝑛𝑖 +𝜅 (𝑧 (𝜒′)) 𝑑𝑧/𝑑𝜒′ +¯𝑛𝑖𝜅 +𝜒′ − 𝜒 +𝜒′ +(11) +and +𝑊𝑖 +𝛿g(𝜒) = 𝑏𝑖(𝑧(𝜒)) +𝑛𝑖 +g(𝑧(𝜒)) +¯𝑛𝑖g +𝑑𝑧 +𝑑𝜒 . +(12) +Here, ¯𝑛𝑖 +g/𝜅 = +∫ +𝑑𝑧 𝑛𝑖 +g/𝜅 (𝑧) is the angular number density of +galaxies in the redshift bin 𝑖, and 𝑏𝑖(𝑧(𝜒)) is the galaxy bias. +Geometryparameters modelthecomovingradial distance[72]. +Being consistent with Eq. 1, 𝑃(𝑘) ∝ (1�Ωgeo +m )2, the matter +density that appears in the 𝑊𝜅 prefactor is Ωgeo +m . This choice +mainly follows the preferences of [72]. We defer the inter- +esting investigation of how changing Ωgeo +m +→ Ωgrowth +m +here +would affect the comparison between growth and geometry +parameters. +The relation between two-point correlation functions and +angular power spectra assumes bin-average curved sky formu- +las in both DES-Y1 and DES-Y3 as shown below +𝑤𝑖 +𝜃 (𝜃) = +∑︁ +ℓ +2ℓ + 1 +4𝜋 +𝑃ℓ𝐶𝑖𝑖 +𝛿 𝛿(ℓ) , +(13) +𝛾𝑖 𝑗 +𝑡 (𝜃) = +∑︁ +ℓ +2ℓ + 1 +4𝜋ℓ(ℓ + 1) 𝑃2 +ℓ𝐶𝑖 𝑗 +𝛿E(ℓ) , +(14) +𝜉𝑖 𝑗 +± (𝜃) = +∑︁ +ℓ +2ℓ + 1 +2𝜋ℓ2(ℓ + 1)2 [𝐺+ +ℓ,2 ± 𝐺− +ℓ,2] +� +𝐶𝑖 𝑗 +𝐸𝐸 (ℓ) ± 𝐶𝑖 𝑗 +𝐵𝐵(ℓ) +� +, +The analytical expressions for Legendre and associated Leg- +endre polynomials 𝑃ℓ and 𝐺± +ℓ,2 can be found in [85]. Further +information about these transformations, including 𝐸/𝐵-mode +projections on the auto and cross power spectra involving shear, +are described in the DES-Y3 methods paper [18]. The com- +putation of non-limber integrals in galaxy position auto power +spectra, and the use of bin-average curved sky formulas for + +5 +cosmic shear, galaxy-galaxy lensing and galaxy clustering in +DES-Y1 is an improvement over modeling choices of [72]. +We use the Tidal Alignment and Tidal Torquing (TATT), +a generalization to the previously DES-Y1 adopted non-linear +alignment model (NLA-IA), to model the intrinsic alignment +of galaxies in DES-Y3 data [18, 86, 87]. Under this framework, +the intrinsic shape of galaxies is written as a collection of terms +depending on the matter overdensity, 𝛿𝑚, and the tidal tensor, +𝑠𝑖 𝑗. These terms describe tidal alignment, tidal torquing, and +density weighting, as shown below: +𝛾𝐼 +𝑖 𝑗 = +𝐶1𝑠𝑖 𝑗 +���� +Tidal Alignment ++ 𝑏TA𝐶1 +�𝛿𝑚 × 𝑠𝑖 𝑗 +� +������������������������������������ +Density Weighting ++ 𝐶2 +� +𝑠 𝑘 +𝑖 𝑠𝑘 𝑗 − 1 +3𝛿𝑖 𝑗𝑠2 +� +���������������������������������������������� +Tidal Torquing +. +(15) +Here, +𝐶1 = − 𝐴1 ¯𝐶Ωgrowth +m +𝑎𝐺growth(𝑧) +� 1 + 𝑧 +1 + 𝑧0 +� 𝜂1 +, +(16) +and +𝐶2 = 5 +𝐴2 ¯𝐶Ωgrowth +m +(𝑎𝐺growth(𝑧))2 +� 1 + 𝑧 +1 + 𝑧0 +� 𝜂2 +. +(17) +The redshift 𝑧0 is to the mean redshift of the source galaxy +sample, and ¯𝐶 = (5 × 10−14𝑀⊙ℎ−2Mpc2) × 𝜌crit. The TATT +model contains five parameters: the amplitude 𝐴𝑖=1,2, power +law index 𝜂𝑖=1,2, and the effective source galaxy bias 𝑏𝑇 𝐴. The +TATT model reduces to NLA-IA when 𝐴2 = 𝑏𝑇 𝐴 = 0. Both +NLA-IA and TATT have an explicit dependence on matter +density and the growth factor; we assume these are both growth +parameters. +IV. +DATA AND ANALYSIS METHOD +Cosmological Parameters +Prior +Ωgeo +m +Flat(0.1, 0.9) +𝑤geo +Flat(-3, -0.01) +𝐴s × 109 +Flat(1.7, 2.5) +𝑛s +Flat(0.92, 1.0) +𝐻0 +Flat(61, 73) +𝜏 +Flat(0.01, 0.8) +Ωgrowth +m +Flat(0.24 , 0.4) +𝑤growth +Flat(-1.7 , -0.7) +TABLE I. Flat priors for the cosmological parameters. +We +take the priors as in the Euclid Emulator for the parameters +(𝐴s, 𝑛s, 𝐻0, Ωgrowth +m +, 𝑤growth). +We only include the optical depth +of reionization, 𝜏, in chains with CMB data. +0.2 +0.3 +0.4 +0.5 +m +60 +70 +80 +H0 +0.5 +1.0 +1.5 +8 +0.5 +1.0 +1.5 +8 +60 +70 +80 +H0 +DES-Y1 (Cosmic Shear) +DES-Y3 (Cosmic Shear) +SN + BAO + BBN +CMBP +SN + BAO + BBN + CMBP +FIG. 4. Cosmic Shear posteriors in ΛCDM model for DES-Y1 and +DES-Y3. Unlike all remaining figures and results in this manuscript, +these constraints assume Halofit for the non-linear matter power +spectrum and the original DES-Y3 priors for the cosmological param- +eters [16]. The red dot-dashed lines are cosmological constraints from +type Ia Supernova, BAO and BBN external data. On the other hand, +the blue dashed lines show posteriors derived from the Cosmic Mi- +crowave Background temperature and polarization Planck 2018 data +with the reduced multipole range 35 < ℓ < 396 in combination with +low-ℓ EE polarization data ℓ < 30. These are the external data com- +binations adopted on priors P2 and P1, respectively (see Sec. IV B). +P1 and P2 priors are not stringent enough to create a significant 𝜎8 +tension, but at the same time, they provide complementary informa- +tion in parameters that DES does not constrain. While both priors +measure Ωm and the Hubble constant 𝐻0, only the CMB data restricts +the inflationary parameters 𝐴s and 𝑛s. As shown in ref. [72], exter- +nal (non-DES) information is necessary to tightly constrain growth +parameters. +A. +DES data +This work presents results using DES-Y1 and DES-Y3 +data; regarding DES-Y1 [72], we have implemented signifi- +cant changes in the choice of external data sets and non-linear +modeling. In both data sets, the collaboration measured 2PCF +via the TreeCorr algorithm [88]. We follow the collaboration +choices when applying scale cuts to remove small-scale infor- +mation. The resulting 3x2pt data vector contains 457 points +for DES-Y1 and 533 points for DES-Y3. +1. +Systematics in galaxy clustering and weak lensing +In this section, we summarize the systematics modelling. +We mainly follow the DES-Y3 key projects and point out the +difference between DES-Y1 and DES-Y3 [18, 87]. + +6 +DES-Y3 Nuisance Parameters +Prior +Linear Galaxy bias +𝑏𝑖𝑔(𝑖 ∈ [1, 5]) +Flat(0.8, 3.0) +Intrinsic Alignment (TATT) +𝐴1 +Flat(-5, 5) +𝐴2 +Flat(-5, 5) +𝜂1 +Flat(-5, 5) +𝜂2 +Flat(-5, 5) +𝑏𝑇 𝐴 +Flat(0 , 2) +Source photo-z +Δ𝑧1s × 102 +Gauss(0, 1.8) +Δ𝑧2s × 102 +Gauss(0, 1.5) +Δ𝑧3s × 102 +Gauss(0, 1.1) +Δ𝑧4s × 102 +Gauss(0, 1.7) +Lens photo-z +Δ𝑧1 +1 × 102 +Gauss(0.6, 0.4) +Δ𝑧2 +1 × 102 +Gauss(0.1, 0.3) +Δ𝑧3 +1 × 102 +Gauss(0.4, 0.3) +Δ𝑧4 +1 × 102 +Gauss(-0.2, 0.5) +Δ𝑧5 +1 × 102 +Gauss(-0.7, 0.1) +Multiplicative shear calibration +𝑚1 × 102 +Gauss(-0.6, 0.9) +𝑚2 × 102 +Gauss(-2.0, 0.8) +𝑚3 × 102 +Gauss(-2.4, 0.8) +𝑚4 × 102 +Gauss(-3.7, 0.8) +Lens magnification +𝐶1 +1 × 102 +Fixed (0.63) +𝐶2 +1 × 102 +Fixed (-3.04) +𝐶3 +1 × 102 +Fixed (-1.33) +𝐶4 +1 × 102 +Fixed (2.50) +𝐶5 +1 × 102 +Fixed (1.93) +Point mass marginalization +𝐵𝑖(𝑖 ∈ [1, 5]) +Flat(-5, 5) +TABLE II. Adopted priors on DES-Y3 nuisance parameters. The pri- +ors are either Flat (min, max) or Gaussian (mean, standard deviation). +Galaxy Bias: +The linear galaxy bias is parameterized +by a scalar for each redshift bin, i.e. 𝑏𝑖(𝑘, 𝑧) = 𝑏𝑖, for five +redshift bins. They are marginalized by a conservative prior +U(0.8, 3.0). We do not consider non-linear galaxy bias in our +analysis. +Intrinsic Alignment of Galaxy: +In our analysis, +we +adopt NLA for DES-Y1 and TATT for DES-Y3; +their +respective parameters are shown in Tables III and II. We fix +the pivot redshift at 𝑧0 = 0.62. +Multiplicative shear calibration: +we model the shear +calibration with a marginalized parameter 𝑚𝑖 for each redshift +DES-Y1 Nuisance Parameters +Prior +Linear Galaxy bias +𝑏𝑖𝑔(𝑖 ∈ [1, 5]) +Flat(0.8, 3.0) +Intrinsic Alignment (NLA) +𝐴1 +Flat(-5, 5) +𝐴2 +Flat(-5, 5) +Source photo-z +Δ𝑧1s × 102 +Gauss(0, 1.8) +Δ𝑧2s × 102 +Gauss(0, 1.5) +Δ𝑧3s × 102 +Gauss(0, 1.1) +Δ𝑧4s × 102 +Gauss(0, 1.7) +Lens photo-z +Δ𝑧1 +1 × 102 +Gauss(0.8, 0.7) +Δ𝑧2 +1 × 102 +Gauss(-0.5, 0.7) +Δ𝑧3 +1 × 102 +Gauss(0.6, 0.6) +Δ𝑧𝑖 +1 × 102(𝑖 ∈ [4, 5]) +Gauss(0, 0.01) +Multiplicative shear calibration +𝑚𝑖 × 102(𝑖 ∈ [1, 4]) +Gauss(1.2, 2.3) +TABLE III. Adopted priors on DES-Y1 nuisance parameters. The +priors are either Flat (min, max) or Gaussian (mean, standard devi- +ation). Note that the parameters not presented here correspond to +systematics not considered for DES-Y1 analysis. +bin, as shown below: +𝜉𝑖 𝑗 +± (𝜃) −→ �1 + 𝑚𝑖� �1 + 𝑚 𝑗� 𝜉𝑖 𝑗 +± (𝜃), +𝛾𝑖 𝑗 +𝑡 (𝜃) −→ +�1 + 𝑚 𝑗� 𝛾𝑖 𝑗 +𝑡 (𝜃). +(18) +DES-Y1 and DES-Y3 have different calibrations from simula- +tions, detailed in Tables III and II. +Photometric redshift uncertainties: +we model the un- +certainties in photometric redshift distribution for both source +and lens galaxies by a shift parameter, Δ𝑧𝑖 +𝑥, unique to each +redshift bin 𝑖, as shown below: +𝑛𝑖 +𝑥(𝑧) = ˆ𝑛𝑖 +𝑥 +�𝑧 − Δ𝑧𝑖 +𝑥 +� , +𝑥 ∈ {source, lens}. +(19) +DES-Y1 priors for Δ𝑧𝑖 +𝑥 differ from DES-Y3 priors and both +are shown in Tables III and II. We do not model stretches in +the photometric redshift distribution of lens galaxies by an +additional free parameter 𝜎𝑖 +𝑧 as in the DES-Y3 key project. +Lensing magnification: +As detailed in [18], a parame- +ter 𝐶𝑖 +𝑙 is defined to describe the foreground mass effects on +the observed number density of lens galaxies. The parameter +is calibrated from data for each redshift bin and held fixed +in our analysis as shown in Table II. This systematic is not +considered for the DES-Y1 data set in this paper. +Non-local effects in galaxy-galaxy lensing: +for DES- +Y3 specifically, we follow the marginalization approach +developed in MacCrann et al. [89] and we adopt an informa- +tive prior of the point-mass parameter 𝐵𝑖 ∈ Flat(−5, 5). Such + +7 +systematic is not considered for the DES-Y1 data set in this +paper. +𝑋lens factor: +A non-physical parameter 𝑋lens was pro- +posed in DES-Y3 to solve the internal inconsistency between +galaxy-galaxy lensing and galaxy clustering 2PCF [16]. The +two lensing samples, redMaGiC and MagLim, show discrep- +ancies between the galaxy bias inferred from galaxy-galaxy +lensing and galaxy clustering in the DES-Y3 analysis [57, 90]. +In this work, we adopt redMaGiC lensing sample with +fixed 𝑋lens = 1 for both DES-Y1 and DES-Y3. +We plan +to follow up this work with a detailed comparison between +redMaGiC and MagLim, including marginalization over +𝑋lens parameter and recent changes to the redMaGiC color +selection algorithm [57]. +B. +Priors and External Data +The split between growth and geometry information is +not unique. +Within our choices, we select external probes +so that DES-Y3 is the only constraining data set on growth +parameters besides the boundaries of validity of the Euclid +Emulator. We don’t present DES-only chains as in [72], since +they have shown that DES needs to be combined with external +data to provide useful constraining power on the difference +between geometry and growth parameters. We combine DES +with external data described below: +CMBP: Planck 2018 low-ℓ EE polarization data (ℓ < +30) in combination with the high-ℓ TTTEEE spectra truncated +right after the first peak (35 < ℓ < 396). +Our choice +removes late-time Integrated Sachs Wolfe information. It also +removes CMB lensing effects as the CMB lensing-induced +smoothing on the temperature power spectrum only affects +constraints on cosmological parameters when including +higher acoustic peaks. We find this prior complementary to +the compressed CMB likelihood adopted [72]. +Our CMB +choices are slightly more conservative on 𝑛s and Ωbℎ2, but +they do constrain the early-Universe inflationary amplitude 𝐴s. +SNIa: +Pantheon Type Ia supernovae sample [91]. +Type +Ia supernovae are a constraint on geometry parameters only; +their likelihood does not require knowledge of the large-scale +structure. There are, however, lensing magnification effects +on the Hubble diagram [92–94] and growth effects on SNIa +peculiar velocity distribution [95, 96] that will need to be +taken into account in future Stage IV surveys; for now, we +disregard modeling these growth effects. +BBN: We use derived constraint on baryon density from +astrophysical probe: 100Ωbℎ2 = 2.208 ± 0.052 [97]. +BAO: We use Baryon Acoustic Oscillation data from +the SDSS DR7 main galaxy sample [98] in combination with +the 6dF galaxy survey [99] at 𝑧eff = 0.15 and 𝑧eff = 0.106 +respectively, and the SDSS BOSS DR12 low-z and CMASS +combined galaxy samples at 𝑧 = 0.38, 0.51, 0.61 [100]. +These constraints come from comparing the observed scale +of the BAO feature and the sound horizon. +As a distance +measurement of the late universe, we consider BAO to be +pure geometry information. +To better understand the effects of these external data sets +on the final results, we adopt the following three sets of priors: +Prior 1 (P1): Emulator prior + CMBP +Prior 2 (P2): Emulator prior + SNIa + BAO + BBN +Prior 3 (All): +Emulator prior + CMBP + SNIa + BAO ++ BBN +Table I summarizes our adopted informative priors on +the cosmological parameters. +Figure 4 compares DES- +Y1/Y3-only chains with the uninformative priors adopted by +the DES collaboration against our P1 and P2 priors. +This +figure assumes a ΛCDM model and Halofit for the non-linear +matter power spectrum. +Our priors are consistent with +DES-only posteriors in all parameters, including 𝜎8. Since +the SNIa + BAO + BBN combination does not provide any +information on inflationary parameters, the only limits on +𝐴s and 𝑛s in P2 comes from the Euclid Emulator bounds. +Therefore, comparing DES + P1 against DES + P2 chains +offers valuable information on how internal DES tensions that +shift 𝐴s and 𝑛s affect our results on growth parameters. +Figures 5 and 6 show that the DES-Y1 and DES-Y3 𝜒2 dis- +tributions are nearly independent of prior P1/P2/All choices +in both split ΛCDM and 𝑤CDM models. The priors are broad +enough not to impact the model’s DES 𝜒2 fit, except for the +DES-Y3 2x2pt in ΛCDM split. As we will see, the internal +tensions on DES-Y3 2x2pt shift the inflationary parameters to +values inconsistent with the CMB prior in both ΛCDM and +𝑤CDM splits. In ΛCDM, ΔΩm cannot restore the goodness- +of-fit; there is a Δ𝜒2 ≈ 5 difference between DES-Y3 2x2pt + +P1 and DES-Y3 2x2pt + P2. Interestingly, ΔΩm and Δ𝑤 can +correct DES-Y3 2x2pt + P1 fit in 𝑤CDM split. +C. +Pipeline +We perform the MCMC analysis using Cocoa, the Cobaya- +Cosmolike Architecture [101]. +Cocoa is a modified ver- +sion of CosmoLike [102] multi-probe analysis software in- +corporated into the Cobaya framework [103]. DES-Y1 and +DES-Y3 covariance matrices were computed using Cosmo- +Cov [104]. +CosmoCov and Cocoa are both derived from +Cosmolike [102], the former pipeline computes covariance +matrices, and the latter evaluates data vectors. CosmoLike +within Cocoa has efficient OpenMP shared-memory paral- +lelization [105] and cache system compatible with the slow- +fast decomposition implemented in the default Cobaya Monte- +Carlo Markov chain sampler (MCMC) The OpenMP efficiency +in CosmoLike is around 50%, i.e., quadrupling the number of +OpenMP cores halves CosmoLike runtime. +CosmoLike has been used in both DES-Y1/Y3 multi-probe +analyses when constraining ΛCDM parameters [18, 87] and + +8 +230 +235 +240 +2 +Y1 ± + CDM +227 +250 +260 +270 +280 +2 +Y1 2x2pt + CDM +230 +300 +310 +320 +2 +Y1 ± + w + CDM +281 +430 +440 +450 +2 +Y1 ± + t + CDM +403 +500 +510 +520 +2 +Y1 3x2pt + CDM +457 +P1 +P2 +All +230 +240 +250 +2 +Y3 ± + CDM +227 +340 +350 +360 +370 +380 +2 +Y3 2x2pt + CDM +306 +300 +310 +320 +330 +2 +Y3 ± + w + CDM +281 +520 +530 +540 +550 +2 +Y3 ± + t + CDM +479 +620 +630 +640 +650 +660 +2 +Y3 3x2pt + CDM +533 +P1 +P2 +All +FIG. 5. The DES 𝜒2 distribution of different combinations of two-point correlation functions in split-ΛCDM chains. The number of data +points is printed in the upper left of each panel, after masking us applied for DES-Y1 and DES-Y3, respectively. This plot demonstrates that the +P1, P2 and All data priors and external data combinations do not degrade the DES fit except for the DES-Y3 2x2pt. This anomalous data vector +combination predicts a small inflationary amplitude, 𝐴s, in ΛCDM, incompatible with CMB data [16]. The cosmic shear cross-correlation +reduces this problem considerably on the 3x2pt fit. However, the detailed comparison between 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 against 3x2pt stands out. +Both 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 combinations show virtually no 𝜒2 changes between all three priors; the same is not true for 3x2pt. +230 +235 +240 +2 +Y1 ± + wCDM +227 +250 +260 +270 +280 +2 +Y1 2x2pt + wCDM +230 +300 +310 +320 +2 +Y1 ± + w + wCDM +281 +430 +440 +450 +2 +Y1 ± + t + wCDM +403 +500 +510 +520 +530 +2 +Y1 3x2pt + wCDM +457 +P1 +P2 +All +230 +240 +250 +2 +Y3 ± + wCDM +227 +340 +350 +360 +370 +380 +2 +Y3 2x2pt + wCDM +306 +300 +310 +320 +330 +2 +Y3 ± + w + wCDM +281 +520 +530 +540 +550 +2 +Y3 ± + t + wCDM +479 +620 +630 +640 +650 +660 +2 +Y3 3x2pt + wCDM +533 +P1 +P2 +All +FIG. 6. The DES 𝜒2 distribution of different combinations of two-point correlation functions in split-𝑤CDM chains. The number of data points +is printed in the upper left of each panel, after masking us applied for DES-Y1 and DES-Y3, respectively. This plot demonstrates that the P1, +P2 and All data priors and external data combinations do not degrade the DES fit. The 2x2pt data vector anomalous data vector combination +prefers a small inflationary amplitude, 𝐴s in the absence of CMB external data, see Fig. 13. However, growth parameters can restore the DES +2x2pt goodness-of-fit when 𝐴s is set by the CMB prior, unlike what we observe in the ΛCDM split, see Fig. 5. Indeed, Figure 13 shows that +DES-Y3 2x2pt data has higher detection of ΔΩm − Δ𝑤 < 0 when combined with CMB data. Unfortunately, both ΛCDM and w-CDM splits +have similar goodness-of-fit on the 3x2pt chains that incorporate cosmic shear, including the slight loss of fit when combining DES and CMB +data, and ΔΩm − Δ𝑤 is consistent with zero. +for calibrating Bayesian evidences [106]. It has also been used +in forecast studies for Rubin Observatory’s LSST and Roman +Space Telescope [107–110]. +We compute the linear power spectrum with the CAMB +Boltzmann code [111, 112]; Cobaya already had implemen- +tations of all external data sets. We adopt Cobaya’s default +adaptive metropolis hasting MCMC sampler, and we employ +the Gelman-Rubin criteria 𝑅 − 1 < 0.02 to establish chain +convergence [113]. We post-process chains and creat figures +using GetDist [114]. +Changes in the growth parameters are only semi-fast; they +do not require CAMB to recompute distances and matter power +spectrum; only CosmoLike must be rerun to update the DES +data vectors. Due to an efficient cache system, CosmoLike +reruns with only modified growth parameters takes about half +the runtime compared with when all parameters are varied. +CAMB and CosmoLike runtimes are roughly equal; the time +ratio between slow and semi-fast parameters is, therefore, ap- +proximately 4:1. The 3x2pt data vector evaluation time with 10 +OpenMP cores is of order 1.5s on modern AMD EPYC 7642 +48-core nodes. This estimation includes CAMB evaluation, +non-limber integration, and TATT modeling. Finally, code +comparisons between the CosmoSiS pipeline and Cosmolike +were presented in [18, 87]. + +9 +0.30 +0.31 +0.32 +0.33 +geo +m +0.26 +0.30 +0.34 +0.38 +growth +m +0.75 +0.80 +0.85 +split +8 +0.75 +0.80 +0.85 +split +8 +0.26 +0.30 +0.34 +0.38 +growth +m +SYNTHETIC + DES + DATA +All + DES-Y3 (Cosmic Shear) +All + DES-Y3 ( ± + t) +All + DES-Y3 ( ±+ w ) +All + DES-Y3 (3x2pt) +FIG. 7. Posteriors derived from different combinations of synthetic +DES-Y3 2PCFs in the split ΛCDM model. As described in Sec. IV B, +the All external data combination consists of CMBP + SNIa + BAO ++ BBN, with CMBP being Planck 2018 low-ℓ EE polarization data +and the high-ℓ TTTEEE spectra truncated right after the first peak +(35 < ℓ < 396). Prior to the growth parameter 0.24 ≤ Ωgrowth +m +≤ 0.4 +is compatible with the Euclid Emulator boundaries. All posteriors +are prior limited, but the plot clarifies the gain in constraining power +when galaxy-galaxy lensing and galaxy-clustering are added to cos- +mic shear. +D. +Validation on Synthetic Data +In +this +section, +we +generate +a +synthetic +noiseless +ΛCDM data vector from Planck best fit cosmological pa- +rameters without lensing: +� +𝐴s ×10−9, 𝑛s, 𝐻0, Ωm, Ωb +� += +� +2.101, 0.965, 67.32, 0.317, 0.049 +� +. +This set of parameters +is compatible with both P1 and P2 priors [115]. +We run +MCMCs, including all nuisance parameters, and see if the +posterior would give equal growth and geometry parameters +at the fiducial value. +1. +Comparison between Cosmic Shear and 3x2pt +Assuming the All prior and external data combination, +there is a significant improvement on Ωgrowth +m +constraints in +ΛCDM split when going from cosmic shear to 3x2pt, as +shown in Fig. 7. In the 3x2pt case, the Ωgrowth +m +posterior is +well centered at the fiducial value, while cosmic shear pro- +vides only marginal improvements compared with the uniform +0.24 < Ωgrowth +m +< 0.4 prior. This narrow prior is informa- +tive in both chains, the boundary coming from the range of +the Euclid Emulator. Improving the small-scale modeling +validity of cosmic shear and 2x2pt may tighten the 95% confi- +dence level of Ωgrowth +m +enough to be within the allowed range; +0.30 +0.31 +0.32 +0.33 +geo +m +0.26 +0.30 +0.34 +0.38 +growth +m +0.75 +0.80 +0.85 +split +8 +0.75 +0.80 +0.85 +split +8 +0.26 +0.30 +0.34 +0.38 +growth +m +SYNTHETIC + DES + DATA +All + DES-Y1 (Cosmic Shear) +All + DES-Y3 (Cosmic Shear) +All + DES-Y1 (3x2pt) +All + DES-Y3 (3x2pt) +FIG. 8. Split ΛCDM posteriors derived from cosmic shear and 3x2pt +combined with the All external data combination. +As described +in Sec. IV B, the All external data combination consists of CMBP ++ SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE +polarization data and the high-ℓ TTTEEE spectra truncated right +after the first peak (35 < ℓ < 396). DES-Y1 3x2pt error bar on +Ωgrowth +m +is approximately 17% larger compared with DES-Y3. On +the other hand, cosmic shear DES-Y1 and DES-Y3 constraints are +similar, and both are prior dominated. +[116] and [117] offer a roadmap on how to implement such +improvements in future work. +DES 3x2pt combinations with P1 and All external data +show nearly identical constraining power on Ωgrowth +m +. Com- +bined priors on the primordial power spectrum (amplitude and +shift) and the shape parameter Γ ≡ Ωmℎ are the needed exter- +nal information so that DES can tightly measure growth. The +CMBP data alone provides both information while the SNIa ++ BAO + BBN measurements on Ωgeo +m +and 𝐻0 only restrict +the shape parameter. Adding more CMB multipoles would +improve constraints on early-Universe parameters even more. +However, CMB temperature and polarization power spectra +are more sensitive to lensing effects on smaller scales, which +would limit our ability to compare DES effects on growth pa- +rameters against CMB lensing. Partial delensing can alleviate +this limitation [118]. Another possibility is to consider all +multipoles up to ℓmax ≈ 1600 where effects from nonlinear +dark matter collapse are negligible. In this case, however, we +would marginalize the chains over lensing principal compo- +nents so that there is no leakage of information on growth +parameters [119]. +In the 𝑤CDM split model, Ωgrowth +m +and 𝑤growth are not well +constrained even in the most informative 3x2pt case. +We +then show real data constraints on the principal component +combination +PC1 = −0.7071Δ𝑤 + 0.7071ΔΩm. +(20) + +10 +In both cosmic shear and 3x2pt chains, PC1 constraints are +prior dominated but well centered around zero. +2. +Comparison between DES-Y1 and DES-Y3 +In all three combinations with external data, posteriors on +Ωgrowth +m +in the ΛCDM split from DES-Y1 and DES-Y3 cosmic +shear are similar, despite the additional nuisance parameters +introduced by the TATT intrinsic alignment model in DES- +Y3 (see Fig. 8). We have yet to check if we can obtain more +constraining power on growth parameters by adopting the more +straightforward NLA model on DES-Y3. On the other hand, +the error bar on Ωgrowth +m +derived from 3x2pt combined with +the All prior is 17% larger in DES-Y1. One caveat to this +result is that we have not tested whether expanding the adopted +priors on point mass marginalization to the more conservative +range Flat(−100, 100) would significantly degrade DES-Y3 +constraints. +The first principle component PC1, defined on Eq. 20, has +nearly identical and prior dominated DES-Y1 and DES-Y3 +posteriors in the 𝑤CDM split. Additional information from ei- +ther smaller scales in the 3x2pt data vector or external growth +information from CMB lensing and RSD are potential oppor- +tunities in future analyses. Figure 2 shows that Ωgrowth +m +and +𝑤growth induce changes on 𝜎split +8 +(𝑧) with different redshift evo- +lution. Including high redshift 𝑧 > 1 lensing samples from +the future Roman Space Telescope may therefore be the key to +disentangling growth parameters in 𝑤CDM split [109]. +There are near-future possibilities that may expand the red- +shift range adopted in this paper. RedMaGiC fifth bin, with +range 0.8 < 𝑧 < 0.9, shows large 𝑋lens biases [117]. The al- +ternative DES-Y3 MagLim sample of lens galaxies does have +an additional redshift bin in the range 0.95 < 𝑧 < 1.05 not ac- +cessible by redMaGiC [20]. However, MagLim high redshift +bins were not adopted in the 3x2pt analysis by the DES col- +laboration and may require further studies on the presence of +potential systematic biases [16]. Finally, there is the emergent +idea of using the same galaxy sample for both clustering and +lensing that could potentially expand DES-Y3 constraints on +𝜎split +8 +(𝑧) beyond 𝑧 > 1 [120]. +V. +RESULTS +We split our results section into three components: start- +ing with a discussion of our results in the ΛCDM parameter +space, we then move to the 𝑤CDM space, and conclude with +quantifying tensions between different probe combinations in +the context of both parameter spaces. +A. +Growth-geometry split results in ΛCDM +For the most constraining probe combination, +DES +3x2pt+All, we show the DES-Y1 and DES-Y3 ΛCDM re- +sults in Fig. 9. In both cases we find no measurable detection +0.29 +0.30 +0.31 +0.32 +geo +m +0.0 +0.1 +m +0.75 +0.80 +0.85 +split +8 +0.75 +0.80 +0.85 +split +8 +0.0 +0.1 +m +All + DES-Y1 (3x2pt) +All + DES-Y3 (3x2pt) +FIG. 9. Split ΛCDM posteriors derived from 3x2pt DES-Y1 and DES- +Y3 data. As described in Sec. IV B, the All external data combination +consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck +2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra +truncated right after the first peak (35 < ℓ < 396). DES-Y1 3x2pt +error bar on growth dark matter density is approximately 10% larger +compared with DES-Y3. +. +of ΔΩm being different from zero. This constitutes the main, +fiducial result of this paper. +We explore subsets of the 3x2pt probe combination in +Fig. 10, where the left panel refers to DES-Y1 and the right +corresponds to DES-Y3. For DES-Y1 we find that in all cases +ΔΩm is compatible with zero even within one sigma. +For +DES-Y3, however, we see shifts from ΔΩm = 0, especially in +the 2x2pt (galaxy clustering+galaxy galaxy lensing) case. +We consider this further in Fig. 11, where we show the one- +dimensional posterior distributions on all relevant ΛCDM split +parameters, finding that except for DES-Y3 𝜉±+P2, 𝜉± + 𝑤 𝜃 +and 2x2pt chains, all combinations of two-point correlation +functions predict ΔΩm compatible with zero within one sigma. +The deviation on 𝜉± + P2 is less than two-sigma. Similarly, +all combinations between DES 2PCFs and the P2 external +data predict 𝐴s and 𝑛s values compatible with CMB data on +P1/All, except for DES-Y3 𝜉± + 𝑤 𝜃 and 2x2pt. +Similarly to what we observe in ΛCDM chains with syn- +thetic data vectors, DES-Y1 and DES-Y3 cosmic shear provide +little information on ΔΩm even with the P1/P2/All priors. +The additional nuisance parameters introduced by the TATT +intrinsic alignment model and point mass marginalization in +DES-Y3 do not reduce constraining power for the growth pa- +rameters. The situation in the 3x2pt chains is different; the +DES-Y1 3x2pt + All error bars are 10% larger than in DES- +Y3, not that far from the predicted 17% improvement in the + +11 +0.29 +0.30 +0.31 +0.32 +geo +m +0.0 +0.1 +m +0.75 +0.80 +0.85 +split +8 +0.75 +0.80 +0.85 +split +8 +0.0 +0.1 +m +All + DES-Y1 ( ±) +All + DES-Y1 ( ± + t) +All + DES-Y1 ( ± + w ) +All + DES-Y1 (2x2pt) +0.29 +0.30 +0.31 +0.32 +geo +m +0.0 +0.1 +m +0.70 +0.75 +0.80 +0.85 +split +8 +0.70 +0.75 +0.80 +0.85 +split +8 +0.0 +0.1 +m +All + DES-Y3 ( ±) +All + DES-Y3 ( ± + t) +All + DES-Y3 ( ± + w ) +All + DES-Y3 (2x2pt) +FIG. 10. Split ΛCDM posteriors derived from multiple 2PCF combinations in DES-Y1 (left panel) and DES-Y3 (right panel). As described in +Sec. IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization +data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396). Right panel shows that the DES-Y3 𝜉± + 𝛾𝑡, 𝜉± + 𝑤(𝜃) +and 2x2pt all prefer lower values for the Ωgrowth +m +with upper limits at 95% confidence level being 0.375, 0.314 and 0.288 respectively. We +emphasize that the apparent constraints at ΔΩm ≡ Ωgrowth +m +− Ωgeo +m +≈ −0.8 is due to effective priors. +synthetic noise-free chains. Priors on Ωgrowth +m +are still informa- +tive, but to a much lesser degree on both DES-Y1 and DES-Y3 +3x2pt compared with their cosmic shear counterpart. +All of the DES-Y1 ΛCDM split chains are compatible with +ΔΩm = 0; Figs. 10 (left panel) and 11 show large consis- +tency between parameter posteriors derived from all 2PCFs +combinations. There are also no appreciable parameter shifts +between chains with and without CMB priors; goodness-of-fit +is identical in these chains (see Fig. 5). As expected, 𝐴s and 𝑛s +constraints are significantly tighter when CMB data is present. +Finally, chains that include galaxy clustering (3x2pt, 2x2pt, +and 𝜉± + 𝑤 𝜃) show a small shift towards ΔΩm > 0, but are still +compatible with zero at 68% confidence level. +For DES-Y3 we see that the 𝜉±+𝑤 𝜃 and 2x2pt chains predict, +in combination with the All prior, ΔΩm ≠ 0 at 1.75𝜎 and +2.60𝜎 in statistical significance (see Fig. 10). We attribute +these findings to the well-known incompatibilities between +galaxy clustering and galaxy-galaxy lensing in DES-Y3 when +using the redMaGiC lens sample. +B. +Growth-geometry split results in 𝑤CDM +For the 𝑤CDM parameter space we summarize our results +in Figs. 12 and 13, where the former again shows selected +results in two dimensions and the latter summarizes all chains +in one-dimensional projections. Qualitatively, we see similar +behaviour as in the ΛCDM case. While DES cosmic shear and +3x2pt data shows ΔΩm − Δ𝑤 being consistent with zero, the +picture becomes more complicated when considering subsets +of the 3x2pt case that involve galaxy clustering of redMaGiC. +In particular, the 2x2pt + All chain favors ΔΩm − Δ𝑤 < 0 +at 4.48𝜎, higher than any ΔΩm ≠ 0 detection in ΛCDM split. +The 𝑤CDM split 2x2pt + All chain also predict quite low +𝜎split +8 += 0.682 ± 0.0243, while in ΛCDM we have 𝜎split +8 += +0.730 ± 0.1813. +While a 4.48𝜎 detection is significant, we again refrain from +claiming new physics in the 𝑤CDM model space, due to the +aforementioned problems with the DES-Y3 redMaGiC sam- +ple. Instead, we plan to further investigate growth-geometry +split with alternative lens samples and when marginalizing +over 𝑋lens. +C. +Quantifying tensions between probes +1. +Method +To evaluate the tension we use the parameter difference +method [121, 122]. +Given two chains 𝜃1 and 𝜃2 and their +corresponding posteriors P1(𝜃1) and P2(𝜃2), begin by com- +puting the difference between these two chains, denoted with +Δ𝜃 = 𝜃1 − 𝜃2. +Using this difference chain we can write +P2(𝜃2) = P2(𝜃1 − Δ𝜃). +By marginalizing over 𝜃1 we get + +12 +0.65 +0.75 +0.85 +split +8 +Y1 2x2pt +Y3 2x2pt +Y1 ± + w +Y3 ± + w +Y1 ± + t +Y3 ± + t +Y1 3x2pt +Y3 3x2pt +Y1 ± +Y3 ± +1.7 +1.9 +2.1 +As +1e 9 +0.94 +0.96 +0.98 +ns +0.29 +0.31 +0.33 +geo +m +0.06 +0.00 +0.06 +m +CDM +All + P1 + P2 +FIG. 11. +One-dimensional posteriors in split ΛCDM for various DES-Y1 and DES-Y3 two-point correlation functions, with error bars +corresponding to marginalized 68% confidence intervals. As described in Sec. IV B, the All external data combination consists of CMBP ++ SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the +first peak (35 < ℓ < 396). The P1 external data combination is restricted to CMBP, while P2 is SNIa + BAO + BBN. Priors on cosmological +parameters are summarized in Table I; we define ΔΩm ≡ Ωgrowth +m +− Ωgeo +m . The grey background separates our primary results from other probe +combinations. +the parameter difference posterior, +P(Δ𝜃) = +∫ +𝑉Π +P1(𝜃1)P2(𝜃1 − Δ𝜃)𝑑𝜃1 , +(21) +where 𝑉Π is the subset of the domain covered by the prior. As +𝜃1 → 𝜃2, the means of each chain approach equality and the +mean of the parameter difference chain approaches 0. Thus +the volume of the regions with P(Δ𝜃) > P(0) approaches 0, +so we can approximate the tension using +Δ = +∫ +P(Δ𝜃)>P(0) +P(Δ𝜃)𝑑Δ𝜃 . +(22) +This volume is interpreted as a probability of parameter shift, +denoted Δ. +If Δ comes from a Gaussian distribution, the +number of standard deviations from 0 is given by +𝑛𝜎 = +√ +2Erf−1(Δ) . +(23) +The resulting 𝑛𝜎 is reported. +To estimate the posterior we use Masked Autoregressive +Flows (MAFs) [121, 123], which is a neural network that learns +an invertible mapping from an arbitrary parameter space to a +gaussianized one. The loss function for MAFs is the negative +log probability from a unit Gaussian. Due to the autoregressive +property, the Jacobian is triangular and thus the determinant +is tractable to compute even for a large number of dimensions. +Thus we can estimate the posterior as a reparameterization of +a Gaussian and find the log-probability of arbitrary points. +Before training the neural network, we follow the imple- +mentation in ref. [121] to apply a linear transformation to Δ𝜃 +given from the Gaussian approximation for P(Δ𝜃) +Δ𝜃′ = 𝐶−1(Δ𝜃 − 𝜇) , +(24) +with 𝐶 the covariance and 𝜇 the mean of P(Δ𝜃), then map Δ𝜃′ +to the fully Gaussianized parameter space. This enhances the +convergence rate of the neural networks. Denoting the learned +mapping as 𝜙(Δ𝜃′) = 𝑦 and the unit Gaussian density as N, + +13 +0.30 +0.32 +geo +m +0.3 +0.0 +0.3 +w+ +m +1.1 +1.0 +wgeo +0.80 +0.90 +split +8 +0.80 +0.90 +split +8 +1.1 +1.0 +wgeo +0.3 +0.0 +0.3 +w+ +m +All + DES-Y1 (3x2pt) +All + DES-Y3 (3x2pt) +0.30 +0.32 +geo +m +0.3 +0.0 +0.3 +w+ +m +1.1 +1.0 +wgeo +0.70 +0.80 +split +8 +0.70 +0.80 +split +8 +1.1 +1.0 +wgeo +0.3 +0.0 +0.3 +w+ +m +All + DES-Y3 ( ±) +All + DES-Y3 ( ± + t) +All + DES-Y3 ( ± + w ) +All + DES-Y3 (2x2pt) +FIG. 12. Split 𝑤CDM posteriors derived from 3x2pt (left panel) and multiple 2PCF combinations (right panel). As described in Sec. IV B, +the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and +the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396). Table I presents the priors on the cosmological parameters; we +define ΔΩm ≡ Ωgrowth +m +− Ωgeo +m +and Δ𝑤 ≡ 𝑤growth − 𝑤geo. All constraints on the combination ΔΩm − Δ𝑤 are prior dominated given the range +limitations of Ωgrowth +m +. +we can then relate the log-probability as +P(Δ𝜃) = N (𝑦) | det(𝐽𝜙(Δ𝜃′))| +| det(𝐶)| +(25) +where 𝐽𝜙 denotes the Jacobian of 𝜙. +To compute the integral in Eq. 22 we use Monte Carlo +integration. Using the MAF we randomly sample from the +posterior and calculate the log probability. The fraction of +generated points that land in the region P(Δ𝜃) > P(0) are +counted. The error of the numerical integration is given by the +Clopper-Pearson interval for a binomial distribution. +2. +Results +We evaluate tensions between different DES 2PCF com- +binations employing the parameter difference method on +� +𝐴s, 𝑛s, 𝐻0, Ωgeo +m , 𝜎split +8 +(𝑧 = 0) +� +set of cosmological param- +eters, with an addition of 𝑤geo in the split 𝑤CDM model. +As a caveat, this metric does not model the existing corre- +lations between the 2PCFs; the precise computation requires +MCMC chains with repeated parameters, which is beyond our +computational capabilities [121]. Figure 14 qualitatively in- +dicates discrepancies; we see, for example, the well-known +redMaGiC problems between 2x2pt and other probe com- +binations. Future utilization of machine learning emulators +will allow the more precise calculation of tensions between +the correlated DES 2PCFs with modest computational re- +sources [124]. +Interestingly, 𝐴s appears to be the culprit of the observed +tensions above two sigmas between 2x2pt and the remaining +combinations of the DES-Y3 data vector. The highest observed +tension in split ΛCDM happens between 2x2pt + P2 and 𝜉± + +𝑤 𝜃 + P1, entirely due to shifts on 𝐴s as both chains favors +ΔΩm < 0. +Figure 5 reveals that CMB priors degrade the +goodness of fit to DES-Y3 2x2pt data by Δ𝜒2 ≈ 5. In all +other DES-Y3 2PCFs, swapping P1 with P2 priors does not +affect 𝜒2 nearly as much. However, the detailed comparison +between 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 against 3x2pt stands out. Both +𝜉±+𝑤 𝜃 and 𝜉±+𝛾𝑡 combinations show virtually no 𝜒2 changes +between all three priors;the same is not true for 3x2pt as there +is a Δ𝜒2 ≈ 1.21 degradation on DES goodness-of-fit when +CMB data is present. +The behavior in 𝑤CDM split is different; growth parameters +can restore the DES 2x2pt goodness-of-fit when 𝐴s is set by the +CMB prior, as shown in Fig. 6. The 𝐴s tension between DES- +Y3 2x2pt + P2 and DES-Y3 2x2pt + P1/All is also smaller on +𝑤CDM when compared with ΛCDM split. The DES-Y3 2x2pt +predicts nonzero values for the principal component ΔΩm−Δ𝑤 +for all external data combinations, the more extreme deviation +from zero happening on DES-Y3 2x2pt + All chain. +The +better fit to DES 2x2pt makes such nonzero detection more +meaningful than the ΛCDM split model. +Finally, the left and right panels on Fig. 14 show that DES- +Y3 2x2pt + P1/All chains have higher tension levels against +other 2PCFs than DES-Y3 2x2pt + P2, the opposite of what we +observe in ΛCDM split. Indeed, when cosmic shear is added +to 2x2pt, the predicted 𝜎split +8 +value shifts by more than three + +14 +0.6 +0.7 +0.8 +0.9 +split +8 +Y1 2x2pt +Y3 2x2pt +Y1 ± + w +Y3 ± + w +Y1 ± + t +Y3 ± + t +Y1 3x2pt +Y3 3x2pt +Y1 ± +Y3 ± +1.7 +1.9 +2.1 +As +1e 9 +0.94 +0.96 +0.98 +ns +0.27 +0.30 +0.33 +geo +m +0.06 +0.00 +0.06 +m +1.2 +1.0 +0.8 +wgeo +0.25 +0.00 +0.25 +w+ +m +wCDM +All + P1 + P2 +FIG. 13. +One-dimensional posteriors in split 𝑤CDM for various DES-Y1 and DES-Y3 two-point correlation functions, with error bars +corresponding to marginalized 68% confidence intervals. As described in Sec. IV B, the All external data combination consists of CMBP ++ SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the +first peak (35 < ℓ < 396). The P1 external data combination is restricted to CMBP, while P2 is SNIa + BAO + BBN. Priors on cosmological +parameters are summarized in Table I; we define ΔΩm ≡ Ωgrowth +m +−Ωgeo +m and Δ𝑤 ≡ 𝑤growth −𝑤geo. The grey background separates our primary +results from other probe combinations. As could be expected, the DES-Y3 2x2pt + P2 predicts lower values for the inflationary amplitude 𝐴s +incompatible with CMB priors. The DES-Y3 2x2pt also predicts non-zero values for the principal component ΔΩm − Δ𝑤 for all external data +combinations; the more extreme deviations being DES-Y3 2x2pt + All with mean −0.296 and standard deviation 0.066. +sigmas. Unfortunately, both Λ and 𝑤CDM split models have +similar DES 3x2pt goodness-of-fit; growth parameters can’t +alleviate the incompatibility between galaxy-galaxy lensing +and galaxy clustering in the 3x2pt chains (see Fig. 5 and 6). +VI. +CONCLUSIONS +This paper studies the growth-geometry split with DES- +Y1 and DES-Y3 data in combination with external data sets. +We utilize the Cobaya-CosmoLike Architecture (Cocoa) +software to efficiently run a large number of MCMC chains +that allow us to explore the variation of results for different +probes and prior combinations. +For DES-Y1 we find that ΔΩm in ΛCDM and ΔΩm − Δ𝑤 in +𝑤CDM are both consistent with 0 for all permutations of DES +2PCFs and external prior combinations. +In the case of DES-Y3, we find that cosmic shear and 3x2pt +results are consistent with equal geometry and growth parame- +ters. Combining cosmic shear and galaxy-galaxy lensing also +does not indicate deviations between growth and geometry pa- +rameters. However, both the 𝜉±+𝑤 𝜃 and 𝛾𝑡 +𝑤 𝜃 combinations +of 2PCF indicate ΔΩm < 0 in ΛCDM and ΔΩm − Δ𝑤 < 0 in +𝑤CDM splits. These results hold with both P1 and P2 pri- +ors, which is interesting as they predict different values for the +primordial power spectrum amplitude 𝐴s. In light of the well- +known DES-Y3 problems of the redMaGiC sample, we do not +interpret these results as a detection but rather assume that it is +a residual of unsolved systematics. We plan to further explore +this with alternative lens samples, in particular the MagLim + +15 +CS,P1 +CS,P2 +CS,All +3x2pt,P1 +3x2pt,P2 +3x2pt,All +CS+ +t,P1 +CS+ +t,P2 +CS+ +t,All +CS+w ,P1 +CS+w ,P2 +CS+w ,All +2x2pt,P1 +2x2pt,P2 +2x2pt,All + CS,P1 + CS,P2 + CS,All + 3x2pt,P1 + 3x2pt,P2 + 3x2pt,All + CS+ +t,P1 + CS+ +t,P2 + CS+ +t,All + CS+w ,P1 + CS+w ,P2 + CS+w ,All + 2x2pt,P1 + 2x2pt,P2 + 2x2pt,All +CDM +Y3 +Y1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +CS,P1 +CS,P2 +CS,All +3x2pt,P1 +3x2pt,P2 +3x2pt,All +CS+ +t,P1 +CS+ +t,P2 +CS+ +t,All +CS+w ,P1 +CS+w ,P2 +CS+w ,All +2x2pt,P1 +2x2pt,P2 +2x2pt,All + CS,P1 + CS,P2 + CS,All + 3x2pt,P1 + 3x2pt,P2 + 3x2pt,All + CS+ +t,P1 + CS+ +t,P2 + CS+ +t,All + CS+w ,P1 + CS+w ,P2 + CS+w ,All + 2x2pt,P1 + 2x2pt,P2 + 2x2pt,All +wCDM +Y3 +Y1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +FIG. 14. Tensions between chains under split ΛCDM (left panel) and 𝑤CDM (right panel) models. As described in Sec. IV B, the P1 external +data combination is composed of Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak +(35 < ℓ < 396). On the other hand, P2 is the combination of SNIa + BAO + BBN. The 2x2pt + P2 chain is the only one with significant tension, +against other DES-Y3 2PCFs, in ΛCDM split; the inflationary amplitude 𝐴s seems to be the culprit of the observed tensions. When 2x2pt is +combined with either P1 or All priors, we see lower tensions at the expense of degradation in goodness-of-fit (see Fig. 5). The 𝑤CDM behavior +is different; the 2x2pt + P1/All chains have the highest tensions, against 3x2pt+All caused by 𝜎split +8 +, and there is no loss in goodness-of-fit +compared with 2x2pt + P2 (see Fig. 6). +sample, and when marginalizing over the 𝑋lens [117]. +Comparing our work with other results in the literature is +unfortunately not straightforward since there are several dif- +ferent ways how ΛCDM parameters can be split into geometry +and growth. This work focuses on additional parameters al- +lowing an anomalous late-time growth-independent evolution +of the matter power spectrum. In [74], on the other hand, +the growth parameters also affect the source function of the +CMB power spectrum. Thus, different values of Ωgrowth +m +af- +fect both early and late-time dynamics and produce significant +changes to the CMB temperature and polarization power spec- +tra. These split parameterizations that affect both early and +late-time dynamics produce ΔΩm ≠ 0 detections at a level +greater than 4𝜎, much higher than what we observe with our +adopted late-time scale-independent modifications to the mat- +ter power spectrum. [125, 126] describe a third possibility for +the split. Their growth parameters affect the growth index 𝛾, +which is a single parameter that approximately describes the +ΛCDM growth history in the late Universe. +Several extensions to this paper come to mind: Firstly, we +already mentioned that it will be important to study the im- +pact of other lens samples, in particular the MagLim sam- +ple. Secondly, additional cosmological information from ex- +ternal datasets, such as including more scales of the CMB +temperature and polarization power spectrum, and adding +CMB lensing are near-term extensions of this work. +The +𝑤CDM split would also benefit from extra information on +𝑤geo from the observed DES Type IA supernova included in +the new Phanteon+ sample [127]. Thirdly, we plan to include +small-scale information to increase the constraining power on +growth-geometry split parameters, e.g. by modeling baryons +in cosmic shear as in [116] or modeling galaxy bias in 2x2pt +via effective field theory [128] or via Halo Occupation Distri- +bution models [102, 129, 130]. +While this paper does not show any hints of new physics +beyond ΛCDM, future datasets from Rubin Observatory’s +LSST [131], the Roman Space Telescope [132], and the Euclid +mission [133], in combination with the Dark Energy Spectro- +scopic Instrument [134], Simons Observatory [135] and the +CMB-S4 mission [136] will significantly tighten the statis- +tical error budget on cosmological models beyond ΛCDM +and 𝑤CDM. It is now timely to develop the theoretical tool- +box to efficiently and consistently explore these models across +datasets. +ACKNOWLEDGEMENTS +Simulations in this paper use High Performance Comput- +ing (HPC) resources supported by the University of Arizona +TRIF, UITS, and RDI and maintained by the UA Research +Technologies department. +The authors would also like to +thank Stony Brook Research Computing and Cyberinfrastruc- +ture, and the Institute for Advanced Computational Science +at Stony Brook University for access to the high-performance +SeaWulf computing system, which was made possible by a +$1.4M National Science Foundation grant (#1531492). TE +and JX are supported by the Department of Energy grant DE- + +16 +SC0020215. EK is supported by the Department of Energy +grant DESC0020247 and an Alfred P. Sloan Research Fellow- +ship. +[1] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, +A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, +S. Jha, R. P. 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Benson, +et al., arXiv e-prints , +arXiv:1610.02743 (2016), arXiv:1610.02743 [astro-ph.CO]. + diff --git a/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/load_file.txt b/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..451ce4aafc8fd182112c1e78d33e615670660c98 --- /dev/null +++ b/ntE2T4oBgHgl3EQfJwYe/content/tmp_files/load_file.txt @@ -0,0 +1,2536 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf,len=2535 +page_content='Growth and Geometry Split in Light of the DES-Y3 Survey Kunhao Zhong,1, ∗ Evan Saraivanov,1 Vivian Miranda,1, 2 Jiachuan Xu,3 Tim Eifler,3 and Elisabeth Krause3, 4 1Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794, USA 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA 3Department of Astronomy/Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA 4Department of Physics, University of Arizona, 1118 E Fourth Str, Tucson, AZ, 85721-0065, USA (Dated: January 11, 2023) We test the smooth dark energy paradigm using Dark Energy Survey (DES) Year 1 and Year 3 weak lensing and galaxy clustering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Within the ΛCDM and 𝑤CDM model we separate the expansion and structure growth history by splitting Ωm (and 𝑤) into two meta-parameters that allow for different evolution of growth and geometry in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We consider three different combinations of priors on geometry from CMB, SNIa, BAO, BBN that differ in constraining power but have been designed such that the growth information comes solely from the DES weak lensing and galaxy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For the DES-Y1 data we find no detectable tension between growth and geometry meta-parameters in both the ΛCDM and 𝑤CDM parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This statement also holds for DES-Y3 cosmic shear and 3x2pt analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For the combination of DES-Y3 galaxy-galaxy lensing and galaxy clustering (2x2pt) we measure a tension between our growth and geometry meta-parameters of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6𝜎 in the ΛCDM and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='48𝜎 in the 𝑤CDM model space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We attribute this tension to residual systematics in the DES-Y3 RedMagic galaxy sample rather than to new physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We plan to investigate our findings further using alternative lens samples in DES-Y3 and future weak lensing and galaxy clustering datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' INTRODUCTION Since the discovery of the accelerated expansion of our Uni- verse [1, 2], the flat ΛCDM, which adopts a late-time Universe dominated by the cosmological constant, has become the stan- dard model of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' From a fundamental physics view- point, the origin of dark energy is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The cosmo- logical constant modeled as vacuum energy is fine-tuned with a value too small to any known quantum field theory [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Dy- namical scalar fields, quintessence and k-essence, have been proposed to solve the fine-tuning problem [4–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Modified gravity is an alternative way to explain the Universe’s acceler- ation without introducing a new component [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' To date, none of these proposed scenarios have been detected by observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' With only six free parameters, the standard model of cos- mology predicts the temperature and polarization anisotropy statistics of the Cosmic Microwave Background (CMB) with remarkable success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Additionally, imaging and spectroscopic surveys show increasing power to constrain ΛCDM’s predic- tions for the late-time evolution of large-scale structures (LSS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' current stage III LSS surveys include the Dark Energy Survey (DES) [10–22], the Kilo-Degree Survey (KiDS) [23–27], the Hyper Suprime-Cam Subaru Strategic Program (HSC) [28– 32], and the Baryon Oscillation Spectroscopic Survey (Boss and eBOSS) [33–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, multiple tensions have arisen in the last few years within the ΛCDM model, particularly between Planck mea- surements of the Cosmic Microwave Background and data from the late-time Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The first tension involves the value of the Hubble constant, 𝐻0 [39–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Local-Universe 𝐻0 estimates from type Ia supernova (SNIa), calibrated us- ing Cepheid variable stars [43, 44], conflict with CMB pre- ∗ kunhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='zhong@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='edu dictions [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Several studies show that this tension is reaching a statistical significance of 5𝜎 [40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Hubble constant predictions from the Cosmic Microwave Background are sensitive to changes in the late-time dark sec- tor [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For example, cold dark matter models decaying to relativistic species can affect the CMB predictions [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These predictions are also sensitive to physics before recombi- nation via the sound horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, observations of SNIa combined with Baryonic Acoustic Oscillations (BAO) show that changes in the late-time Universe dark sector cannot solve the 𝐻0 tension without creating additional problems [51–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These constraints suggest that the new physics should come from the time before recombination [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The Dark Energy Survey year one (DES-Y1) and year three (DES-Y3) analysis conclude that the parameter 𝑆8 is in mild tension with the ΛCDM model predicted by Planck CMB data [20, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Multiple independent surveys have inde- pendently discovered this discrepancy [24, 35, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The projected one-dimensional 𝑆8 tension is not large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' however, in- vestigations of the multi-dimensional degeneracy directions in ΛCDM parameter space offers a more complete picture [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The generalizations of the late-time dark sector can reduce this discrepancy, but the 𝑆8 tension generally increases with statistical significance when an early-dark energy component is added in the ΛCDM model [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this work, we split the matter density, Ωm, and the dark energy equation of state, 𝑤, to test the consistency of smooth- dark-energy between the background evolution and the late- time scale-independent growth of structures [64–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Using different data sets containing geometry or growth information, we can verify such consistency in ΛCDM and 𝑤CDM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Parameter splitting has been extensively applied in multiple contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For example, baryon density can be divided into two parts with one only affecting ionization history [69], cold matter density can be split into parts representing different aspects of type Ia supernova [70], or the primordial inflationary amplitude can be separated into one that affects the CMB and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='03694v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='CO] 9 Jan 2023 2 another that only affects predictions from the effective field theory of large-scale structure [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This work is a follow-up investigation of two previous anal- yses, one employing DES-Y1 data [72], and the other adopting older weak lensing data from the Canada-France Hawaii Tele- scope Lensing Survey [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this work, we employ the new DES-Y3 3x2pt data, including different data combina- tions that clarify some internal aspects of the galaxy-galaxy lensing and galaxy clustering combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The Kilo-Degree Survey (KiDS) collaboration also analyzed their data with the growth-geometry split type of parameters [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In addition to weak lensing and galaxy clustering, redshift space distortion (RSD) and clusters data are used to extract growth informa- tion [73, 74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' VI for the discussion of different split methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The structure of the paper is as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' II, we ex- plain the geometry-growth split and the 3x2pt combination of two-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We summarize DES analy- sis choices and the external data sets in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV, which also contains a detailed description of our adopted pipeline and the validation tests we performed based on synthetic ΛCDM DES-Y1 and DES-Y3 data vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We present the results and discussions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' V, and conclusions, including an exposition on planned follow-up improvements, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' THEORY AND METHODOLOGY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Split Matter Power Spectrum The linear matter power spectrum quantifies the inhomo- geneity of matter distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' and it can be written as the product of the inflationary primordial spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the transfer function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' and the growth function: 𝑃linear(𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 𝑘) = 2𝜋2 𝑘3 4 25 𝐴s � 𝑘 𝑘norm �𝑛s−1 � 𝑘 𝐻0 �4 × × 𝑇2(𝑘) � 𝐺(𝑧) Ωm(1 + 𝑧) �2 (1) The growth function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 𝐺(𝑧) = (1 + 𝑧)𝐷(𝑧) = (1 + 𝑧) 𝛿m(𝑧) 𝛿m(𝑧ini) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (2) describes the scale-independent time evolution of matter over- density from initial conditions defined at redshift 𝑧ini = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In smooth dark energy cosmologies, the growth-factor evolu- tion obeys the following ordinary differential equation: 𝐺′′ + � 4 + 𝐻′ 𝐻 � 𝐺′ + � 3 + 𝐻′ 𝐻 − 3 2Ωm(𝑧) � 𝐺 = 0, (3) where the prime denotes derivative with respect to the log- arithm of the scale factor, ln 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The initial conditions are 𝐺ini = 1 and 𝐺′ ini = −(3/5)(1 − 𝑤)ΩDE(𝑧ini) [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Mod- els that introduce clustering of dark energy break this scale- independent relation between growth factor and dark energy parameters [75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this work, we confine our study to the case of smooth dark energy with a constant equation of state (𝑤CDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Our results can be generalized, for example, by considering instead principal component based 𝑤(𝑧) parame- terizations [68, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We split the Ωm and 𝑤 parameters into geometry, {Ωgeo m , 𝑤geo}, and growth counterparts {Ωgrowth m , 𝑤growth}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The growth parameters affect the late-time growth factor evolu- tion via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The remaining parameters, {Ωb, 𝐻0, 𝐴s, 𝑛s, 𝜏}, are not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The split ΛCDM cosmology assumes 𝑤geo = 𝑤growth = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Since the linear power spectrum 𝑃linear(𝑧, 𝑘) ∝ 𝐺2(𝑧), we can define the split linear matter power spectrum to be: 𝑃linear split (𝑘, 𝑧) = 𝑃linear−camb geo (𝑘, 𝑧) 𝐺camb geo (𝑧)2 × 𝐺growth(𝑧)2 , (4) with 𝐺growth(𝑧) = 𝐺camb geo (𝑧) × �𝐺ODE growth(𝑧) 𝐺ODE geo (𝑧) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (5) 𝑃linear−camb geo and 𝐺camb geo are respectively the linear power spec- trum and the growth factor both computed by the Boltzmann code CAMB [78, 79] assuming the geometry parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 𝐺ODE geo and 𝐺ODE growth are solutions of the differential Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 3 given geometry and growth parameters respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Our slightly convoluted definition is analytically equivalent to 𝑃linear split (𝑘, 𝑧) = 𝑃linear−camb geo (𝑘, 𝑧) × �𝐺ODE growth(𝑧) 𝐺ODE geo (𝑧) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (6) Definition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5 resolves the small numerical error between the growth factor calculated by CAMB versus the solution from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 3 with no radiation and accurate background evolution of massive neutrinos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the adopted multi-probe lensing pipeline requires 𝐺growth(𝑧) itself when computing intrinsic alignment contributions for cosmic shear and galaxy-galaxy lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We follow the naming convention in the parameter split literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In our parameter split distance probes heavily con- strain geometry parameters while growth parameters allow the late-time growth factor to vary with extra degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However Ωgeo m and 𝑤geo can also affect structure growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Specifically, the split matter power spectrum in our definition is proportional to (Ωgeo m )−2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Additionally, early universe physics that affect both background expansion and structure formation are also modeled by Ωgeo m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The split between geometry and growth is not uniquely defined, and we defer to future work examining the impact of these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The root mean variance within 8 Mpc/h is defined as: 𝜎2 8 (𝑧) = 1 2𝜋2 ∫ d log 𝑘𝑊2(𝑘𝑅)𝑘3𝑃(𝑘, 𝑧), (7) 3 k [h/Mpc] 101 103 P(k) [Mpc/h]3 P(k), z = 0 growth m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 growth m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='35 geo m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 geo m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='35 m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 10 4 10 3 10 2 10 1 100 k [h/Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 P(k)/P(k)fid FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Geometry and growth effects on the matter power spectrum in the split-ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' When changing one parameter, we keep the remaining matter density at ΩXm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The choice of employing Ωgrowth m in the Euclid Emulator has the effect of roughly maintaining the scale-independent amplitude shift induced by changes in Ωgrowth m on mildly non-linear scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' On 𝑘 ≳ 10−2 scales that are within DES reach, changes in the shape parameter Γ = Ωgeo m ℎ induced by varying Ωgeo m are degenerate with the primordial power spectrum tilt, 𝑛s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This degeneracy motivates our choice of priors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CMB brings external constraints on the inflationary and shape parameters, while BAO and SNIa indirectly limits the shape parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' where 𝑊(𝑘𝑅) is a top-hat filter function in Fourier space with radius 𝑅 = 8 Mpc/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The split 𝜎split 8 (𝑧) is then given by: 𝜎split 8 (𝑧) = 𝜎geo−camb 8 (𝑧) × �𝐺ODE growth(𝑧) 𝐺ODE geo (𝑧) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (8) The different behavior of 𝜎split 8 (𝑧) versus 𝜎8(𝑧) with respect to the change of growth and geometry parameters is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In the split ΛCDM case, the change of Ωgrowth m will give a smaller change on 𝜎8(𝑧) compared with the unsplit case, namely when changing Ωgrowth m and Ωgeo m simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' It is the opposite in the split 𝑤CDM case where a change in 𝑤growth gives a larger change on 𝜎8(𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In both cases, the change is larger at low redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' To account for non-linearities in the split power spec- trum, we utilize the Euclid Emulator to compute the factor 𝐵(𝑘, 𝑧) ≡ 𝑃(𝑘, 𝑧)�𝑃linear(𝑘, 𝑧) [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this work, we defined the 𝐵(𝑘, 𝑧) as dependent on the growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We then define the split matter power spectrum as 𝑃split(𝑘, 𝑧) = 𝑃linear split (𝑘, 𝑧) × 𝐵growth(𝑘, 𝑧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (9) The official DES-Y1 and DES-Y3 analyses adopt Halofit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figure 3 shows that Halofit and Euclid Emulator differ- ences are within 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This disagreement doesn’t affect infer- ences on the ΛCDM parameters as shown in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, our definition has practical advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 8(z) m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='35 m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 growth m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='35 growth m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 z 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 8(z) w = 1 w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 wgrowth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 wgrowth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Both panels show 𝜎8 changes under the variations of the unsplit and split Ωm and 𝑤 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The solid lines show shifts on the unsplit ΛCDM and 𝑤CDM models, and the dashed lines change growth while keeping the geometry parameters at their fiducial Ωgeo m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 and 𝑤geo = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We can see that changes in the growth parameter Ωgrowth m results in a minor shift in 𝜎8(𝑧), whereas 𝑤growth gives a more considerable change compared with the unsplit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Massive neutrinos break the scale-independent evolution of dark matter perturbations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' neutrinos transition from relativis- tic to non-relativistic behavior as the universe cools down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The scale-dependant changes in the matter spectrum are absorbed in 𝑃linear geo (𝑘, 𝑧) calculated by the Boltzmann code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For this pa- per, we only study fixed neutrino mass with � 𝜈 𝑚𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='06 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' TWO-POINT CORRELATION FUNCTIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Weak Lensing and Galaxy clustering The dark matter distribution of the universe is traced by two fields: i) the galaxy density field, and ii) the weak lensing shear field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These fields generate three two-point cor- relation functions (2PCF) as a function of angular separation 𝜃: 4 10 1 100 k [h/Mpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='04 P(k)Halofit/P(k)Emulator 1 z = 0 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 m 64 66 68 70 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 8 64 66 68 70 H0 P1 + DES-Y1 ( ±) - Emul P1 + DES-Y3 ( ±) - Emul P1 + DES-Y1 ( ±) - Halofit P1 + DES-Y3 ( ±) - Halofit FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Top panel: Fractional difference of non-linear power spec- trum between Halofit and the Euclid Emulator at three different redshifts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Fiducial parameter values are the same adopted in the syn- thetic DES chains (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Bottom panel: posterior compari- son between Halofit and the Euclid Emulator on P1 + DES-Y1/Y3 cosmic shear combinations (P1 prior is defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We assumed the ΛCDM model in all four chains, and the DES-Y1/Y3 data vector was synthetic with the same fiducial model adopted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Both figures illustrate that within the prior ranges adopted in this manuscript, the few percent differences between Halofit and the Euclid Emulator do not impact our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Cosmic shear: 𝜉𝑖 𝑗 ± (𝜃): the correlation, ⟨𝜅𝜅⟩, between source galaxy shear in redshift bins 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Galaxy-Galaxy lensing 𝛾𝑖 𝑗 (𝜃): the correlation, ⟨𝛿𝑔𝜅⟩, between lens galaxy positions and source galaxy tangential shear in redshift bins 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Galaxy Clustering 𝑤𝑖 𝑗 (𝜃): the correlation, ⟨𝛿𝑔𝛿𝑔⟩, be- tween lens galaxy position in redshift bins 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In combination, these probes significantly increase the information about the matter distribution and improve the systematics mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Throughout this paper, "3x2pt" refers to the multi-probe analysis involving the combination of the three 2PCF, and "2x2pt" refers to the multi-probe combination of galaxy-galaxy lensing and galaxy clustering (𝛾𝑡 + 𝑤 𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Theory predictions and 2PCF are related by the angular power spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In both DES-Y1 and DES-Y3, we calculate the full non-Limber integral on large angles only in the galaxy position auto power spectra, following Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Using Limber approximation, the angular power spectra of tracer 𝐴 at redshift bin 𝑖 and tracer 𝐵 at redshift bin 𝑗 is [82, 83]: 𝐶𝑖 𝑗 𝐴𝐵(ℓ) = ∫ 𝑑𝑧 𝑊𝑖 𝐴(𝑧)𝑊 𝑗 𝐵(𝑧) 𝜒2(𝑧) 𝑃split (𝑘, 𝑧) ��� 𝑘=(𝑙+1/2)/𝜒(𝑧), (10) where 𝜒 is the comoving radial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The weighting func- tion of weak lensing shear 𝜅 and galaxy number density 𝛿𝑔 are respectively [84] 𝑊𝑖 𝜅 (𝜒) = 3𝐻2 0Ωgeo m 2c2 𝜒 𝑎(𝜒) ∫ d𝜒′ 𝑛𝑖 𝜅 (𝑧 (𝜒′)) 𝑑𝑧/𝑑𝜒′ ¯𝑛𝑖𝜅 𝜒′ − 𝜒 𝜒′ (11) and 𝑊𝑖 𝛿g(𝜒) = 𝑏𝑖(𝑧(𝜒)) 𝑛𝑖 g(𝑧(𝜒)) ¯𝑛𝑖g 𝑑𝑧 𝑑𝜒 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (12) Here, ¯𝑛𝑖 g/𝜅 = ∫ 𝑑𝑧 𝑛𝑖 g/𝜅 (𝑧) is the angular number density of galaxies in the redshift bin 𝑖, and 𝑏𝑖(𝑧(𝜒)) is the galaxy bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Geometryparameters modelthecomovingradial distance[72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Being consistent with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 1, 𝑃(𝑘) ∝ (1�Ωgeo m )2, the matter density that appears in the 𝑊𝜅 prefactor is Ωgeo m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This choice mainly follows the preferences of [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We defer the inter- esting investigation of how changing Ωgeo m → Ωgrowth m here would affect the comparison between growth and geometry parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The relation between two-point correlation functions and angular power spectra assumes bin-average curved sky formu- las in both DES-Y1 and DES-Y3 as shown below 𝑤𝑖 𝜃 (𝜃) = ∑︁ ℓ 2ℓ + 1 4𝜋 𝑃ℓ𝐶𝑖𝑖 𝛿 𝛿(ℓ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (13) 𝛾𝑖 𝑗 𝑡 (𝜃) = ∑︁ ℓ 2ℓ + 1 4𝜋ℓ(ℓ + 1) 𝑃2 ℓ𝐶𝑖 𝑗 𝛿E(ℓ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (14) 𝜉𝑖 𝑗 ± (𝜃) = ∑︁ ℓ 2ℓ + 1 2𝜋ℓ2(ℓ + 1)2 [𝐺+ ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 ± 𝐺− ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2] � 𝐶𝑖 𝑗 𝐸𝐸 (ℓ) ± 𝐶𝑖 𝑗 𝐵𝐵(ℓ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The analytical expressions for Legendre and associated Leg- endre polynomials 𝑃ℓ and 𝐺± ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 can be found in [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Further information about these transformations, including 𝐸/𝐵-mode projections on the auto and cross power spectra involving shear, are described in the DES-Y3 methods paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The com- putation of non-limber integrals in galaxy position auto power spectra, and the use of bin-average curved sky formulas for 5 cosmic shear, galaxy-galaxy lensing and galaxy clustering in DES-Y1 is an improvement over modeling choices of [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We use the Tidal Alignment and Tidal Torquing (TATT), a generalization to the previously DES-Y1 adopted non-linear alignment model (NLA-IA), to model the intrinsic alignment of galaxies in DES-Y3 data [18, 86, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Under this framework, the intrinsic shape of galaxies is written as a collection of terms depending on the matter overdensity, 𝛿𝑚, and the tidal tensor, 𝑠𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These terms describe tidal alignment, tidal torquing, and density weighting, as shown below: 𝛾𝐼 𝑖 𝑗 = 𝐶1𝑠𝑖 𝑗 ���� Tidal Alignment + 𝑏TA𝐶1 �𝛿𝑚 × 𝑠𝑖 𝑗 � ������������������������������������ Density Weighting + 𝐶2 � 𝑠 𝑘 𝑖 𝑠𝑘 𝑗 − 1 3𝛿𝑖 𝑗𝑠2 � ���������������������������������������������� Tidal Torquing .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (15) Here, 𝐶1 = − 𝐴1 ¯𝐶Ωgrowth m 𝑎𝐺growth(𝑧) � 1 + 𝑧 1 + 𝑧0 � 𝜂1 , (16) and 𝐶2 = 5 𝐴2 ¯𝐶Ωgrowth m (𝑎𝐺growth(𝑧))2 � 1 + 𝑧 1 + 𝑧0 � 𝜂2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (17) The redshift 𝑧0 is to the mean redshift of the source galaxy sample, and ¯𝐶 = (5 × 10−14𝑀⊙ℎ−2Mpc2) × 𝜌crit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The TATT model contains five parameters: the amplitude 𝐴𝑖=1,2, power law index 𝜂𝑖=1,2, and the effective source galaxy bias 𝑏𝑇 𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The TATT model reduces to NLA-IA when 𝐴2 = 𝑏𝑇 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Both NLA-IA and TATT have an explicit dependence on matter density and the growth factor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we assume these are both growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DATA AND ANALYSIS METHOD Cosmological Parameters Prior Ωgeo m Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9) 𝑤geo Flat(-3, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='01) 𝐴s × 109 Flat(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5) 𝑛s Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='92, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0) 𝐻0 Flat(61, 73) 𝜏 Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) Ωgrowth m Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='24 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4) 𝑤growth Flat(-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7 , -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Flat priors for the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We take the priors as in the Euclid Emulator for the parameters (𝐴s, 𝑛s, 𝐻0, Ωgrowth m , 𝑤growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We only include the optical depth of reionization, 𝜏, in chains with CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 m 60 70 80 H0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 8 60 70 80 H0 DES-Y1 (Cosmic Shear) DES-Y3 (Cosmic Shear) SN + BAO + BBN CMBP SN + BAO + BBN + CMBP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Cosmic Shear posteriors in ΛCDM model for DES-Y1 and DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Unlike all remaining figures and results in this manuscript, these constraints assume Halofit for the non-linear matter power spectrum and the original DES-Y3 priors for the cosmological param- eters [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The red dot-dashed lines are cosmological constraints from type Ia Supernova, BAO and BBN external data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' On the other hand, the blue dashed lines show posteriors derived from the Cosmic Mi- crowave Background temperature and polarization Planck 2018 data with the reduced multipole range 35 < ℓ < 396 in combination with low-ℓ EE polarization data ℓ < 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These are the external data com- binations adopted on priors P2 and P1, respectively (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' P1 and P2 priors are not stringent enough to create a significant 𝜎8 tension, but at the same time, they provide complementary informa- tion in parameters that DES does not constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' While both priors measure Ωm and the Hubble constant 𝐻0, only the CMB data restricts the inflationary parameters 𝐴s and 𝑛s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [72], exter- nal (non-DES) information is necessary to tightly constrain growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DES data This work presents results using DES-Y1 and DES-Y3 data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' regarding DES-Y1 [72], we have implemented signifi- cant changes in the choice of external data sets and non-linear modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In both data sets, the collaboration measured 2PCF via the TreeCorr algorithm [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We follow the collaboration choices when applying scale cuts to remove small-scale infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The resulting 3x2pt data vector contains 457 points for DES-Y1 and 533 points for DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Systematics in galaxy clustering and weak lensing In this section, we summarize the systematics modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We mainly follow the DES-Y3 key projects and point out the difference between DES-Y1 and DES-Y3 [18, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 6 DES-Y3 Nuisance Parameters Prior Linear Galaxy bias 𝑏𝑖𝑔(𝑖 ∈ [1, 5]) Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0) Intrinsic Alignment (TATT) 𝐴1 Flat(-5, 5) 𝐴2 Flat(-5, 5) 𝜂1 Flat(-5, 5) 𝜂2 Flat(-5, 5) 𝑏𝑇 𝐴 Flat(0 , 2) Source photo-z Δ𝑧1s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) Δ𝑧2s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5) Δ𝑧3s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1) Δ𝑧4s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7) Lens photo-z Δ𝑧1 1 × 102 Gauss(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4) Δ𝑧2 1 × 102 Gauss(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3) Δ𝑧3 1 × 102 Gauss(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3) Δ𝑧4 1 × 102 Gauss(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5) Δ𝑧5 1 × 102 Gauss(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1) Multiplicative shear calibration 𝑚1 × 102 Gauss(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9) 𝑚2 × 102 Gauss(-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) 𝑚3 × 102 Gauss(-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) 𝑚4 × 102 Gauss(-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) Lens magnification 𝐶1 1 × 102 Fixed (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='63) 𝐶2 1 × 102 Fixed (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='04) 𝐶3 1 × 102 Fixed (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='33) 𝐶4 1 × 102 Fixed (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='50) 𝐶5 1 × 102 Fixed (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='93) Point mass marginalization 𝐵𝑖(𝑖 ∈ [1, 5]) Flat(-5, 5) TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Adopted priors on DES-Y3 nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The pri- ors are either Flat (min, max) or Gaussian (mean, standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Galaxy Bias: The linear galaxy bias is parameterized by a scalar for each redshift bin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 𝑏𝑖(𝑘, 𝑧) = 𝑏𝑖, for five redshift bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' They are marginalized by a conservative prior U(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We do not consider non-linear galaxy bias in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Intrinsic Alignment of Galaxy: In our analysis, we adopt NLA for DES-Y1 and TATT for DES-Y3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' their respective parameters are shown in Tables III and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We fix the pivot redshift at 𝑧0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Multiplicative shear calibration: we model the shear calibration with a marginalized parameter 𝑚𝑖 for each redshift DES-Y1 Nuisance Parameters Prior Linear Galaxy bias 𝑏𝑖𝑔(𝑖 ∈ [1, 5]) Flat(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0) Intrinsic Alignment (NLA) 𝐴1 Flat(-5, 5) 𝐴2 Flat(-5, 5) Source photo-z Δ𝑧1s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8) Δ𝑧2s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5) Δ𝑧3s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1) Δ𝑧4s × 102 Gauss(0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7) Lens photo-z Δ𝑧1 1 × 102 Gauss(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7) Δ𝑧2 1 × 102 Gauss(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7) Δ𝑧3 1 × 102 Gauss(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6) Δ𝑧𝑖 1 × 102(𝑖 ∈ [4, 5]) Gauss(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='01) Multiplicative shear calibration 𝑚𝑖 × 102(𝑖 ∈ [1, 4]) Gauss(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3) TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Adopted priors on DES-Y1 nuisance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The priors are either Flat (min, max) or Gaussian (mean, standard devi- ation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Note that the parameters not presented here correspond to systematics not considered for DES-Y1 analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' bin, as shown below: 𝜉𝑖 𝑗 ± (𝜃) −→ �1 + 𝑚𝑖� �1 + 𝑚 𝑗� 𝜉𝑖 𝑗 ± (𝜃), 𝛾𝑖 𝑗 𝑡 (𝜃) −→ �1 + 𝑚 𝑗� 𝛾𝑖 𝑗 𝑡 (𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (18) DES-Y1 and DES-Y3 have different calibrations from simula- tions, detailed in Tables III and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Photometric redshift uncertainties: we model the un- certainties in photometric redshift distribution for both source and lens galaxies by a shift parameter, Δ𝑧𝑖 𝑥, unique to each redshift bin 𝑖, as shown below: 𝑛𝑖 𝑥(𝑧) = ˆ𝑛𝑖 𝑥 �𝑧 − Δ𝑧𝑖 𝑥 � , 𝑥 ∈ {source, lens}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (19) DES-Y1 priors for Δ𝑧𝑖 𝑥 differ from DES-Y3 priors and both are shown in Tables III and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We do not model stretches in the photometric redshift distribution of lens galaxies by an additional free parameter 𝜎𝑖 𝑧 as in the DES-Y3 key project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Lensing magnification: As detailed in [18], a parame- ter 𝐶𝑖 𝑙 is defined to describe the foreground mass effects on the observed number density of lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The parameter is calibrated from data for each redshift bin and held fixed in our analysis as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This systematic is not considered for the DES-Y1 data set in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Non-local effects in galaxy-galaxy lensing: for DES- Y3 specifically, we follow the marginalization approach developed in MacCrann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [89] and we adopt an informa- tive prior of the point-mass parameter 𝐵𝑖 ∈ Flat(−5, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Such 7 systematic is not considered for the DES-Y1 data set in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 𝑋lens factor: A non-physical parameter 𝑋lens was pro- posed in DES-Y3 to solve the internal inconsistency between galaxy-galaxy lensing and galaxy clustering 2PCF [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The two lensing samples, redMaGiC and MagLim, show discrep- ancies between the galaxy bias inferred from galaxy-galaxy lensing and galaxy clustering in the DES-Y3 analysis [57, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this work, we adopt redMaGiC lensing sample with fixed 𝑋lens = 1 for both DES-Y1 and DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We plan to follow up this work with a detailed comparison between redMaGiC and MagLim, including marginalization over 𝑋lens parameter and recent changes to the redMaGiC color selection algorithm [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Priors and External Data The split between growth and geometry information is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Within our choices, we select external probes so that DES-Y3 is the only constraining data set on growth parameters besides the boundaries of validity of the Euclid Emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We don’t present DES-only chains as in [72], since they have shown that DES needs to be combined with external data to provide useful constraining power on the difference between geometry and growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We combine DES with external data described below: CMBP: Planck 2018 low-ℓ EE polarization data (ℓ < 30) in combination with the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Our choice removes late-time Integrated Sachs Wolfe information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' It also removes CMB lensing effects as the CMB lensing-induced smoothing on the temperature power spectrum only affects constraints on cosmological parameters when including higher acoustic peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We find this prior complementary to the compressed CMB likelihood adopted [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Our CMB choices are slightly more conservative on 𝑛s and Ωbℎ2, but they do constrain the early-Universe inflationary amplitude 𝐴s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' SNIa: Pantheon Type Ia supernovae sample [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Type Ia supernovae are a constraint on geometry parameters only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' their likelihood does not require knowledge of the large-scale structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' There are, however, lensing magnification effects on the Hubble diagram [92–94] and growth effects on SNIa peculiar velocity distribution [95, 96] that will need to be taken into account in future Stage IV surveys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' for now, we disregard modeling these growth effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' BBN: We use derived constraint on baryon density from astrophysical probe: 100Ωbℎ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='208 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='052 [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' BAO: We use Baryon Acoustic Oscillation data from the SDSS DR7 main galaxy sample [98] in combination with the 6dF galaxy survey [99] at 𝑧eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='15 and 𝑧eff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='106 respectively, and the SDSS BOSS DR12 low-z and CMASS combined galaxy samples at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='38, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='51, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='61 [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These constraints come from comparing the observed scale of the BAO feature and the sound horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As a distance measurement of the late universe, we consider BAO to be pure geometry information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' To better understand the effects of these external data sets on the final results, we adopt the following three sets of priors: Prior 1 (P1): Emulator prior + CMBP Prior 2 (P2): Emulator prior + SNIa + BAO + BBN Prior 3 (All): Emulator prior + CMBP + SNIa + BAO + BBN Table I summarizes our adopted informative priors on the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figure 4 compares DES- Y1/Y3-only chains with the uninformative priors adopted by the DES collaboration against our P1 and P2 priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This figure assumes a ΛCDM model and Halofit for the non-linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Our priors are consistent with DES-only posteriors in all parameters, including 𝜎8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Since the SNIa + BAO + BBN combination does not provide any information on inflationary parameters, the only limits on 𝐴s and 𝑛s in P2 comes from the Euclid Emulator bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Therefore, comparing DES + P1 against DES + P2 chains offers valuable information on how internal DES tensions that shift 𝐴s and 𝑛s affect our results on growth parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figures 5 and 6 show that the DES-Y1 and DES-Y3 𝜒2 dis- tributions are nearly independent of prior P1/P2/All choices in both split ΛCDM and 𝑤CDM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The priors are broad enough not to impact the model’s DES 𝜒2 fit, except for the DES-Y3 2x2pt in ΛCDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As we will see, the internal tensions on DES-Y3 2x2pt shift the inflationary parameters to values inconsistent with the CMB prior in both ΛCDM and 𝑤CDM splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In ΛCDM, ΔΩm cannot restore the goodness- of-fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' there is a Δ𝜒2 ≈ 5 difference between DES-Y3 2x2pt + P1 and DES-Y3 2x2pt + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Interestingly, ΔΩm and Δ𝑤 can correct DES-Y3 2x2pt + P1 fit in 𝑤CDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Pipeline We perform the MCMC analysis using Cocoa, the Cobaya- Cosmolike Architecture [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Cocoa is a modified ver- sion of CosmoLike [102] multi-probe analysis software in- corporated into the Cobaya framework [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DES-Y1 and DES-Y3 covariance matrices were computed using Cosmo- Cov [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CosmoCov and Cocoa are both derived from Cosmolike [102], the former pipeline computes covariance matrices, and the latter evaluates data vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CosmoLike within Cocoa has efficient OpenMP shared-memory paral- lelization [105] and cache system compatible with the slow- fast decomposition implemented in the default Cobaya Monte- Carlo Markov chain sampler (MCMC) The OpenMP efficiency in CosmoLike is around 50%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=', quadrupling the number of OpenMP cores halves CosmoLike runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CosmoLike has been used in both DES-Y1/Y3 multi-probe analyses when constraining ΛCDM parameters [18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 87] and 8 230 235 240 2 Y1 ± CDM 227 250 260 270 280 2 Y1 2x2pt CDM 230 300 310 320 2 Y1 ± + w CDM 281 430 440 450 2 Y1 ± + t CDM 403 500 510 520 2 Y1 3x2pt CDM 457 P1 P2 All 230 240 250 2 Y3 ± CDM 227 340 350 360 370 380 2 Y3 2x2pt CDM 306 300 310 320 330 2 Y3 ± + w CDM 281 520 530 540 550 2 Y3 ± + t CDM 479 620 630 640 650 660 2 Y3 3x2pt CDM 533 P1 P2 All FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The DES 𝜒2 distribution of different combinations of two-point correlation functions in split-ΛCDM chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The number of data points is printed in the upper left of each panel, after masking us applied for DES-Y1 and DES-Y3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This plot demonstrates that the P1, P2 and All data priors and external data combinations do not degrade the DES fit except for the DES-Y3 2x2pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This anomalous data vector combination predicts a small inflationary amplitude, 𝐴s, in ΛCDM, incompatible with CMB data [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The cosmic shear cross-correlation reduces this problem considerably on the 3x2pt fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, the detailed comparison between 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 against 3x2pt stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Both 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 combinations show virtually no 𝜒2 changes between all three priors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the same is not true for 3x2pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 230 235 240 2 Y1 ± wCDM 227 250 260 270 280 2 Y1 2x2pt wCDM 230 300 310 320 2 Y1 ± + w wCDM 281 430 440 450 2 Y1 ± + t wCDM 403 500 510 520 530 2 Y1 3x2pt wCDM 457 P1 P2 All 230 240 250 2 Y3 ± wCDM 227 340 350 360 370 380 2 Y3 2x2pt wCDM 306 300 310 320 330 2 Y3 ± + w wCDM 281 520 530 540 550 2 Y3 ± + t wCDM 479 620 630 640 650 660 2 Y3 3x2pt wCDM 533 P1 P2 All FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The DES 𝜒2 distribution of different combinations of two-point correlation functions in split-𝑤CDM chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The number of data points is printed in the upper left of each panel, after masking us applied for DES-Y1 and DES-Y3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This plot demonstrates that the P1, P2 and All data priors and external data combinations do not degrade the DES fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 2x2pt data vector anomalous data vector combination prefers a small inflationary amplitude, 𝐴s in the absence of CMB external data, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, growth parameters can restore the DES 2x2pt goodness-of-fit when 𝐴s is set by the CMB prior, unlike what we observe in the ΛCDM split, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Indeed, Figure 13 shows that DES-Y3 2x2pt data has higher detection of ΔΩm − Δ𝑤 < 0 when combined with CMB data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Unfortunately, both ΛCDM and w-CDM splits have similar goodness-of-fit on the 3x2pt chains that incorporate cosmic shear, including the slight loss of fit when combining DES and CMB data, and ΔΩm − Δ𝑤 is consistent with zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' for calibrating Bayesian evidences [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' It has also been used in forecast studies for Rubin Observatory’s LSST and Roman Space Telescope [107–110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We compute the linear power spectrum with the CAMB Boltzmann code [111, 112];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Cobaya already had implemen- tations of all external data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We adopt Cobaya’s default adaptive metropolis hasting MCMC sampler, and we employ the Gelman-Rubin criteria 𝑅 − 1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='02 to establish chain convergence [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We post-process chains and creat figures using GetDist [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Changes in the growth parameters are only semi-fast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' they do not require CAMB to recompute distances and matter power spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' only CosmoLike must be rerun to update the DES data vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Due to an efficient cache system, CosmoLike reruns with only modified growth parameters takes about half the runtime compared with when all parameters are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CAMB and CosmoLike runtimes are roughly equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the time ratio between slow and semi-fast parameters is, therefore, ap- proximately 4:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 3x2pt data vector evaluation time with 10 OpenMP cores is of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5s on modern AMD EPYC 7642 48-core nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This estimation includes CAMB evaluation, non-limber integration, and TATT modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Finally, code comparisons between the CosmoSiS pipeline and Cosmolike were presented in [18, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='33 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='38 growth m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='38 growth m SYNTHETIC DES DATA All + DES-Y3 (Cosmic Shear) All + DES-Y3 ( ± + t) All + DES-Y3 ( ±+ w ) All + DES-Y3 (3x2pt) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Posteriors derived from different combinations of synthetic DES-Y3 2PCFs in the split ΛCDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Prior to the growth parameter 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='24 ≤ Ωgrowth m ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4 is compatible with the Euclid Emulator boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' All posteriors are prior limited, but the plot clarifies the gain in constraining power when galaxy-galaxy lensing and galaxy-clustering are added to cos- mic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Validation on Synthetic Data In this section, we generate a synthetic noiseless ΛCDM data vector from Planck best fit cosmological pa- rameters without lensing: � 𝐴s ×10−9, 𝑛s, 𝐻0, Ωm, Ωb � = � 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='101, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='965, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='317, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='049 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This set of parameters is compatible with both P1 and P2 priors [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We run MCMCs, including all nuisance parameters, and see if the posterior would give equal growth and geometry parameters at the fiducial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Comparison between Cosmic Shear and 3x2pt Assuming the All prior and external data combination, there is a significant improvement on Ωgrowth m constraints in ΛCDM split when going from cosmic shear to 3x2pt, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In the 3x2pt case, the Ωgrowth m posterior is well centered at the fiducial value, while cosmic shear pro- vides only marginal improvements compared with the uniform 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='24 < Ωgrowth m < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4 prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This narrow prior is informa- tive in both chains, the boundary coming from the range of the Euclid Emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Improving the small-scale modeling validity of cosmic shear and 2x2pt may tighten the 95% confi- dence level of Ωgrowth m enough to be within the allowed range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='33 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='38 growth m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='38 growth m SYNTHETIC DES DATA All + DES-Y1 (Cosmic Shear) All + DES-Y3 (Cosmic Shear) All + DES-Y1 (3x2pt) All + DES-Y3 (3x2pt) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Split ΛCDM posteriors derived from cosmic shear and 3x2pt combined with the All external data combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DES-Y1 3x2pt error bar on Ωgrowth m is approximately 17% larger compared with DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' On the other hand, cosmic shear DES-Y1 and DES-Y3 constraints are similar, and both are prior dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [116] and [117] offer a roadmap on how to implement such improvements in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DES 3x2pt combinations with P1 and All external data show nearly identical constraining power on Ωgrowth m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Com- bined priors on the primordial power spectrum (amplitude and shift) and the shape parameter Γ ≡ Ωmℎ are the needed exter- nal information so that DES can tightly measure growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The CMBP data alone provides both information while the SNIa + BAO + BBN measurements on Ωgeo m and 𝐻0 only restrict the shape parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Adding more CMB multipoles would improve constraints on early-Universe parameters even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, CMB temperature and polarization power spectra are more sensitive to lensing effects on smaller scales, which would limit our ability to compare DES effects on growth pa- rameters against CMB lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Partial delensing can alleviate this limitation [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Another possibility is to consider all multipoles up to ℓmax ≈ 1600 where effects from nonlinear dark matter collapse are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In this case, however, we would marginalize the chains over lensing principal compo- nents so that there is no leakage of information on growth parameters [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In the 𝑤CDM split model, Ωgrowth m and 𝑤growth are not well constrained even in the most informative 3x2pt case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We then show real data constraints on the principal component combination PC1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7071Δ𝑤 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7071ΔΩm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (20) 10 In both cosmic shear and 3x2pt chains, PC1 constraints are prior dominated but well centered around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Comparison between DES-Y1 and DES-Y3 In all three combinations with external data, posteriors on Ωgrowth m in the ΛCDM split from DES-Y1 and DES-Y3 cosmic shear are similar, despite the additional nuisance parameters introduced by the TATT intrinsic alignment model in DES- Y3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We have yet to check if we can obtain more constraining power on growth parameters by adopting the more straightforward NLA model on DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' On the other hand, the error bar on Ωgrowth m derived from 3x2pt combined with the All prior is 17% larger in DES-Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' One caveat to this result is that we have not tested whether expanding the adopted priors on point mass marginalization to the more conservative range Flat(−100, 100) would significantly degrade DES-Y3 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The first principle component PC1, defined on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 20, has nearly identical and prior dominated DES-Y1 and DES-Y3 posteriors in the 𝑤CDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Additional information from ei- ther smaller scales in the 3x2pt data vector or external growth information from CMB lensing and RSD are potential oppor- tunities in future analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figure 2 shows that Ωgrowth m and 𝑤growth induce changes on 𝜎split 8 (𝑧) with different redshift evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Including high redshift 𝑧 > 1 lensing samples from the future Roman Space Telescope may therefore be the key to disentangling growth parameters in 𝑤CDM split [109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' There are near-future possibilities that may expand the red- shift range adopted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' RedMaGiC fifth bin, with range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9, shows large 𝑋lens biases [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The al- ternative DES-Y3 MagLim sample of lens galaxies does have an additional redshift bin in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='95 < 𝑧 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='05 not ac- cessible by redMaGiC [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, MagLim high redshift bins were not adopted in the 3x2pt analysis by the DES col- laboration and may require further studies on the presence of potential systematic biases [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Finally, there is the emergent idea of using the same galaxy sample for both clustering and lensing that could potentially expand DES-Y3 constraints on 𝜎split 8 (𝑧) beyond 𝑧 > 1 [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' RESULTS We split our results section into three components: start- ing with a discussion of our results in the ΛCDM parameter space, we then move to the 𝑤CDM space, and conclude with quantifying tensions between different probe combinations in the context of both parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Growth-geometry split results in ΛCDM For the most constraining probe combination, DES 3x2pt+All, we show the DES-Y1 and DES-Y3 ΛCDM re- sults in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In both cases we find no measurable detection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m All + DES-Y1 (3x2pt) All + DES-Y3 (3x2pt) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Split ΛCDM posteriors derived from 3x2pt DES-Y1 and DES- Y3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' DES-Y1 3x2pt error bar on growth dark matter density is approximately 10% larger compared with DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' of ΔΩm being different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This constitutes the main, fiducial result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We explore subsets of the 3x2pt probe combination in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 10, where the left panel refers to DES-Y1 and the right corresponds to DES-Y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For DES-Y1 we find that in all cases ΔΩm is compatible with zero even within one sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For DES-Y3, however, we see shifts from ΔΩm = 0, especially in the 2x2pt (galaxy clustering+galaxy galaxy lensing) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We consider this further in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 11, where we show the one- dimensional posterior distributions on all relevant ΛCDM split parameters, finding that except for DES-Y3 𝜉±+P2, 𝜉± + 𝑤 𝜃 and 2x2pt chains, all combinations of two-point correlation functions predict ΔΩm compatible with zero within one sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The deviation on 𝜉± + P2 is less than two-sigma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Similarly, all combinations between DES 2PCFs and the P2 external data predict 𝐴s and 𝑛s values compatible with CMB data on P1/All, except for DES-Y3 𝜉± + 𝑤 𝜃 and 2x2pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Similarly to what we observe in ΛCDM chains with syn- thetic data vectors, DES-Y1 and DES-Y3 cosmic shear provide little information on ΔΩm even with the P1/P2/All priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The additional nuisance parameters introduced by the TATT intrinsic alignment model and point mass marginalization in DES-Y3 do not reduce constraining power for the growth pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The situation in the 3x2pt chains is different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the DES-Y1 3x2pt + All error bars are 10% larger than in DES- Y3, not that far from the predicted 17% improvement in the 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m All + DES-Y1 ( ±) All + DES-Y1 ( ± + t) All + DES-Y1 ( ± + w ) All + DES-Y1 (2x2pt) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='32 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 m All + DES-Y3 ( ±) All + DES-Y3 ( ± + t) All + DES-Y3 ( ± + w ) All + DES-Y3 (2x2pt) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Split ΛCDM posteriors derived from multiple 2PCF combinations in DES-Y1 (left panel) and DES-Y3 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Right panel shows that the DES-Y3 𝜉± + 𝛾𝑡, 𝜉± + 𝑤(𝜃) and 2x2pt all prefer lower values for the Ωgrowth m with upper limits at 95% confidence level being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='375, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='314 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='288 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We emphasize that the apparent constraints at ΔΩm ≡ Ωgrowth m − Ωgeo m ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 is due to effective priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' synthetic noise-free chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Priors on Ωgrowth m are still informa- tive, but to a much lesser degree on both DES-Y1 and DES-Y3 3x2pt compared with their cosmic shear counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' All of the DES-Y1 ΛCDM split chains are compatible with ΔΩm = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 10 (left panel) and 11 show large consis- tency between parameter posteriors derived from all 2PCFs combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' There are also no appreciable parameter shifts between chains with and without CMB priors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' goodness-of-fit is identical in these chains (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As expected, 𝐴s and 𝑛s constraints are significantly tighter when CMB data is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Finally, chains that include galaxy clustering (3x2pt, 2x2pt, and 𝜉± + 𝑤 𝜃) show a small shift towards ΔΩm > 0, but are still compatible with zero at 68% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For DES-Y3 we see that the 𝜉±+𝑤 𝜃 and 2x2pt chains predict, in combination with the All prior, ΔΩm ≠ 0 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75𝜎 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='60𝜎 in statistical significance (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We attribute these findings to the well-known incompatibilities between galaxy clustering and galaxy-galaxy lensing in DES-Y3 when using the redMaGiC lens sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Growth-geometry split results in 𝑤CDM For the 𝑤CDM parameter space we summarize our results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 12 and 13, where the former again shows selected results in two dimensions and the latter summarizes all chains in one-dimensional projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Qualitatively, we see similar behaviour as in the ΛCDM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' While DES cosmic shear and 3x2pt data shows ΔΩm − Δ𝑤 being consistent with zero, the picture becomes more complicated when considering subsets of the 3x2pt case that involve galaxy clustering of redMaGiC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In particular, the 2x2pt + All chain favors ΔΩm − Δ𝑤 < 0 at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='48𝜎, higher than any ΔΩm ≠ 0 detection in ΛCDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 𝑤CDM split 2x2pt + All chain also predict quite low 𝜎split 8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0243, while in ΛCDM we have 𝜎split 8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='730 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' While a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='48𝜎 detection is significant, we again refrain from claiming new physics in the 𝑤CDM model space, due to the aforementioned problems with the DES-Y3 redMaGiC sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Instead, we plan to further investigate growth-geometry split with alternative lens samples and when marginalizing over 𝑋lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Quantifying tensions between probes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Method To evaluate the tension we use the parameter difference method [121, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Given two chains 𝜃1 and 𝜃2 and their corresponding posteriors P1(𝜃1) and P2(𝜃2), begin by com- puting the difference between these two chains, denoted with Δ𝜃 = 𝜃1 − 𝜃2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Using this difference chain we can write P2(𝜃2) = P2(𝜃1 − Δ𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' By marginalizing over 𝜃1 we get 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='85 split 8 Y1 2x2pt Y3 2x2pt Y1 ± + w Y3 ± + w Y1 ± + t Y3 ± + t Y1 3x2pt Y3 3x2pt Y1 ± Y3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 As 1e 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='98 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='33 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='06 m CDM All P1 P2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' One-dimensional posteriors in split ΛCDM for various DES-Y1 and DES-Y3 two-point correlation functions, with error bars corresponding to marginalized 68% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The P1 external data combination is restricted to CMBP, while P2 is SNIa + BAO + BBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Priors on cosmological parameters are summarized in Table I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we define ΔΩm ≡ Ωgrowth m − Ωgeo m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The grey background separates our primary results from other probe combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the parameter difference posterior, P(Δ𝜃) = ∫ 𝑉Π P1(𝜃1)P2(𝜃1 − Δ𝜃)𝑑𝜃1 , (21) where 𝑉Π is the subset of the domain covered by the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As 𝜃1 → 𝜃2, the means of each chain approach equality and the mean of the parameter difference chain approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Thus the volume of the regions with P(Δ𝜃) > P(0) approaches 0, so we can approximate the tension using Δ = ∫ P(Δ𝜃)>P(0) P(Δ𝜃)𝑑Δ𝜃 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (22) This volume is interpreted as a probability of parameter shift, denoted Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' If Δ comes from a Gaussian distribution, the number of standard deviations from 0 is given by 𝑛𝜎 = √ 2Erf−1(Δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' (23) The resulting 𝑛𝜎 is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' To estimate the posterior we use Masked Autoregressive Flows (MAFs) [121, 123], which is a neural network that learns an invertible mapping from an arbitrary parameter space to a gaussianized one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The loss function for MAFs is the negative log probability from a unit Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Due to the autoregressive property, the Jacobian is triangular and thus the determinant is tractable to compute even for a large number of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Thus we can estimate the posterior as a reparameterization of a Gaussian and find the log-probability of arbitrary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Before training the neural network, we follow the imple- mentation in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [121] to apply a linear transformation to Δ𝜃 given from the Gaussian approximation for P(Δ𝜃) Δ𝜃′ = 𝐶−1(Δ𝜃 − 𝜇) , (24) with 𝐶 the covariance and 𝜇 the mean of P(Δ𝜃), then map Δ𝜃′ to the fully Gaussianized parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This enhances the convergence rate of the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Denoting the learned mapping as 𝜙(Δ𝜃′) = 𝑦 and the unit Gaussian density as N, 13 0.' metadata={'source': 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m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 wgeo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 split 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='80 split 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 wgeo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='3 w+ m All + DES-Y3 ( ±) All + DES-Y3 ( ± + t) All + DES-Y3 ( ± + w ) All + DES-Y3 (2x2pt) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Split 𝑤CDM posteriors derived from 3x2pt (left panel) and multiple 2PCF combinations (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Table I presents the priors on the cosmological parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we define ΔΩm ≡ Ωgrowth m − Ωgeo m and Δ𝑤 ≡ 𝑤growth − 𝑤geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' All constraints on the combination ΔΩm − Δ𝑤 are prior dominated given the range limitations of Ωgrowth m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we can then relate the log-probability as P(Δ𝜃) = N (𝑦) | det(𝐽𝜙(Δ𝜃′))| | det(𝐶)| (25) where 𝐽𝜙 denotes the Jacobian of 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' To compute the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 22 we use Monte Carlo integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Using the MAF we randomly sample from the posterior and calculate the log probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The fraction of generated points that land in the region P(Δ𝜃) > P(0) are counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The error of the numerical integration is given by the Clopper-Pearson interval for a binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Results We evaluate tensions between different DES 2PCF com- binations employing the parameter difference method on � 𝐴s, 𝑛s, 𝐻0, Ωgeo m , 𝜎split 8 (𝑧 = 0) � set of cosmological param- eters, with an addition of 𝑤geo in the split 𝑤CDM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As a caveat, this metric does not model the existing corre- lations between the 2PCFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the precise computation requires MCMC chains with repeated parameters, which is beyond our computational capabilities [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figure 14 qualitatively in- dicates discrepancies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we see, for example, the well-known redMaGiC problems between 2x2pt and other probe com- binations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Future utilization of machine learning emulators will allow the more precise calculation of tensions between the correlated DES 2PCFs with modest computational re- sources [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Interestingly, 𝐴s appears to be the culprit of the observed tensions above two sigmas between 2x2pt and the remaining combinations of the DES-Y3 data vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The highest observed tension in split ΛCDM happens between 2x2pt + P2 and 𝜉± + 𝑤 𝜃 + P1, entirely due to shifts on 𝐴s as both chains favors ΔΩm < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Figure 5 reveals that CMB priors degrade the goodness of fit to DES-Y3 2x2pt data by Δ𝜒2 ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In all other DES-Y3 2PCFs, swapping P1 with P2 priors does not affect 𝜒2 nearly as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, the detailed comparison between 𝜉± + 𝑤 𝜃 and 𝜉± + 𝛾𝑡 against 3x2pt stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Both 𝜉±+𝑤 𝜃 and 𝜉±+𝛾𝑡 combinations show virtually no 𝜒2 changes between all three priors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='the same is not true for 3x2pt as there is a Δ𝜒2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='21 degradation on DES goodness-of-fit when CMB data is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The behavior in 𝑤CDM split is different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' growth parameters can restore the DES 2x2pt goodness-of-fit when 𝐴s is set by the CMB prior, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 𝐴s tension between DES- Y3 2x2pt + P2 and DES-Y3 2x2pt + P1/All is also smaller on 𝑤CDM when compared with ΛCDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The DES-Y3 2x2pt predicts nonzero values for the principal component ΔΩm−Δ𝑤 for all external data combinations, the more extreme deviation from zero happening on DES-Y3 2x2pt + All chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The better fit to DES 2x2pt makes such nonzero detection more meaningful than the ΛCDM split model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Finally, the left and right panels on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 14 show that DES- Y3 2x2pt + P1/All chains have higher tension levels against other 2PCFs than DES-Y3 2x2pt + P2, the opposite of what we observe in ΛCDM split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Indeed, when cosmic shear is added to 2x2pt, the predicted 𝜎split 8 value shifts by more than three 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9 split 8 Y1 2x2pt Y3 2x2pt Y1 ± + w Y3 ± + w Y1 ± + t Y3 ± + t Y1 3x2pt Y3 3x2pt Y1 ± Y3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='1 As 1e 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='98 ns 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='33 geo m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='06 m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='8 wgeo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='25 w+ m wCDM All P1 P2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' One-dimensional posteriors in split 𝑤CDM for various DES-Y1 and DES-Y3 two-point correlation functions, with error bars corresponding to marginalized 68% confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the All external data combination consists of CMBP + SNIa + BAO + BBN, with CMBP being Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The P1 external data combination is restricted to CMBP, while P2 is SNIa + BAO + BBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Priors on cosmological parameters are summarized in Table I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' we define ΔΩm ≡ Ωgrowth m −Ωgeo m and Δ𝑤 ≡ 𝑤growth −𝑤geo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The grey background separates our primary results from other probe combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As could be expected, the DES-Y3 2x2pt + P2 predicts lower values for the inflationary amplitude 𝐴s incompatible with CMB priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The DES-Y3 2x2pt also predicts non-zero values for the principal component ΔΩm − Δ𝑤 for all external data combinations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the more extreme deviations being DES-Y3 2x2pt + All with mean −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='296 and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' sigmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Unfortunately, both Λ and 𝑤CDM split models have similar DES 3x2pt goodness-of-fit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' growth parameters can’t alleviate the incompatibility between galaxy-galaxy lensing and galaxy clustering in the 3x2pt chains (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' CONCLUSIONS This paper studies the growth-geometry split with DES- Y1 and DES-Y3 data in combination with external data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We utilize the Cobaya-CosmoLike Architecture (Cocoa) software to efficiently run a large number of MCMC chains that allow us to explore the variation of results for different probes and prior combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' For DES-Y1 we find that ΔΩm in ΛCDM and ΔΩm − Δ𝑤 in 𝑤CDM are both consistent with 0 for all permutations of DES 2PCFs and external prior combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In the case of DES-Y3, we find that cosmic shear and 3x2pt results are consistent with equal geometry and growth parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Combining cosmic shear and galaxy-galaxy lensing also does not indicate deviations between growth and geometry pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' However, both the 𝜉±+𝑤 𝜃 and 𝛾𝑡 +𝑤 𝜃 combinations of 2PCF indicate ΔΩm < 0 in ΛCDM and ΔΩm − Δ𝑤 < 0 in 𝑤CDM splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These results hold with both P1 and P2 pri- ors, which is interesting as they predict different values for the primordial power spectrum amplitude 𝐴s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In light of the well- known DES-Y3 problems of the redMaGiC sample, we do not interpret these results as a detection but rather assume that it is a residual of unsolved systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' We plan to further explore this with alternative lens samples, in particular the MagLim 15 CS,P1 CS,P2 CS,All 3x2pt,P1 3x2pt,P2 3x2pt,All CS+ t,P1 CS+ t,P2 CS+ t,All CS+w ,P1 CS+w ,P2 CS+w ,All 2x2pt,P1 2x2pt,P2 2x2pt,All CS,P1 CS,P2 CS,All 3x2pt,P1 3x2pt,P2 3x2pt,All CS+ t,P1 CS+ t,P2 CS+ t,All CS+w ,P1 CS+w ,P2 CS+w ,All 2x2pt,P1 2x2pt,P2 2x2pt,All CDM Y3 Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 CS,P1 CS,P2 CS,All 3x2pt,P1 3x2pt,P2 3x2pt,All CS+ t,P1 CS+ t,P2 CS+ t,All CS+w ,P1 CS+w ,P2 CS+w ,All 2x2pt,P1 2x2pt,P2 2x2pt,All CS,P1 CS,P2 CS,All 3x2pt,P1 3x2pt,P2 3x2pt,All CS+ t,P1 CS+ t,P2 CS+ t,All CS+w ,P1 CS+w ,P2 CS+w ,All 2x2pt,P1 2x2pt,P2 2x2pt,All wCDM Y3 Y1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Tensions between chains under split ΛCDM (left panel) and 𝑤CDM (right panel) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' IV B, the P1 external data combination is composed of Planck 2018 low-ℓ EE polarization data and the high-ℓ TTTEEE spectra truncated right after the first peak (35 < ℓ < 396).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' On the other hand, P2 is the combination of SNIa + BAO + BBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 2x2pt + P2 chain is the only one with significant tension, against other DES-Y3 2PCFs, in ΛCDM split;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the inflationary amplitude 𝐴s seems to be the culprit of the observed tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' When 2x2pt is combined with either P1 or All priors, we see lower tensions at the expense of degradation in goodness-of-fit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 𝑤CDM behavior is different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' the 2x2pt + P1/All chains have the highest tensions, against 3x2pt+All caused by 𝜎split 8 , and there is no loss in goodness-of-fit compared with 2x2pt + P2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' sample, and when marginalizing over the 𝑋lens [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Comparing our work with other results in the literature is unfortunately not straightforward since there are several dif- ferent ways how ΛCDM parameters can be split into geometry and growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' This work focuses on additional parameters al- lowing an anomalous late-time growth-independent evolution of the matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' In [74], on the other hand, the growth parameters also affect the source function of the CMB power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Thus, different values of Ωgrowth m af- fect both early and late-time dynamics and produce significant changes to the CMB temperature and polarization power spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' These split parameterizations that affect both early and late-time dynamics produce ΔΩm ≠ 0 detections at a level greater than 4𝜎, much higher than what we observe with our adopted late-time scale-independent modifications to the mat- ter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [125, 126] describe a third possibility for the split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Their growth parameters affect the growth index 𝛾, which is a single parameter that approximately describes the ΛCDM growth history in the late Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Several extensions to this paper come to mind: Firstly, we already mentioned that it will be important to study the im- pact of other lens samples, in particular the MagLim sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Secondly, additional cosmological information from ex- ternal datasets, such as including more scales of the CMB temperature and polarization power spectrum, and adding CMB lensing are near-term extensions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The 𝑤CDM split would also benefit from extra information on 𝑤geo from the observed DES Type IA supernova included in the new Phanteon+ sample [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Thirdly, we plan to include small-scale information to increase the constraining power on growth-geometry split parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' by modeling baryons in cosmic shear as in [116] or modeling galaxy bias in 2x2pt via effective field theory [128] or via Halo Occupation Distri- bution models [102, 129, 130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' While this paper does not show any hints of new physics beyond ΛCDM, future datasets from Rubin Observatory’s LSST [131], the Roman Space Telescope [132], and the Euclid mission [133], in combination with the Dark Energy Spectro- scopic Instrument [134], Simons Observatory [135] and the CMB-S4 mission [136] will significantly tighten the statis- tical error budget on cosmological models beyond ΛCDM and 𝑤CDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' It is now timely to develop the theoretical tool- box to efficiently and consistently explore these models across datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Simulations in this paper use High Performance Comput- ing (HPC) resources supported by the University of Arizona TRIF, UITS, and RDI and maintained by the UA Research Technologies department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' The authors would also like to thank Stony Brook Research Computing and Cyberinfrastruc- ture, and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system, which was made possible by a $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content='4M National Science Foundation grant (#1531492).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' TE and JX are supported by the Department of Energy grant DE- 16 SC0020215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' EK is supported by the Department of Energy grant DESC0020247 and an Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Sloan Research Fellow- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf'} +page_content=' V.' metadata={'source': 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Shaping +Lina Mezghani +Meta AI, Inria∗ +linamezghani@fb.com +Sainbayar Sukhbaatar +Meta AI +Piotr Bojanowski +Meta AI +Alessandro Lazaric +Meta AI +Karteek Alahari +Inria∗ +Abstract: +Developing agents that can execute multiple skills by learning from pre-collected +datasets is an important problem in robotics, where online interaction with the en- +vironment is extremely time-consuming. Moreover, manually designing reward +functions for every single desired skill is prohibitive. Prior works [1, 2] targeted +these challenges by learning goal-conditioned policies from offline datasets with- +out manually specified rewards, through hindsight relabeling. +These methods +suffer from the issue of sparsity of rewards, and fail at long-horizon tasks. In +this work, we propose a novel self-supervised learning phase on the pre-collected +dataset to understand the structure and the dynamics of the model, and shape a +dense reward function for learning policies offline. We evaluate our method on +three continuous control tasks, and show that our model significantly outperforms +existing approaches [1, 2], especially on tasks that involve long-term planning. +Keywords: Offline RL, Self-Supervised Learning, Goal-Conditioned RL +1 +Introduction +While the goal of realizing general autonomous agents requires mastery of a large and diverse set +of skills, achieving this by focusing on each skill individually with standard reinforcement learning +(RL) frameworks is prohibitive. This is primarily due to the need for manually designed reward func- +tions and environment interactions for each skill. Unsupervised RL has opened a way for learning +agents that can execute diverse abilities without supervision (i.e., hand-crafted rewards), and then be +further adapted to downstream tasks through few-shot or zero-shot generalization [3, 4, 5, 6]. How- +ever, learning policies with such methods is impractical with real robots as they require millions of +interactions when trained online. +Recently, a line of study has emerged that uses pre-collected datasets of trajectories and trains poli- +cies offline (i.e., without additional interactions with the environment) [7, 8]. More precisely, given +a dataset of reward-free trajectories and a reward function designed to solve a specific task, the agent +learns offline by relabeling the transitions in the dataset with the reward function. This setting is par- +ticularly relevant in robotics, where data collection is extremely time-consuming: disentangling data +collection and policy learning in this context allows for faster policy iteration. However, it would +require designing one specific reward function and learning one policy for each individual task. +An important question to scale offline robot learning is therefore to find ways of learning multi-task +policies from already collected datasets. Recent works [1, 9, 10], have targeted this problem from a +goal-conditioned perspective: given a dataset of previously collected trajectories, the objective is to +learn a goal-oriented agent that can reach any state in the dataset. The advantages of this formulation +are two-fold: first, it makes it easy to interpret skills, and second it does not require any adaptation at +∗Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France +Project page: https://linamezghani.github.io/go-fresh +6th Conference on Robot Learning (CoRL 2022), Auckland, New Zealand. +arXiv:2301.02099v1 [cs.RO] 5 Jan 2023 + +test time. Making this framework unsupervised requires to break free from hand-crafted rewards, as +proposed by Chebotar et al. [1], where they learn goal-conditioned policies offline through hindsight +relabeling [2]. However, their approach is subject to the pitfall of learning from sparse rewards, and +can be inefficient in long-horizon tasks. +In this work, we present a self-supervised reward shaping method that enables building an offline +dataset with dense rewards. To this end, we develop a self-supervised learning phase that aims +at learning the structure and dynamics of the environment before training the policy. During this +phase, we: (i) train a reachability network [11] to estimate the local distance in the state space S, +then (ii) extract a set of representative states that covers S, and finally (iii) build a graph on this +set to approximate the global distance in S. When training the goal-conditioned policy, we use the +graph in two ways: to compute rewards through shortest path distance, and to create transitions of +intermediate difficulty on the path to the goal. +We evaluate our method on complex continuous control tasks, and compare it to previous state- +of-the-art offline [1, 2] approaches. We show that our graph-based reward method learns good +goal-conditioned policies by leveraging transitions from a dataset of past experience with neither +any additional interactions with the environment nor manually-designed rewards. Moreover, we +show that, contrary to prior work that uses datasets collected with a policy trained with supervised +rewards [1], our method allows for learning goal-conditioned policies even from datasets of poor +quality, e.g. containing trajectories sampled with a random policy. Our work is thus the first to learn +goal-conditioned policies from offline datasets without any supervision, as it does not require any +hand-crafted reward function at any stage: data collection, policy training and evaluation. +2 +Related Work +Goal-conditioned RL. +In its original formulation, goal-conditioned reinforcement learning was +tackled by several methods [12, 13, 2, 14]. The policy learning process is supervised in these works: +the set of evaluation goals is available at train time as well as a reward function that guides the agent +to the goal. Several works propose solutions for generating goals automatically when training goal- +conditioned policies, including self-play [15, 16, 17], and adversarial student-teacher policies [18]. +A recent line of research [19, 20, 21, 22, 23, 24, 25, 26] focuses on learning goal-conditioned policies +in an unsupervised fashion. The objective is to train general agents that can reach any goal state in +the environment without any supervision (reward, goal-reaching function) at train time. In particular, +Mendonca et al. [25] trains a model-based agent that learns to discover novel goals with an explorer +model, and reach them with an achiever policy via imagined rollouts. +Offline RL. +The data collection technique is an important aspect when studying the training of +policies from pre-collected datasets. In this context, the first works assumed access to policies +trained with task-specific rewards [27, 28]. More recently, methods proposed to leverage unsuper- +vised exploration to collect datasets for offline RL [7, 8]. In particular, Yarats et al. [7] creates a +dataset of pre-collected trajectories, ExoRL, on the DeepMind control suite [29] generated without +any hand-crafted rewards. Similar to URLB [30], ExoRL benchmarks a number of exploration al- +gorithms [3, 6, 31, 5], and evaluates the performance of a policy trained on the corresponding offline +datasets relabeled with task-specific rewards. +Multi-task Offline RL. +Recent works proposed to learn multiple tasks from pre-collected datasets, +starting with methods [32] that generate goals to improve the offline data collection process in a self- +supervised way. This connection has also been studied in the supervised setting [9, 33] and when +learning hierarchical policies [10]. In a setting closely related to our work, Actionable Models [1] +considers the problem of learning goal-conditioned policies from offline datasets without interacting +with the environment, and with no task-specific rewards. They employ goal-conditioned Q-learning +with hindsight relabeling [2]. As opposed to their work that relies on learning from sparse rewards, +we propose to leverage a self-supervised training stage to densely shape rewards. +3 +Preliminaries +Let E = (S, A, P, p0, γ, T) define a reward-free Markov decision process (MDP), where S and A +are state and action spaces respectively, P : S × A × S → IR+ is a state-transition probability +2 + +add edge +add node +Figure 1: Overview of the graph building algo- +rithm. Given a transition (si, si+1) ∈ D, we add +si as node if it is distant enough from existing +nodes in the graph. Moreover, we add an edge in +the graph between the incoming nearest neighbor +of si and the outgoing nearest neighbor of si+1. +function, p0 : S → IR+ is an initial state distribution, γ is the discount factor, and T is the task +horizon. In the goal-conditioned setting, the objective is to learn a policy π : S × G → A that +maximizes the expectation of the cumulative return over the goal distribution, where G denotes the +goal space. Here, we make the common assumption that states and goals are defined in the same +form, i.e., G ⊂ S. +We assume that we have access to a dataset D of pre-collected episodes generated by using any data +collection algorithm in E. Each episode is stored in D as a series of (s, a, s′) tuples, where s, s′ ∈ S +and a ∈ A. In the general offline formulation introduced by Yarats et al. [7], the dataset D can be +relabeled by evaluating any reward function r : S × A → IR at each tuple in D, and adding the +resulting tuple (s, a, r(s, a), s′) in the relabeled dataset Dr. We can extend this protocol to the goal- +oriented setting by considering a goal distribution pG in the goal space, and any goal-conditioned +reward function r : S × A × G → IR. Given a tuple (s, a, s′) in D, we relabel it by sampling a goal +g ∼ pG, computing r(s, a, g) and adding the resulting tuple (s, a, g, r(s, a, g), s′) in the relabeled +dataset Dr,pG. +Once the relabeled dataset Dr,pG is generated, we can learn a goal-conditioned policy by executing +any offline RL algorithm. The algorithm runs completely offline, by sampling tuples from Dr,pG and +without any interaction with the environment. The goal-conditioned policy is then evaluated online +in E on a set of fixed evaluation goals that is not known during training. +4 +Self-supervised Reward Shaping +We now describe our self-supervised reward shaping method. It comprises three stages that we will +detail below. In the first stage, we train a Reachability Network (RNet) [11] on the trajectories in D +to predict whether two states are reachable from one another. The second stage consists in building +a directed graph M whose nodes are a subset of states in D, and edges connect reachable states. +We employ the RNet as a criterion to avoid adding similar states to M so that its nodes cover the +states in D uniformly. The final stage consists in training the goal-conditioned policy on transitions +and goals sampled from D. It is trained with dense rewards computed as the sum of a global (based +on the graph distance in M) and local (based on the RNet) distance terms. The important aspect of +our method is that the whole training only uses trajectories from the pre-collected dataset D without +running a single action in the environment. We now describe each component in more detail. +4.1 +Reachability network +In order to learn a good local distance between states in D, we adopt an asymmetric version of +the Reachability Network (RNet) [11]. The general idea of RNet is to approximate the distance +between states in the environment by the average number of steps it takes for a random policy to +go from one state to another. We adapted the original formulation with two modifications: first, +we use exploration trajectories from D instead of random trajectories and second, we leverage the +temporal direction because a state can be reachable from another without the converse being true. +Let (sa +1, ..., sa +T ) denote a trajectory in D, where a is a trajectory index. We define a reachability label +yab +ij for each pair of observations (sa +i , sb +j) by +yab +ij = +�1 +if a = b and 0 ≤ j − i ≤ τreach, +0 +otherwise, +for 1 ≤ i, j ≤ T, +(1) +where the reachability threshold τreach is a hyperparameter. The reachability label is equal to 1 iff +the states are in the same trajectory and the number of steps from sa +i to sb +j is below τreach, as shown +3 + +(a) Training labels for RNet +1 +0.2 +0.5 +0.8 +shortest path +(b) Visualisation of the reward computation +Figure 2: Visualization of our dense reward shaping method. (a) shows how training labels are +generated for training the RNet: given a state si, positive pairs are sampled in the same trajectory +within a threshold τreach, and the rest of the trajectory forms negative pairs. (b) presents how rewards +are implemented as a combination of a global distance term (green), computed with the shortest +path in the graph between the outgoing nearest neighbor (NNout) of the state st+1 and the incoming +nearest neighbor of the goal (NNin), and a local distance term (red) computed using the RNet value +between NNin and g. +in Figure 2a. Note that yab +ij ̸= yab +ji . We train a siamese neural network R, the RNet, to predict the +reachability label yab +ij from a pair of observations (sa +i , sb +j) in D. The RNet consists of an embedding +network g, and a fully-connected network f to compare the embeddings, i.e., +R(sa +i , sb +j) = σ +� +f(g(sa +i ), g(sb +j)) +� +, +(2) +where σ is a sigmoid function. A higher R value indicates two states reachable easily with random +walk, so they can be considered close in the environment. More precisely, R takes values in (0, 1) +and s′ is reachable from s if R(s, s′) ≥ 0.5. RNet is learned in a self-supervised fashion, as the +ground-truth labels needed to train the network are generated automatically. +4.2 +Directed graph +In the next phase, we use trajectories in D to build a directed graph M that captures high-level +dynamics of the environment, as illustrated in Figure 1. We want the nodes of M to evenly represent +the states in D. This is achieved by filtering the states in D: a state is added to M only if it is distant +enough from all the other nodes in M. More precisely, a state s ∈ D is added to M if and only if +R(s, n) < 0.5 and R(n, s) < 0.5, +for all n ∈ M. +(3) +Note that we require both the directions to be novel. This filtering avoids redundancy by preventing +similar states to be added to the memory. It also has a balancing effect because it limits the number +of states that can be added from a certain area even if it is visited by the agent many times in D. +Once the nodes are selected, we connect pairs that are reachable from one to another. To this end, we +employ trajectories in D because they contain actual feasible transitions. Given a transition si → sj +in D, we add edge ni → nj if si can be reached from node ni and node nj can be reached from +sj. This way, we have a chain ni → si → sj → nj and can assume nj is reachable from ni. +Concretely, we select node ni to be the incoming nearest neighbor (NNin) to si, and nj to be the +outgoing nearest neighbor (NNout) from sj, i.e., +ni = NNin(si) = argmax +n∈M +R(n, si), +nj = NNout(sj) = argmax +n∈M +R(sj, n). +(4) +By performing this action over all the transitions in D, we turn M into a directed graph where edges +represent reachability from one node to another. +4.3 +Distance function for policy training +We then use the obtained directed graph to compute a global distance in the state space. Indeed, +RNet predicts reachability between si and sj so we can directly use it as a distance metric +dl(si, sj) = 1 − R(si, sj), +∀si, sj ∈ S. +(5) +4 + +R(Si, Si) +Sirt = - global distance da - local distance dHowever, this reachability metric is confined to a certain threshold, so there is no guarantee that the +RNet predictions will have good global properties. +In contrast, the directed graph M captures high-level global dynamics of the environment. We can +easily derive a distance function dM(ni, nj) between any pair of nodes in M by computing the +length of the shortest path in this graph, provided the graph is connected. In practice, we can use a +trick to connect the graph if necessary, by adding an edge between the pair of nodes from different +connected components with the maximum RNet value. Moreover, we can extend this distance dM +to a global distance function dg in the state space S by finding, for any pair si and sj in S their +nearest neighbors in the corresponding direction. More precisely, +dg(si, sj) = dM(NNout(si), NNin(sj)), +∀si, sj ∈ S. +(6) +The distance dg between two states in the state space becomes the length of the shortest path between +their respective closest nodes in the graph. This process, summarized in Figure 2b, propagates +the good local properties of RNet to get a well-shaped distance function for states that are further +away. Since dg captures global distances while dl captures local fine-grained distance, we use their +combination as a final distance function: ∀si, sj ∈ S, +d(si, sj) = dg(si, sj) + dl(si, sj). +4.4 +Policy training +The last phase of our method is training the goal-conditioned policy offline. Here, we create an of- +fline replay buffer B that is filled with relabeled data. We randomly sample a transition (st, at, st+1) +from D as well as a goal g and relabel the transition with reward rt = −d(st+1, g). We then push the +relabeled transition (st, at, g, rt, st+1) to B. In order to create a curriculum that artificially guides +the agent towards the goal, we experimented with two different transition augmentation techniques: +Sub-goal augmentation. +Let (st, at, g, rt, st+1) denote a relabeled transition and (n0, ..., nP −1) +the shortest path in the graph M between n0 = NNout(st) and nP −1 = NNin(g). The augmen- +tation technique consists in adding to the replay buffer every transition (st, at, ni, ri +t, st+1) for all +i ∈ {0, P − 1}, where ri +t = −d(st+1, ni). In other words, given a transition (st, at, st+1) and +a goal g from D, we push to the replay buffer a set of relabeled transitions with all goals on the +shortest path from st to g (and their corresponding rewards). +Edge augmentation. +Similar to the subgoal augmentation technique, we consider a relabeled tran- +sition (st, at, g, rt, st+1) and the associated shortest path (n0, ..., nP −1). This time, we keep the +same goal g for every augmented transition, but for every edge (ni−1, ni), i ∈ {1, P − 1}, we +add the relabeled transition (si +t, ai +t, g, ri +t, si +t+1) to B where (si +t, ai +t, si +t+1) ∈ D, NNout(si +t) = ni−1, +NNin(si +t+1) = ni and ri +t = −d(si +t, g). Note that the existence of such a transition in D is guaranteed +by construction: an edge is added to the graph from one node to another iff there exist a transition +in D whose corresponding nearest neighbors are these two nodes (in the same order). +Once the replay buffer B is filled, the goal-conditioned policy can be trained using any off-policy +algorithm. In our implementation, we chose Soft Actor-Critic [34], as it is known to require few +hyper-parameter tuning, and is widely used in the literature. +5 +Experiments +5.1 +Environments & data collection +We perform experiments on three continuous control tasks with state-based inputs. +UMaze [35]. +The first environment, shown in Figure 3a, is a two-dimensional U-shaped maze +with continuous action space and a fixed initial position. We generate the training data for this +environment by deploying a random policy with randomized start position in the maze. We collect +10k trajectories of length 1k. We evaluate the goal-conditioned agent by giving the agent a goal +sampled at random in the environment and computing the final euclidean distance to the goal. +RoboYoga Walker [25]. +Introduced by Mendonca et al. [25], the challenging RoboYoga bench- +mark is based on the Walker domain of the DeepMind control suite [29], and consists of 12 goals +5 + +(a) UMaze +(b) RNet distance +(c) Graph distance +Avg Room1Room2Room3Room4 +0.0 +0.25 +0.5 +Success Rate +Graph Reward +RNet Reward +(d) RNet vs. Graph Rewards +Figure 3: (a) UMaze environment, Heatmap of rewards computed with RNet (b) and graph (c) +distances, and (d) Performance of the goal-conditioned policy trained with RNet and graph-based +rewards on UMaze. In (b) and (c), high rewards are shown in yellow, and low rewards in black. +that correspond to body poses inspired from yoga (e.g. lying down, raising one leg or balancing). We +consider the state-based version of the task, and use the task-agnostic dataset from Yarats et al. [7] +generated with an unsupervised exploration policy. It contains 10k trajectories of length 1k obtained +by deploying the “proto” [5] algorithm in the Walker domain. The success metric of the evaluation +policy is assessed by the pose of the humanoid at the end of the episode. +Pusher [20]. +We also apply our method on Pusher, a realistic robotic environment shown in Fig- +ure 7 (left), where a robot arm (red) needs to push a puck (blue) to a specified location on a table. +To build the offline dataset, we generated 10k random trajectories of length 200. Similar to prior +works [20, 22, 26], we generated 500 goals at random in the state space, and we measured the +performance as the final Euclidean distance between the puck and its target location. +5.2 +Ablation & design choices +We first show that the graph structure is necessary for long-term planning. Then, we explain the +importance of the directness of the graph on tasks with asymmetric behaviours. Finally, we show +the impact of transition augmentation techniques when labeling data for the goal-conditioned policy. +Necessity of graph-based rewards. +An important component of our method is the construction of +the graph M that enables computing a distance with good global properties. To empirically validate +this hypothesis, we performed a comparison between the goal-conditioned policy trained with RNet +rewards (i.e., by using the distance dl from equation (5)) and the one trained with both distance +terms as reward. We run this experiment on the UMaze environment, and show results in Figure 3d. +We note that the model trained with graph rewards outperforms the one trained with RNet rewards +overall, particularly for distant goals (ie. rooms 3 and 4). We also notice that the model trained with +RNet rewards is slightly better for goals that are close to the initial position. This highlights the +fact that RNet is good at estimating local distances. The qualitative visualization in Figure 3b & 3c +confirms this observation, as it shows low values between states in the first and fourth rooms. +Importance of graph directness. +We then investigate the importance of the asymmetry of the +RNet and the directness of the graph. To this end, we implement an undirected version of our +method where the RNet is symmetric and the graph is undirected. All other components of our +method are unchanged. First, we compare the performance of both variants in the UMaze task in +Figure 4a, and note that asymmetric RNet and directed graph in our approach significantly improve +the goal-conditioned policy performance (+11% on success rate), especially on goals close to the +initial location, i.e., goals in rooms 1 and 2. We then analyze qualitative visualizations of the shortest +path in the undirected and directed graphs in the RoboYoga task, as shown in Figure 4b. In the +undirected case, the humanoid defies the laws of gravity and is encouraged to stand its head by +flipping backwards, which might be extremely difficult, or even infeasible. In the directed case, the +shortest path fosters the agent to first get back on its legs, and then lean forward. In this exemple, +the gravity makes the dynamics of the environment non-symmetric and non-fully reversible, which +justifies the directed formulation described in our method. +Transition sampling strategy. +As a final ablation study, we study the utility of the transition aug- +mentation techniques described in subsection 4.4. We evaluate four possible variants of our method: +6 + +Avg +Room1 +Room2 +Room3 +Room4 +0.0 +0.25 +0.5 +Success Rate +Directed +Undirected +(a) Comparison on the UMaze task +(b) Shortest Path visualization for undirected (top) and directed +(bottom) graphs +Figure 4: Importance of graph directness on (a) the UMaze task and (b) the RoboYoga Walker task. +(i) without any augmentation, (ii) with edge augmentation only, (iii) with subgoal augmentation +only, and (iv) with both augmentations. We execute this experiment on the RoboYoga task, and +show results in Figure 6b. We observe that both of the augmentation techniques improve the per- +formance of the goal-conditioned agent, with subgoal augmentation showing greater improvement. +Moreover, we note that combining both augmentations improves the performance further. For the +reminder of the experiments, we use both these augmentation techniques. +0 +500 +1000 +epoch +0.0 +0.2 +0.4 +0.6 +Success Rate +Average +0 +500 +1000 +epoch +Room 1 +0 +500 +1000 +epoch +Room 2 +0 +500 +1000 +epoch +Room 3 +0 +500 +1000 +epoch +Room 4 +Ours +HER +HER + random negative action +Actionable Models +Figure 5: Performance on the UMaze task. We show the success rate for goals sampled at random +in each of the four rooms, as well as the average over all rooms. +5.3 +Comparison to prior work +Baselines. We compare our method to prior work on unsupervised goal-conditioned policy learning. +We perform an apples-to-apples comparison by implementing the baselines using the same learn- +ing framework as our method, and changing the reward relabeling process. We compare with the +following baselines: +• Hindsight Experience Replay [HER] [2] This is a re-implementation of the standard un- +supervised RL technique, adapted to the offline setting. More precisely, we relabel sub- +trajectories from D with a sparse reward, which is equal to 1 only for the final transition +of the sub-trajectory, and 0 everywhere else. Following Chebotar et al. [1], we also label +sub-trajectories with goals sampled at random in D and zero reward. +• HER [2] with random negative action is a variant of HER where, for a transition in D +we sample an action uniformly at random in the action space and label it with zero reward. +This helps overcoming the problem of over-estimation of the Q-values for unseen actions +mentioned in Chebotar et al. [1]. +• Actionable Models [1] This approach is based on goal-conditioned Q-learning with hind- +sight relabeling. We re-implemented the goal relabeling procedure that uses the Q-value at +the final state of sub-trajectories in D to enable goal chaining, as well as the negative action +sampling trick. +Comparison on UMaze. We compare our method to the baselines on the UMaze task, and show +results in Figure 5. We observe that our model outperforms all baselines overall, and shows greater +improvements on challenging goals that are far from the initial position. Interestingly, we note +that Actionable Models reaches goals in the first room only. This confirms the intuition that sparse +rewards make it difficult for the policy to learn long-horizon tasks. +7 + +0 +500 +1000 +epoch +0.0 +0.2 +0.4 +Success Rate +Ours +HER +HER + random neg action +Actionable Models +(a) Comparison to baselines +0.0 +0.2 +0.4 +Success Rate +Subgoal + Edge +Subgoal Only +Edge Only +No Augmentation +(b) Impact of Transition Augmentation +Figure 6: Performance on the RoboYoga Walker task +Comparison on RoboYoga Walker. In a second experiment, we compare our method to baselines +on the RoboYoga task, as shown in Figure 6a. Here again, our method outperforms prior work, and +Actionable Models does not make any significant improvement over HER. The results broken down +by goal are shown in the supplementary material. Overall these results suggest that our dense reward +shaping method allows for faster and more robust offline goal-conditioned policy training. +0 +500 +1000 +epoch +0.00 +0.04 +0.08 +0.12 Average Distance +0 +500 +1000 +epoch +Hand Distance +0 +500 +1000 +epoch +Puck Distance +Ours +HER +HER + random neg action +Actionable Models +Figure 7: Performance on the Pusher task (lower is better). We report the final average, hand, and +puck distance to the goal for our model and all baselines. +Comparison on Pusher. As a final experiment, we compared our method to prior work on a realistic +robotic environment, as shown in Figure 7. Our policy trained offline is evaluated by sampling +a goal at random in the state space, and measuring three different metrics: (i) the hand distance, +which corresponds to the final distance between the end of the robot arm and the target, (ii) the puck +distance, which measures the distance between the final puck location and the target, and (iii) the +average distance, the average of the first two metrics. Our method outperforms the baselines on this +task, and our goal-conditioned agent is able to sequentially place the puck at the goal location, and +then place the hand at its target location. On the contrary, HER [2] places the puck at the target +location with a performance similar to our method, but lacks precision on the hand location. +6 +Conclusion: Summary and Limitations +We proposed a method for learning multi-task policies from pre-generated datasets in an offline +and unsupervised fashion, i.e., without requiring any additional interaction with the environment, +nor manually designed rewards. Our method leverages a self-supervised stage that aims at learn- +ing the dynamics of the environment from the offline dataset, and that allows for shaping a dense +reward function. It shows significant improvement over prior works based on hindsight relabeling, +especially on long-horizon tasks, where dense rewards are crucial for learning a good policy. +The main limitation of our method is that it relies on the availability of a pre-collected dataset of +trajectories, with a sufficiently large coverage of the state space for proper policy learning. 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Efficient self-supervised data +collection for offline robot learning. In 2021 IEEE International Conference on Robotics and +Automation (ICRA), pages 4650–4656. IEEE, 2021. +[33] Y. J. Ma, J. Yan, D. Jayaraman, and O. Bastani. How far i’ll go: Offline goal-conditioned +reinforcement learning via f-advantage regression. arXiv preprint arXiv:2206.03023, 2022. +[34] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine. Soft actor-critic: Off-policy maximum entropy +deep reinforcement learning with a stochastic actor. In International conference on machine +learning, pages 1861–1870. PMLR, 2018. +[35] Y. Kanagawa. mujoco-maze. https://github.com/kngwyu/mujoco-maze, 2020. +10 + +A +Implementation details +A.1 +Self-supervised reward shaping +We provide pseudo-code for the two stages of our approach: the graph building process is shown in +Algorithm 1, and the steps for filling the replay buffer are shown in Algorithm 2. Here we use the +notation R(s, M) as the maximum RNet value between s and all nodes in M i.e., +R(s, M) := max +m∈M R(s, m) +Algorithm 1 Building the directed graph M +Input: pre-collected dataset D, Reachability Network R +Initialize: M = {} +/ * Build the set of nodes * / +for each state s in D do +if R(s, M) < 0.5 and R(M, s) < 0.5 then +Update M := M ∪ {s} +end if +end for +/ * Build edges * / +for each transition (st, st+1) in D do +Let nt := NNin(st) = argmaxn∈M R(n, st) +Let nt+1 := NNout(st+1) = argmaxn∈M R(st+1, n) +Create directed edge from nt to nt+1 +end for +Algorithm 2 Building replay buffer B for offline policy training +Input: pre-collected dataset D, Reachability Network R, di- +rected graph M +Initialize: B = {} +while B is not full do +Sample a transition (st, at, st+1) at random in D +Sample a goal g at random in D +Compute dl(st+1, g) := 1 − R(st+1, g) +Let nt+1 := NNout(st+1) = argmaxn∈M R(st+1, n) +Let ng := NNin(g) = argmaxn∈M R(n, g) +Compute dg(st+1, g) := ShortestPathLength(nt+1, ng) +Compute reward rt := −(dl(st+1, g) + dg(st+1, g)) +Relabel transition with goal g and reward rt, and +Push (st, at, g, rt, st+1) to B +end while +A.2 +Actionable Models baselines re-implementation +In this section, we provide details for our re-implementation of Actionable Models [1] and HER [2]. +Since we are using the same optimization algorithm for offline policy training for these baselines +and our method, the only difference lies in how the transitions in the pre-collected dataset D are +relabeled to build the replay buffer B. +In HER [2], the idea is to sample a trajectory and a goal g at random in D, and to cut the trajectory +at a step i. Each transition in the trajectory (until step i) is then relabelled twice: once with the goal +g and reward 0 for all transitions, and once with goal si (the final state of the trajectory) and reward +11 + +0 for all transitions except the final one that gets a reward of 1. The pseudo-code for this method is +shown in Figure 8. +Actionable Models [1] relies on a similar idea as in HER [2], but contains two additional steps to +improve the method. The first step is a form of goal chaining and it consists in using the Q-value at +the final state of the trajectory to compute the reward for the final transition. The second step aims at +balancing the unseen action effect, in order to regularize the action space. In practice, it consists in +sampling negative actions from the policy and label zero-reward transitions with these actions. The +implementation of both tricks is shown in Figure 8. +For the implementation of the third baseline, HER + random negative action, the overall algorithm +is the same as HER, except that we also generate transitions with negative actions, similar to Ac- +tionable Models. This time, the negative actions are not sample from the policy, but are generated +uniformly at random from the action space. +Algorithm 3 HER +Input: dataset D +Initialize: B = {} +while B is not full do +Sample trajectory τ ∈ D +Sample goal g ∈ D +Randomly cut τ at step i +for j ∈ {0, ..., i − 2} do +(sj, aj, g, 0, sj+1) → B +(sj, aj, si, 0, sj+1) → B +end for +(si−1, ai−1, g, 0, si) → B +(si−1, ai−1, si, 1, si) → B +end while +Algorithm 4 Actionable Models +Input: dataset D, +goal-conditioned critic network Q, +goal-conditioned policy π +Initialize: B = {} +while B is not full do +Sample trajectory τ ∈ D +Sample goal g ∈ D +Randomly cut τ at step i +for j ∈ {0, ..., i − 2} do +(sj, aj, g, 0, sj+1) → B +a1 ∼ π(sj, g) +(sj, a1, g, 0, sj+1) → B +(sj, aj, si, 0, sj+1) → B +a2 ∼ π(sj, si) +(sj, a2, si, 0, sj+1) → B +end for +(si−1, ai−1, g, Q(si, ai−1, g), si) → B +a3 ∼ π(si−1, g) +(si−1, a3, g, 0, si) → B +(si−1, ai−1, si, 1, si) → B +a4 ∼ π(si−1, si) +(si−1, a4, si, 0, si) → B +end while +Figure 8: Pseudo-code for replay buffer filling with HER [2] and Actionable Models [1] methods. +We compare both implementations by showing in red modifications related to goal chaining, and in +blue edits related to unseen action regularization. +12 + +A.3 +Hyper-parameters +We first list the hyper-parameters for the self-supervised reward shaping phase in Table 1. Table 2 +details the hyper-parameters for the offline policy training stage with SAC [34]. For the Action- +able Models [1] and HER [2] baselines, we used the same parameters as in our approach, with the +exception of some parameters specific to these methods, shown in Table 3. +These hyper-parameters were obtained by performing a random search for all the methods over +several parameter values. All the experiments in this work were performed on 3 random seeds. +Common hyper-parameter +Value +Task +UMaze +RoboYoga +Pusher +Number of training pairs +5 × 105 +5 × 105 +5 × 105 +Ratio of negatives +0.5 +0.5 +0.5 +Ratio of negatives from same trajectory +0.5 +0.5 +0.5 +Reachability threshold (τreach) +5 +2 +10 +Weight of local distance in reward +1 +1 +100 +Batch size +512 +512 +512 +Learning rate +0.001 +0.0003 +0.001 +Weight decay +0.00001 +0.00001 +0.00001 +Total number of training epochs +100 +100 +100 +Capacity of the directed graph +1000 +10000 +1000 +Table 1: Hyper-parameters for reachability network training and directed graph construction. +Hyper-parameter +Value +Task +UMaze +RoboYoga +Pusher +Replay buffer capacity +106 +106 +106 +Batch size +2048 +2048 +2048 +Discount (γ) +0.90 +0.95 +0.95 +Number of updates per epoch +1000 +1000 +1000 +Total number of epochs +1000 +1000 +1000 +Target update interval +1 +1 +1 +Soft update coefficient (τ) +0.005 +0.005 +0.005 +SAC entropy parameter (α) +0.05 +0.01 +0.0001 +Optimizer +Adam +Adam +Adam +Learning rate +0.0003 +0.0003 +0.0005 +Action repeat +1 +2 +1 +Reward scaling factor +0.1 +0.5 +0.01 +Table 2: Hyper-parameters for offline policy learning with SAC [34] with our method. +Hyper-parameter +Value +Task +UMaze +RoboYoga +Pusher +Discount (γ) +0.99 +0.99 +0.99 +SAC entropy parameter (α) +0.01 +0.001 +0.005 +Learning rate +0.0001 +0.0001 +0.0001 +Reward scaling factor +1 +10 +1 +Table 3: Hyper-parameters for offline policy learning with SAC [34] specific to Actionable Mod- +els [1] and HER [2] baselines. +13 + +A.4 +Architecture details +Reachability Network [11] +The RNet has a siamese architecture with two embedding heads (one +for each observation of the pair) with tied weights, and a comparator network that compares both +embeddings and returns a reachability score. For the UMaze task, we used an embedding head with +3 fully-connected layers with batch normalization and Tanh activations, with a hidden size of 64 +and an embedding size of 16. For the Roboyoga Walker task, the embedding network has the same +architecture, but we increased both the hidden and embedding sizes to 128. The comparator network +is also a fully-connected network. It contains batch normalization and ReLU activations. The hidden +size for the UMaze (respectively the RoboYoga Walker) task is set to 16 (resp. 128) and the number +of layers is 2 (resp. 4). +Policy Network +The goal-conditioned policy network takes as input both the observation and +the goal, in separate heads with the same architecture but independent weights. These heads are +implemented as 3-layer fully-connected networks with Tanh activations, hidden size of dimension +64, and 16 dimensions for the feature size. The output from both the heads is then concatenated and +fed into a 2-layer fully-connected network of width 256. The critic network has the same architecture +for both observation and goal heads, and is followed by 3 fully connected-layers of width 256. +B +Full results on RoboYoga Walker task +We show the comparison of our method against the aforementioned baselines on each of the 12 +goals of the RoboYoga Walker task in Figure 9. These goals are illustrated in Figure 10. We see that +our method masters most of the goals that do not require balancing (Lie Back & Front, Legs Up, +Lunge), and succeeds quite well at more complicated goals like Side Angle, Lean Back and Bridge, +but is unable to progress in complex goals like Head Stand or Arabesque. +0.0 +0.5 +1.0 +Success Rate +Lie Back +Lie Front +Legs Up +Lunge +0.0 +0.5 +1.0 +Success Rate +Side Angle +Stand +Lean Back +Boat +0 +500 +1000 +epoch +0.0 +0.5 +1.0 +Success Rate +Bridge +0 +500 +1000 +epoch +Stand One Feet +0 +500 +1000 +epoch +Head Stand +0 +500 +1000 +epoch +Arabesque +Ours +HER +HER + random neg action +Actionable Models +Figure 9: Performance on the RoboYoga Walker talk for each of the 12 goals. +14 + +Kitchen +Quadruped +Walker +Bins +Reach Left +Reach Right +Push Front +Push Back +Push Both Front +Push Both Back +Lie Back +Lie Front +Legs Up +Lunge +Side Angle +Stand +Burner +Light +Slide +Hinge +Microwave +Kettle +Lean Back +Boat +Bridge +Stand One Feet +Head Stand +Arabesque +Lie Back +Stretch +Lie Back 2 +Legs Up +Lie Side +Lie Side 2 +Stand +Stand 2 +Point +Attack +Balance +Balance 2 +Light + Slide +Light + Hinge +Light + Kettle +Slide + Hinge +Slide + Kettle +Hinge + Kettle +Place Front +Place Both Front +Kitchen +Quadruped +Walker +Bins +Reach Left +Reach Right +Push Front +Push Back +Push Both Front +Push Both Back +Lie Back +Lie Front +Legs Up +Lunge +Side Angle +Stand +Burner +Light +Slide +Hinge +Microwave +Kettle +Lean Back +Boat +Bridge +Stand One Feet +Head Stand +Arabesque +Lie Back +Stretch +Lie Back 2 +Legs Up +Lie Side +Lie Side 2 +Stand +Stand 2 +Point +Attack +Balance +Balance 2 +Light + Slide +Light + Hinge +Light + Kettle +Slide + Hinge +Slide + Kettle +Hinge + Kettle +Place Front +Place Both Front +Figure 10: Visualization of the 12 evaluation goals for the RoboYoga Walker task. +15 + +人人 \ No newline at end of file diff --git a/odA0T4oBgHgl3EQfJ__J/content/tmp_files/load_file.txt b/odA0T4oBgHgl3EQfJ__J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e3e3a1a33e34ee0e0b675af5de589b1fba29c8d --- /dev/null +++ b/odA0T4oBgHgl3EQfJ__J/content/tmp_files/load_file.txt @@ -0,0 +1,834 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf,len=833 +page_content='Learning Goal-Conditioned Policies Offline with Self-Supervised Reward Shaping Lina Mezghani Meta AI, Inria∗ linamezghani@fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='com Sainbayar Sukhbaatar Meta AI Piotr Bojanowski Meta AI Alessandro Lazaric Meta AI Karteek Alahari Inria∗ Abstract: Developing agents that can execute multiple skills by learning from pre-collected datasets is an important problem in robotics, where online interaction with the en- vironment is extremely time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Moreover, manually designing reward functions for every single desired skill is prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Prior works [1, 2] targeted these challenges by learning goal-conditioned policies from offline datasets with- out manually specified rewards, through hindsight relabeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' These methods suffer from the issue of sparsity of rewards, and fail at long-horizon tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In this work, we propose a novel self-supervised learning phase on the pre-collected dataset to understand the structure and the dynamics of the model, and shape a dense reward function for learning policies offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We evaluate our method on three continuous control tasks, and show that our model significantly outperforms existing approaches [1, 2], especially on tasks that involve long-term planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Keywords: Offline RL, Self-Supervised Learning, Goal-Conditioned RL 1 Introduction While the goal of realizing general autonomous agents requires mastery of a large and diverse set of skills, achieving this by focusing on each skill individually with standard reinforcement learning (RL) frameworks is prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This is primarily due to the need for manually designed reward func- tions and environment interactions for each skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Unsupervised RL has opened a way for learning agents that can execute diverse abilities without supervision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', hand-crafted rewards), and then be further adapted to downstream tasks through few-shot or zero-shot generalization [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' How- ever, learning policies with such methods is impractical with real robots as they require millions of interactions when trained online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Recently, a line of study has emerged that uses pre-collected datasets of trajectories and trains poli- cies offline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', without additional interactions with the environment) [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More precisely, given a dataset of reward-free trajectories and a reward function designed to solve a specific task, the agent learns offline by relabeling the transitions in the dataset with the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This setting is par- ticularly relevant in robotics, where data collection is extremely time-consuming: disentangling data collection and policy learning in this context allows for faster policy iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' However, it would require designing one specific reward function and learning one policy for each individual task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' An important question to scale offline robot learning is therefore to find ways of learning multi-task policies from already collected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Recent works [1, 9, 10], have targeted this problem from a goal-conditioned perspective: given a dataset of previously collected trajectories, the objective is to learn a goal-oriented agent that can reach any state in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The advantages of this formulation are two-fold: first, it makes it easy to interpret skills, and second it does not require any adaptation at ∗Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France Project page: https://linamezghani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='io/go-fresh 6th Conference on Robot Learning (CoRL 2022), Auckland, New Zealand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='02099v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='RO] 5 Jan 2023 test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Making this framework unsupervised requires to break free from hand-crafted rewards, as proposed by Chebotar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [1], where they learn goal-conditioned policies offline through hindsight relabeling [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' However, their approach is subject to the pitfall of learning from sparse rewards, and can be inefficient in long-horizon tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In this work, we present a self-supervised reward shaping method that enables building an offline dataset with dense rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' To this end, we develop a self-supervised learning phase that aims at learning the structure and dynamics of the environment before training the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' During this phase, we: (i) train a reachability network [11] to estimate the local distance in the state space S, then (ii) extract a set of representative states that covers S, and finally (iii) build a graph on this set to approximate the global distance in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' When training the goal-conditioned policy, we use the graph in two ways: to compute rewards through shortest path distance, and to create transitions of intermediate difficulty on the path to the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We evaluate our method on complex continuous control tasks, and compare it to previous state- of-the-art offline [1, 2] approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We show that our graph-based reward method learns good goal-conditioned policies by leveraging transitions from a dataset of past experience with neither any additional interactions with the environment nor manually-designed rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Moreover, we show that, contrary to prior work that uses datasets collected with a policy trained with supervised rewards [1], our method allows for learning goal-conditioned policies even from datasets of poor quality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' containing trajectories sampled with a random policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Our work is thus the first to learn goal-conditioned policies from offline datasets without any supervision, as it does not require any hand-crafted reward function at any stage: data collection, policy training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 2 Related Work Goal-conditioned RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In its original formulation, goal-conditioned reinforcement learning was tackled by several methods [12, 13, 2, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The policy learning process is supervised in these works: the set of evaluation goals is available at train time as well as a reward function that guides the agent to the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Several works propose solutions for generating goals automatically when training goal- conditioned policies, including self-play [15, 16, 17], and adversarial student-teacher policies [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' A recent line of research [19, 20, 21, 22, 23, 24, 25, 26] focuses on learning goal-conditioned policies in an unsupervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The objective is to train general agents that can reach any goal state in the environment without any supervision (reward, goal-reaching function) at train time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In particular, Mendonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [25] trains a model-based agent that learns to discover novel goals with an explorer model, and reach them with an achiever policy via imagined rollouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The data collection technique is an important aspect when studying the training of policies from pre-collected datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In this context, the first works assumed access to policies trained with task-specific rewards [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More recently, methods proposed to leverage unsuper- vised exploration to collect datasets for offline RL [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In particular, Yarats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [7] creates a dataset of pre-collected trajectories, ExoRL, on the DeepMind control suite [29] generated without any hand-crafted rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Similar to URLB [30], ExoRL benchmarks a number of exploration al- gorithms [3, 6, 31, 5], and evaluates the performance of a policy trained on the corresponding offline datasets relabeled with task-specific rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Multi-task Offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Recent works proposed to learn multiple tasks from pre-collected datasets, starting with methods [32] that generate goals to improve the offline data collection process in a self- supervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This connection has also been studied in the supervised setting [9, 33] and when learning hierarchical policies [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In a setting closely related to our work, Actionable Models [1] considers the problem of learning goal-conditioned policies from offline datasets without interacting with the environment, and with no task-specific rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' They employ goal-conditioned Q-learning with hindsight relabeling [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' As opposed to their work that relies on learning from sparse rewards, we propose to leverage a self-supervised training stage to densely shape rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 3 Preliminaries Let E = (S, A, P, p0, γ, T) define a reward-free Markov decision process (MDP), where S and A are state and action spaces respectively, P : S × A × S → IR+ is a state-transition probability 2 add edge add node Figure 1: Overview of the graph building algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Given a transition (si, si+1) ∈ D, we add si as node if it is distant enough from existing nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Moreover, we add an edge in the graph between the incoming nearest neighbor of si and the outgoing nearest neighbor of si+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' function, p0 : S → IR+ is an initial state distribution, γ is the discount factor, and T is the task horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In the goal-conditioned setting, the objective is to learn a policy π : S × G → A that maximizes the expectation of the cumulative return over the goal distribution, where G denotes the goal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Here, we make the common assumption that states and goals are defined in the same form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', G ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We assume that we have access to a dataset D of pre-collected episodes generated by using any data collection algorithm in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Each episode is stored in D as a series of (s, a, s′) tuples, where s, s′ ∈ S and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In the general offline formulation introduced by Yarats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [7], the dataset D can be relabeled by evaluating any reward function r : S × A → IR at each tuple in D, and adding the resulting tuple (s, a, r(s, a), s′) in the relabeled dataset Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We can extend this protocol to the goal- oriented setting by considering a goal distribution pG in the goal space, and any goal-conditioned reward function r : S × A × G → IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Given a tuple (s, a, s′) in D, we relabel it by sampling a goal g ∼ pG, computing r(s, a, g) and adding the resulting tuple (s, a, g, r(s, a, g), s′) in the relabeled dataset Dr,pG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Once the relabeled dataset Dr,pG is generated, we can learn a goal-conditioned policy by executing any offline RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The algorithm runs completely offline, by sampling tuples from Dr,pG and without any interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The goal-conditioned policy is then evaluated online in E on a set of fixed evaluation goals that is not known during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4 Self-supervised Reward Shaping We now describe our self-supervised reward shaping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It comprises three stages that we will detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In the first stage, we train a Reachability Network (RNet) [11] on the trajectories in D to predict whether two states are reachable from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The second stage consists in building a directed graph M whose nodes are a subset of states in D, and edges connect reachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We employ the RNet as a criterion to avoid adding similar states to M so that its nodes cover the states in D uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The final stage consists in training the goal-conditioned policy on transitions and goals sampled from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It is trained with dense rewards computed as the sum of a global (based on the graph distance in M) and local (based on the RNet) distance terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The important aspect of our method is that the whole training only uses trajectories from the pre-collected dataset D without running a single action in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We now describe each component in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='1 Reachability network In order to learn a good local distance between states in D, we adopt an asymmetric version of the Reachability Network (RNet) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The general idea of RNet is to approximate the distance between states in the environment by the average number of steps it takes for a random policy to go from one state to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We adapted the original formulation with two modifications: first, we use exploration trajectories from D instead of random trajectories and second, we leverage the temporal direction because a state can be reachable from another without the converse being true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Let (sa 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', sa T ) denote a trajectory in D, where a is a trajectory index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We define a reachability label yab ij for each pair of observations (sa i , sb j) by yab ij = �1 if a = b and 0 ≤ j − i ≤ τreach, 0 otherwise, for 1 ≤ i, j ≤ T, (1) where the reachability threshold τreach is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The reachability label is equal to 1 iff the states are in the same trajectory and the number of steps from sa i to sb j is below τreach, as shown 3 (a) Training labels for RNet 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='8 shortest path (b) Visualisation of the reward computation Figure 2: Visualization of our dense reward shaping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (a) shows how training labels are generated for training the RNet: given a state si, positive pairs are sampled in the same trajectory within a threshold τreach, and the rest of the trajectory forms negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (b) presents how rewards are implemented as a combination of a global distance term (green), computed with the shortest path in the graph between the outgoing nearest neighbor (NNout) of the state st+1 and the incoming nearest neighbor of the goal (NNin), and a local distance term (red) computed using the RNet value between NNin and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Note that yab ij ̸= yab ji .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We train a siamese neural network R, the RNet, to predict the reachability label yab ij from a pair of observations (sa i , sb j) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The RNet consists of an embedding network g, and a fully-connected network f to compare the embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', R(sa i , sb j) = σ � f(g(sa i ), g(sb j)) � , (2) where σ is a sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' A higher R value indicates two states reachable easily with random walk, so they can be considered close in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More precisely, R takes values in (0, 1) and s′ is reachable from s if R(s, s′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' RNet is learned in a self-supervised fashion, as the ground-truth labels needed to train the network are generated automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 Directed graph In the next phase, we use trajectories in D to build a directed graph M that captures high-level dynamics of the environment, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We want the nodes of M to evenly represent the states in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This is achieved by filtering the states in D: a state is added to M only if it is distant enough from all the other nodes in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More precisely, a state s ∈ D is added to M if and only if R(s, n) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 and R(n, s) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5, for all n ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (3) Note that we require both the directions to be novel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This filtering avoids redundancy by preventing similar states to be added to the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It also has a balancing effect because it limits the number of states that can be added from a certain area even if it is visited by the agent many times in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Once the nodes are selected, we connect pairs that are reachable from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' To this end, we employ trajectories in D because they contain actual feasible transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Given a transition si → sj in D, we add edge ni → nj if si can be reached from node ni and node nj can be reached from sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This way, we have a chain ni → si → sj → nj and can assume nj is reachable from ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Concretely, we select node ni to be the incoming nearest neighbor (NNin) to si, and nj to be the outgoing nearest neighbor (NNout) from sj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', ni = NNin(si) = argmax n∈M R(n, si), nj = NNout(sj) = argmax n∈M R(sj, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (4) By performing this action over all the transitions in D, we turn M into a directed graph where edges represent reachability from one node to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='3 Distance function for policy training We then use the obtained directed graph to compute a global distance in the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Indeed, RNet predicts reachability between si and sj so we can directly use it as a distance metric dl(si, sj) = 1 − R(si, sj), ∀si, sj ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (5) 4 R(Si, Si) Sirt = - global distance da - local distance dHowever, this reachability metric is confined to a certain threshold, so there is no guarantee that the RNet predictions will have good global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In contrast, the directed graph M captures high-level global dynamics of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We can easily derive a distance function dM(ni, nj) between any pair of nodes in M by computing the length of the shortest path in this graph, provided the graph is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In practice, we can use a trick to connect the graph if necessary, by adding an edge between the pair of nodes from different connected components with the maximum RNet value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Moreover, we can extend this distance dM to a global distance function dg in the state space S by finding, for any pair si and sj in S their nearest neighbors in the corresponding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More precisely, dg(si, sj) = dM(NNout(si), NNin(sj)), ∀si, sj ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (6) The distance dg between two states in the state space becomes the length of the shortest path between their respective closest nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This process, summarized in Figure 2b, propagates the good local properties of RNet to get a well-shaped distance function for states that are further away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Since dg captures global distances while dl captures local fine-grained distance, we use their combination as a final distance function: ∀si, sj ∈ S, d(si, sj) = dg(si, sj) + dl(si, sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4 Policy training The last phase of our method is training the goal-conditioned policy offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Here, we create an of- fline replay buffer B that is filled with relabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We randomly sample a transition (st, at, st+1) from D as well as a goal g and relabel the transition with reward rt = −d(st+1, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We then push the relabeled transition (st, at, g, rt, st+1) to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In order to create a curriculum that artificially guides the agent towards the goal, we experimented with two different transition augmentation techniques: Sub-goal augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Let (st, at, g, rt, st+1) denote a relabeled transition and (n0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', nP −1) the shortest path in the graph M between n0 = NNout(st) and nP −1 = NNin(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The augmen- tation technique consists in adding to the replay buffer every transition (st, at, ni, ri t, st+1) for all i ∈ {0, P − 1}, where ri t = −d(st+1, ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In other words, given a transition (st, at, st+1) and a goal g from D, we push to the replay buffer a set of relabeled transitions with all goals on the shortest path from st to g (and their corresponding rewards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Edge augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Similar to the subgoal augmentation technique, we consider a relabeled tran- sition (st, at, g, rt, st+1) and the associated shortest path (n0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', nP −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This time, we keep the same goal g for every augmented transition, but for every edge (ni−1, ni), i ∈ {1, P − 1}, we add the relabeled transition (si t, ai t, g, ri t, si t+1) to B where (si t, ai t, si t+1) ∈ D, NNout(si t) = ni−1, NNin(si t+1) = ni and ri t = −d(si t, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Note that the existence of such a transition in D is guaranteed by construction: an edge is added to the graph from one node to another iff there exist a transition in D whose corresponding nearest neighbors are these two nodes (in the same order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Once the replay buffer B is filled, the goal-conditioned policy can be trained using any off-policy algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In our implementation, we chose Soft Actor-Critic [34], as it is known to require few hyper-parameter tuning, and is widely used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 5 Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='1 Environments & data collection We perform experiments on three continuous control tasks with state-based inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' UMaze [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The first environment, shown in Figure 3a, is a two-dimensional U-shaped maze with continuous action space and a fixed initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We generate the training data for this environment by deploying a random policy with randomized start position in the maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We collect 10k trajectories of length 1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We evaluate the goal-conditioned agent by giving the agent a goal sampled at random in the environment and computing the final euclidean distance to the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' RoboYoga Walker [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Introduced by Mendonca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [25], the challenging RoboYoga bench- mark is based on the Walker domain of the DeepMind control suite [29], and consists of 12 goals 5 (a) UMaze (b) RNet distance (c) Graph distance Avg Room1Room2Room3Room4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 Success Rate Graph Reward RNet Reward (d) RNet vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Graph Rewards Figure 3: (a) UMaze environment, Heatmap of rewards computed with RNet (b) and graph (c) distances, and (d) Performance of the goal-conditioned policy trained with RNet and graph-based rewards on UMaze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In (b) and (c), high rewards are shown in yellow, and low rewards in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' that correspond to body poses inspired from yoga (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' lying down, raising one leg or balancing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We consider the state-based version of the task, and use the task-agnostic dataset from Yarats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [7] generated with an unsupervised exploration policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It contains 10k trajectories of length 1k obtained by deploying the “proto” [5] algorithm in the Walker domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The success metric of the evaluation policy is assessed by the pose of the humanoid at the end of the episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Pusher [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We also apply our method on Pusher, a realistic robotic environment shown in Fig- ure 7 (left), where a robot arm (red) needs to push a puck (blue) to a specified location on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' To build the offline dataset, we generated 10k random trajectories of length 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Similar to prior works [20, 22, 26], we generated 500 goals at random in the state space, and we measured the performance as the final Euclidean distance between the puck and its target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 Ablation & design choices We first show that the graph structure is necessary for long-term planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Then, we explain the importance of the directness of the graph on tasks with asymmetric behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Finally, we show the impact of transition augmentation techniques when labeling data for the goal-conditioned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Necessity of graph-based rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' An important component of our method is the construction of the graph M that enables computing a distance with good global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' To empirically validate this hypothesis, we performed a comparison between the goal-conditioned policy trained with RNet rewards (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', by using the distance dl from equation (5)) and the one trained with both distance terms as reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We run this experiment on the UMaze environment, and show results in Figure 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We note that the model trained with graph rewards outperforms the one trained with RNet rewards overall, particularly for distant goals (ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' rooms 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We also notice that the model trained with RNet rewards is slightly better for goals that are close to the initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This highlights the fact that RNet is good at estimating local distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The qualitative visualization in Figure 3b & 3c confirms this observation, as it shows low values between states in the first and fourth rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Importance of graph directness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We then investigate the importance of the asymmetry of the RNet and the directness of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' To this end, we implement an undirected version of our method where the RNet is symmetric and the graph is undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' All other components of our method are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' First, we compare the performance of both variants in the UMaze task in Figure 4a, and note that asymmetric RNet and directed graph in our approach significantly improve the goal-conditioned policy performance (+11% on success rate), especially on goals close to the initial location, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', goals in rooms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We then analyze qualitative visualizations of the shortest path in the undirected and directed graphs in the RoboYoga task, as shown in Figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In the undirected case, the humanoid defies the laws of gravity and is encouraged to stand its head by flipping backwards, which might be extremely difficult, or even infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In the directed case, the shortest path fosters the agent to first get back on its legs, and then lean forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In this exemple, the gravity makes the dynamics of the environment non-symmetric and non-fully reversible, which justifies the directed formulation described in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Transition sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' As a final ablation study, we study the utility of the transition aug- mentation techniques described in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We evaluate four possible variants of our method: 6 Avg Room1 Room2 Room3 Room4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 Success Rate Directed Undirected (a) Comparison on the UMaze task (b) Shortest Path visualization for undirected (top) and directed (bottom) graphs Figure 4: Importance of graph directness on (a) the UMaze task and (b) the RoboYoga Walker task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' (i) without any augmentation, (ii) with edge augmentation only, (iii) with subgoal augmentation only, and (iv) with both augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We execute this experiment on the RoboYoga task, and show results in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We observe that both of the augmentation techniques improve the per- formance of the goal-conditioned agent, with subgoal augmentation showing greater improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Moreover, we note that combining both augmentations improves the performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' For the reminder of the experiments, we use both these augmentation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 0 500 1000 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='6 Success Rate Average 0 500 1000 epoch Room 1 0 500 1000 epoch Room 2 0 500 1000 epoch Room 3 0 500 1000 epoch Room 4 Ours HER HER + random negative action Actionable Models Figure 5: Performance on the UMaze task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We show the success rate for goals sampled at random in each of the four rooms, as well as the average over all rooms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='3 Comparison to prior work Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We compare our method to prior work on unsupervised goal-conditioned policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We perform an apples-to-apples comparison by implementing the baselines using the same learn- ing framework as our method, and changing the reward relabeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We compare with the following baselines: Hindsight Experience Replay [HER] [2] This is a re-implementation of the standard un- supervised RL technique, adapted to the offline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' More precisely, we relabel sub- trajectories from D with a sparse reward, which is equal to 1 only for the final transition of the sub-trajectory, and 0 everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Following Chebotar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [1], we also label sub-trajectories with goals sampled at random in D and zero reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' HER [2] with random negative action is a variant of HER where, for a transition in D we sample an action uniformly at random in the action space and label it with zero reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This helps overcoming the problem of over-estimation of the Q-values for unseen actions mentioned in Chebotar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Actionable Models [1] This approach is based on goal-conditioned Q-learning with hind- sight relabeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We re-implemented the goal relabeling procedure that uses the Q-value at the final state of sub-trajectories in D to enable goal chaining, as well as the negative action sampling trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Comparison on UMaze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We compare our method to the baselines on the UMaze task, and show results in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We observe that our model outperforms all baselines overall, and shows greater improvements on challenging goals that are far from the initial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Interestingly, we note that Actionable Models reaches goals in the first room only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This confirms the intuition that sparse rewards make it difficult for the policy to learn long-horizon tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 7 0 500 1000 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4 Success Rate Ours HER HER + random neg action Actionable Models (a) Comparison to baselines 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4 Success Rate Subgoal + Edge Subgoal Only Edge Only No Augmentation (b) Impact of Transition Augmentation Figure 6: Performance on the RoboYoga Walker task Comparison on RoboYoga Walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In a second experiment, we compare our method to baselines on the RoboYoga task, as shown in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Here again, our method outperforms prior work, and Actionable Models does not make any significant improvement over HER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The results broken down by goal are shown in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Overall these results suggest that our dense reward shaping method allows for faster and more robust offline goal-conditioned policy training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 0 500 1000 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='12 Average Distance 0 500 1000 epoch Hand Distance 0 500 1000 epoch Puck Distance Ours HER HER + random neg action Actionable Models Figure 7: Performance on the Pusher task (lower is better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We report the final average, hand, and puck distance to the goal for our model and all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Comparison on Pusher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' As a final experiment, we compared our method to prior work on a realistic robotic environment, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Our policy trained offline is evaluated by sampling a goal at random in the state space, and measuring three different metrics: (i) the hand distance, which corresponds to the final distance between the end of the robot arm and the target, (ii) the puck distance, which measures the distance between the final puck location and the target, and (iii) the average distance, the average of the first two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Our method outperforms the baselines on this task, and our goal-conditioned agent is able to sequentially place the puck at the goal location, and then place the hand at its target location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' On the contrary, HER [2] places the puck at the target location with a performance similar to our method, but lacks precision on the hand location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 6 Conclusion: Summary and Limitations We proposed a method for learning multi-task policies from pre-generated datasets in an offline and unsupervised fashion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', without requiring any additional interaction with the environment, nor manually designed rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Our method leverages a self-supervised stage that aims at learn- ing the dynamics of the environment from the offline dataset, and that allows for shaping a dense reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It shows significant improvement over prior works based on hindsight relabeling, especially on long-horizon tasks, where dense rewards are crucial for learning a good policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The main limitation of our method is that it relies on the availability of a pre-collected dataset of trajectories, with a sufficiently large coverage of the state space for proper policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Although such data can be already available, as for the RoboYoga Walker task, or that offline dataset collection could be done with random policies, as we did on the UMaze and Pusher tasks, this step can be challenging for other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Another limitation is that we evaluated our method exclusively on simulated environments, and we did not perform any experiments on real robots, for which pre- collected dataset with expert demonstrations can be available [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 8 HAcknowledgments Karteek Alahari is supported in part by the ANR grant AVENUE (ANR-18-CE23-0011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Chebotar, K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In International conference on machine learning, pages 1861–1870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' [35] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Kanagawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' mujoco-maze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='com/kngwyu/mujoco-maze, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 10 A Implementation details A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='1 Self-supervised reward shaping We provide pseudo-code for the two stages of our approach: the graph building process is shown in Algorithm 1, and the steps for filling the replay buffer are shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Here we use the notation R(s, M) as the maximum RNet value between s and all nodes in M i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', R(s, M) := max m∈M R(s, m) Algorithm 1 Building the directed graph M Input: pre-collected dataset D, Reachability Network R Initialize: M = {} / * Build the set of nodes * / for each state s in D do if R(s, M) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 and R(M, s) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 then Update M := M ∪ {s} end if end for / * Build edges * / for each transition (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' st+1) in D do Let nt := NNin(st) = argmaxn∈M R(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' st) Let nt+1 := NNout(st+1) = argmaxn∈M R(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' n) Create directed edge from nt to nt+1 end for Algorithm 2 Building replay buffer B for offline policy training Input: pre-collected dataset D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Reachability Network R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' di- rected graph M Initialize: B = {} while B is not full do Sample a transition (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' st+1) at random in D Sample a goal g at random in D Compute dl(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g) := 1 − R(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g) Let nt+1 := NNout(st+1) = argmaxn∈M R(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' n) Let ng := NNin(g) = argmaxn∈M R(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g) Compute dg(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g) := ShortestPathLength(nt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' ng) Compute reward rt := −(dl(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g) + dg(st+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g)) Relabel transition with goal g and reward rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' and Push (st,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' at,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' rt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' st+1) to B end while A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='2 Actionable Models baselines re-implementation In this section, we provide details for our re-implementation of Actionable Models [1] and HER [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Since we are using the same optimization algorithm for offline policy training for these baselines and our method, the only difference lies in how the transitions in the pre-collected dataset D are relabeled to build the replay buffer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In HER [2], the idea is to sample a trajectory and a goal g at random in D, and to cut the trajectory at a step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Each transition in the trajectory (until step i) is then relabelled twice: once with the goal g and reward 0 for all transitions, and once with goal si (the final state of the trajectory) and reward 11 0 for all transitions except the final one that gets a reward of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The pseudo-code for this method is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Actionable Models [1] relies on a similar idea as in HER [2], but contains two additional steps to improve the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The first step is a form of goal chaining and it consists in using the Q-value at the final state of the trajectory to compute the reward for the final transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The second step aims at balancing the unseen action effect, in order to regularize the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' In practice, it consists in sampling negative actions from the policy and label zero-reward transitions with these actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The implementation of both tricks is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' For the implementation of the third baseline, HER + random negative action, the overall algorithm is the same as HER, except that we also generate transitions with negative actions, similar to Ac- tionable Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' This time, the negative actions are not sample from the policy, but are generated uniformly at random from the action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Algorithm 3 HER Input: dataset D Initialize: B = {} while B is not full do Sample trajectory τ ∈ D Sample goal g ∈ D Randomly cut τ at step i for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', i − 2} do (sj, aj, g, 0, sj+1) → B (sj, aj, si, 0, sj+1) → B end for (si−1, ai−1, g, 0, si) → B (si−1, ai−1, si, 1, si) → B end while Algorithm 4 Actionable Models Input: dataset D, goal-conditioned critic network Q, goal-conditioned policy π Initialize: B = {} while B is not full do Sample trajectory τ ∈ D Sample goal g ∈ D Randomly cut τ at step i for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=', i − 2} do (sj, aj, g, 0, sj+1) → B a1 ∼ π(sj, g) (sj, a1, g, 0, sj+1) → B (sj, aj, si, 0, sj+1) → B a2 ∼ π(sj, si) (sj, a2, si, 0, sj+1) → B end for (si−1, ai−1, g, Q(si, ai−1, g), si) → B a3 ∼ π(si−1, g) (si−1, a3, g, 0, si) → B (si−1, ai−1, si, 1, si) → B a4 ∼ π(si−1, si) (si−1, a4, si, 0, si) → B end while Figure 8: Pseudo-code for replay buffer filling with HER [2] and Actionable Models [1] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We compare both implementations by showing in red modifications related to goal chaining, and in blue edits related to unseen action regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='3 Hyper-parameters We first list the hyper-parameters for the self-supervised reward shaping phase in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Table 2 details the hyper-parameters for the offline policy training stage with SAC [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' For the Action- able Models [1] and HER [2] baselines, we used the same parameters as in our approach, with the exception of some parameters specific to these methods, shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' These hyper-parameters were obtained by performing a random search for all the methods over several parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' All the experiments in this work were performed on 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Common hyper-parameter Value Task UMaze RoboYoga Pusher Number of training pairs 5 × 105 5 × 105 5 × 105 Ratio of negatives 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 Ratio of negatives from same trajectory 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 Reachability threshold (τreach) 5 2 10 Weight of local distance in reward 1 1 100 Batch size 512 512 512 Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='001 Weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='00001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='00001 Total number of training epochs 100 100 100 Capacity of the directed graph 1000 10000 1000 Table 1: Hyper-parameters for reachability network training and directed graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Hyper-parameter Value Task UMaze RoboYoga Pusher Replay buffer capacity 106 106 106 Batch size 2048 2048 2048 Discount (γ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='95 Number of updates per epoch 1000 1000 1000 Total number of epochs 1000 1000 1000 Target update interval 1 1 1 Soft update coefficient (τ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='005 SAC entropy parameter (α) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0001 Optimizer Adam Adam Adam Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0005 Action repeat 1 2 1 Reward scaling factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='01 Table 2: Hyper-parameters for offline policy learning with SAC [34] with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Hyper-parameter Value Task UMaze RoboYoga Pusher Discount (γ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='99 SAC entropy parameter (α) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='005 Learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0001 Reward scaling factor 1 10 1 Table 3: Hyper-parameters for offline policy learning with SAC [34] specific to Actionable Mod- els [1] and HER [2] baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='4 Architecture details Reachability Network [11] The RNet has a siamese architecture with two embedding heads (one for each observation of the pair) with tied weights, and a comparator network that compares both embeddings and returns a reachability score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' For the UMaze task, we used an embedding head with 3 fully-connected layers with batch normalization and Tanh activations, with a hidden size of 64 and an embedding size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' For the Roboyoga Walker task, the embedding network has the same architecture, but we increased both the hidden and embedding sizes to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The comparator network is also a fully-connected network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' It contains batch normalization and ReLU activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The hidden size for the UMaze (respectively the RoboYoga Walker) task is set to 16 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 128) and the number of layers is 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' Policy Network The goal-conditioned policy network takes as input both the observation and the goal, in separate heads with the same architecture but independent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' These heads are implemented as 3-layer fully-connected networks with Tanh activations, hidden size of dimension 64, and 16 dimensions for the feature size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The output from both the heads is then concatenated and fed into a 2-layer fully-connected network of width 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' The critic network has the same architecture for both observation and goal heads, and is followed by 3 fully connected-layers of width 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' B Full results on RoboYoga Walker task We show the comparison of our method against the aforementioned baselines on each of the 12 goals of the RoboYoga Walker task in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' These goals are illustrated in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' We see that our method masters most of the goals that do not require balancing (Lie Back & Front, Legs Up, Lunge), and succeeds quite well at more complicated goals like Side Angle, Lean Back and Bridge, but is unable to progress in complex goals like Head Stand or Arabesque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 Success Rate Lie Back Lie Front Legs Up Lunge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 Success Rate Side Angle Stand Lean Back Boat 0 500 1000 epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='0 Success Rate Bridge 0 500 1000 epoch Stand One Feet 0 500 1000 epoch Head Stand 0 500 1000 epoch Arabesque Ours HER HER + random neg action Actionable Models Figure 9: Performance on the RoboYoga Walker talk for each of the 12 goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Kitchen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Quadruped ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Walker ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Bins ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Reach Left ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Reach Right ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Push Front ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Push Back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Push Both Front ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Push Both Back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Lie Back ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Lie Front ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Legs Up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Lunge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Side Angle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Stand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Burner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Light ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Slide ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} +page_content='Hinge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odA0T4oBgHgl3EQfJ__J/content/2301.02099v1.pdf'} 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We develop an algebraic formalism for perturbative quantum field theory (pQFT) which +is based on Joyal’s combinatorial species. We show that certain basic structures of pQFT are +correctly viewed as algebraic structures internal to species, constructed with respect to the Cauchy +monoidal product. Aspects of this formalism have appeared in the physics literature, particularly in +the work of Bogoliubov-Shirkov, Steinmann, Ruelle, and Epstein-Glaser-Stora. In this paper, we +give a fully explicit account in terms of modern theory developed by Aguiar-Mahajan. We describe +the central construction of causal perturbation theory as a homomorphism from the Hopf monoid of +set compositions, decorated with local observables, into the Wick algebra of microcausal polynomial +observables. The operator-valued distributions called (generalized) time-ordered products and +(generalized) retarded products are obtained as images of fundamental elements of this Hopf monoid +under the curried homomorphism. The perturbative S-matrix scheme corresponds to the so-called +universal series, and the property of causal factorization is naturally expressed in terms of the action +of the Hopf monoid on itself by Hopf powers, called the Tits product. Given a system of fully +renormalized time-ordered products, the perturbative construction of the corresponding interacting +products is via an up biderivation of the Hopf monoid, which recovers Bogoliubov’s formula. +Contents +Introduction +1 +Part 1. +Hopf Monoids +9 +1. +The Algebras +9 +2. +Σ as a Hopf E-Algebra +18 +3. +Products and Series +24 +4. +Perturbation of T-Products +27 +Part 2. +Perturbative Algebraic Quantum Field Theory +30 +5. +Spacetime and Field Configurations +30 +6. +Observables +31 +7. +Time-Ordered Products and S-Matrix Schemes +33 +8. +Interactions +35 +9. +Scattering Amplitudes +35 +References +37 +Introduction +The theory of species is a richer, categorified version of analyzing combinatorial structures in terms +of generating functions, going back to André Joyal [Joy81], [Joy86], [BLL98]. In this approach, +This paper is an abridged version of ‘Species-theoretic foundations of perturbative quantum field theory’, +arXiv:2009.09969. +1 +arXiv:2301.00702v1 [math-ph] 2 Jan 2023 + +2 +WILLIAM NORLEDGE +one sees additional structure by encoding processes of relabeling combinatorial objects, that is by +modeling combinatorial objects as presheaves on the category S of finite sets I (the labels) and +bijections σ (relabelings). In this paper, we are concerned with species p valued in complex vector +spaces, i.e. functors of the form +p : Sop → Vec, +I �→ p[I], +σ �→ p[σ] +where Vec is the category of complex vector spaces. Explicitly, p consists of a complex vector space +p[I] for each finite set I, and a bijective linear map p[σ] : p[I] → p[J] for each bijection σ : J → I +such that composition of bijections is preserved. +A highly structured theory of gebras1 internal to vector species has been developed by +Aguiar-Mahajan [AM10], [AM13], building on the work of Barratt [Bar78], Joyal [Joy86], Schmitt +[Sch93], Stover [Sto93b], and others. For the internalization, one uses the Day convolution monoidal +product p • q with respect to disjoint union and tensor product, given by +p • q[I] = p ⊗Day q[I] = +� +S⊔T=I +p[S] ⊗ q[T]. +This may be viewed as a categorification of the Cauchy product of formal power series.2 Various +decategorifications of Aguiar-Mahajan’s theory recovers the plethora of graded combinatorial Hopf +algebras which have been studied [AM10, Chapter 15]. +On the other hand, quantum field theory (QFT) may be viewed as a kind of modern infinite +dimensional calculus. Perturbative quantum field theory (pQFT) is the part of QFT which considers +Taylor series approximations of smooth functions. By an argument of Dyson [Dys52], Taylor series +of realistic pQFTs are expected to have vanishing radius of convergence. Nevertheless, if an actual +smooth function of a non-perturbative quantum field theory is being approximated, then they are +asymptotic series, and so one might expect their truncations to agree to reasonable precision with +experiment. This is indeed the case. +There are two main synthetic approaches to (non-perturbative) QFT, which grew out of the failure +to make sense of the path integral analytically. There is functorial quantum field theory (FQFT), +which formalizes the Schrödinger picture by assigning time evolution operators to cobordisms between +spacetimes. There is also algebraic quantum field theory (AQFT), going back to [HK64], which +formalizes the Heisenberg picture by assigning C∗-algebras of observables to regions of spacetime. +Low dimension examples of AQFTs/Wightman field theories were rigorously constructed in seminal +work of Glimm-Jaffe and others [GJ68], [CJ70], [GJS74]. +Perturbative algebraic quantum field theory (pAQFT) [Rej16], [Düt19], [Sch20, nLab], due to +Brunetti, Dütsch, Fredenhagen, Hollands, Rejzner, Wald, and others, is (mathematically precise, +realistic) pQFT based on causal perturbation theory [Ste71], [EG73], [Sch95], due to Stückelberg, +Bogoliubov, Steinmann, Epstein, Glaser, Stora, and others. See [Düt19, Foreword] for an account of +the history. Following [IS78], [BF00], [DF01], in which one takes the algebraic adiabatic limit to +handle IR-divergences, pAQFT satisfies the Haag-Kastler axioms of AQFT, but with C∗-algebras +replaced by formal power series ∗-algebras, reflecting the fact that pQFT deals with Taylor series +approximations. In this paper, we show that the construction and structure of these formal power +series algebras is naturally described in terms of gebra theory internal to species. +1 meaning (co/bi/Hopf)algebras and Lie (co)algebras +2 from the perspective of S-colored (co)operads, as defined in e.g. [Pet13, Section 3], there is an equivalent +description of these gebras as (co)algebras over the left (co)action (co)monads of the (co)operads Com(∗), Ass(∗), +Lie(∗) [AM10, Appendix B.5], which relates the gebras of this paper to structures such as cyclic operads, which already +appear in mathematical physics + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +3 +For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on Minkowski spacetime +X ∼= Rp,1, p ∈ N (pAQFT may be applied in more general settings, see e.g. [Hol08]). Therefore for +us, an off-shell field configuration Φ is a smooth function +Φ : X → R, +x �→ Φ(x). +In particular, we do not impose conditions on the asymptotic behaviour of Φ at infinite times. Let +Floc denote the space of local observables A ∈ Floc; these are functionals of field configurations +which are obtained by integrating polynomials in Φ and its derivatives against bump functions on +X. Let F denote the commutative ∗-algebra of microcausal polynomial observables O ∈ F; these are +polynomial functionals of field configurations satisfying a microlocal-theoretic condition known as +microcausality, with multiplication the pointwise multiplication of functionals, sometimes called the +normal-ordered product. Then F[[ℏ]] is a formal power series ∗-algebra in formal Planck’s constant +ℏ, called the (abstract, off-shell) Wick algebra, with multiplication the Moyal star product for the +Wightman propagator ∆H of the Klein-Gordan field +F[[ℏ]] ⊗ F[[ℏ]] → F[[ℏ]], +O1 ⊗ O2 �→ O1 ⋆HO2, +sometimes called the operator product. +Perhaps the most fundamental Hopf monoid of Aguiar-Mahajan’s theory is the cocommutative +Hopf algebra3 of compositions Σ, see Section 1.2, which is a Hopf monoid internal to vector species +defined with respect to the Day convolution. (More familiar is perhaps a certain decategorification +of Σ, which is the graded Hopf algebra of noncommutative symmetric functions NSym, see [AM10, +Section 17.3].) A composition F of I is a surjective function of the form +F : I → {1, . . . , k}, +for some +k ∈ N. +The ordering 1 > · · · > k is understood, so that F models the kth ordinal with I-marked points. +We let Sj = F −1(j), called the lumps of F, and write F = (S1, . . . , Sk). Each component Σ[I] is +the space of formal linear combinations of compositions F of I, +Σ[I] = +� +a = +� +F +cF HF +�� cF ∈ C +� +. +The multiplication +µS,T : Σ[S] ⊗ Σ[T] → Σ[I], +HF ⊗ HG �→ HFG +is the linearization of concatenating compositions (‘gluing’ via ordinal sum), and the comultiplication +∆S,T : Σ[I] → Σ[S] ⊗ Σ[T], +HF �→ HF|S ⊗ HF|T +is the linearization of restricting compositions to subsets (‘forgetting marked points’), where S⊔T = I. +Aspects of Σ have appeared in the physics literature as follows. Firstly, Epstein-Glaser-Stora’s +algebra of proper sequences [EGS75, Section 4.1] is the action of Σ on itself by Hopf powers, called +the Tits product [AM13, Section 13], going back to Tits [Tit74]. Secondly, the primitive part +Zie = P(Σ)4, which is a Lie algebra internal to species, is essentially the Steinmann algebra from +e.g. [Rue61, Section 6], [BL75, Section III.1]. More precisely, the Steinmann algebra is a graded +Lie algebra based on the structure map of the adjoint realization of Zie, see Section 1.7. Thirdly +and fourthly, and outside the scope of this paper, see below regarding work of Losev-Manin and +Feynman integrals. +3 we say ‘algebra’ and not ‘monoid’ since vector species form a linear category +4 the name ‘Zie’ comes from [AM17] + +4 +WILLIAM NORLEDGE +The central idea of this paper is to formalize the construction of a system of interacting +time-ordered products in causal perturbation theory as the construction of a homomorphism +�T of algebras internal to species of the form +�T : Σ ⊗ EFloc[[ℏ]] → UF[[ℏ,g]]. +We describe this construction in a clean abstract setting in Section 3.1, and then specialize to QFT +in Section 7. Here, ⊗ is the Hadamard monoidal product (=componentwise tensoring), EFloc[[ℏ]] +is the species given by I �→ (Floc[[ℏ]])⊗I, and UF[[ℏ,g]] is the algebra in species which has the Wick +algebra, with formal coupling constant g adjoined, in each I-component, +EFloc[[ℏ]][I] = (Floc[[ℏ]])⊗I, +UF[[ℏ,g]][I] = F[[ℏ, g]]. +It follows that the data of a system of products �T is equivalently a homomorphism of C-algebras +ˆΣ(Floc[[ℏ]]) → F[[ℏ, g]] +where ˆΣ(−) : Vec → Vec is the analytic endofunctor, or Schur functor, on vector spaces +associated to Σ [AM10, Section 19.1.2].5 +Decategorified versions of this formalization appear +in graded Hopf algebra approaches to pQFT [Bro09], [Bor11, p. 635]. In particular, there is an +interpretation of the Moyal deformation quantization in terms of Laplace pairings (=coquasitriangular +structures) [Fau01], [Bro09, Section 2.4]. +Also related is the notion of a Losev-Manin cohomological field theory [LM00, Theorem +3.3.1], [SZ11, Definition 1.3], where finite ordinals are replaced by strings of Riemann spheres +glued at the poles, giving a Hopf monoid structure on the toric variety of the permutohedron, and +Σ is replaced by the ordinary homology of this toric variety. The Hopf monoid structure of this +toric variety is also central to modern approaches to Feynman integrals [Bro17, p.6], [Sch18]. We +shall study this Hopf monoid in future work. +Explicitly, the homomorphism �T consists of component linear maps +�TI : Σ[I] ⊗ (Floc[[ℏ]])⊗I → F[[ℏ, g]], +HF ⊗ Ai1 ⊗ · · · ⊗ Ain �→ �TI(HF ⊗ Ai1 ⊗ · · · ⊗ Ain) +for each finite set I = {i1, . . . , in}. This homomorphism should also satisfy causal factorization, +which says +�TI(a ⊗ Ai1 ⊗ . . . Ain) = �TI( a ▷ HG +� �� � +Tits product +⊗Ai1 ⊗ · · · ⊗ Ain) +for all +a ∈ Σ[I] +whenever the local observables Ai1, . . . , Ain respect the ordering of I induced by the composition G, +see Proposition 7.1. Additional properties are often included, such as translation equivariance. +We can curry �T with respect to the internal hom H(−, −) for the Hadamard product, giving a +homomorphism of algebras +Σ → H(EFloc[[ℏ]], UF[[ℏ,g]]), +HF = H(S1,...,Sk) �→ �T(S1) . . . �T(Sk). +The resulting linear maps +�T(S1) . . . �T(Sk) : (Floc[[ℏ]])⊗I → F[[ℏ, g]] +are called interacting generalized time-ordered products. For each choice of a field polynomial, the +curried homomorphism is a ‘representation’ of Σ as F[[ℏ, g]]-valued generalized functions on X I, +5 the hat ˆΣ is meant to suggest a kind of categorified Fourier transform + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +5 +called operator-valued distributions since the Wick algebra is often represented on a Hilbert space. +The composition of the time-ordered products �T(I) with the Hadamard vacuum state +⟨−⟩0 : F[[ℏ, g]] → C[[ℏ, g]], +O �→ O(Φ = 0) +are then translation invariant C[[ℏ, g]]-valued generalized functions +GI : X I → C[[ℏ, g]], +(xi1, . . . , xin) �→ GI(xi1, . . . , xin)6 +called time-ordered n-point correlation functions. After taking the adiabatic limit, and in the +presence of vacuum stability, these functions may be interpreted as the probabilistic predictions +made by the pQFT of the outcomes of scattering experiments, called scattering amplitudes, see +Section 9. However, their values are formal power series in ℏ and g, and so have to be truncated. +Central to Aguiar-Mahajan’s work is the interpretation of Σ (and other Hopf monoids) in terms +of the geometry of the type A reflection hyperplane arrangement, called the (essentialized) braid +arrangement +Br[I] = +�{xi1 − xi2 = 0} ⊆ RI/R ↞ RI +� +�� +� +quotient by translations +: (i1, i2) ∈ I2, i1 ̸= i2 +�. +In causal perturbation theory, the braid arrangement appears as the space of time components of +configurations X I modulo translational symmetry [Rue61, Section 2], and the reflection hyperplanes +are the coinciding interaction points. Every real hyperplane arrangement A has a corresponding +adjoint hyperplane arrangement A∨ [AM17, Section 1.9.2]. The free vector space RI on I is naturally +Hom(RI, R), and so the adjoint of the braid arrangement is given by +Br∨[I] = +�� � +i∈S +xi = +� +i∈T +xi = 0 +� +⊆ Hom(RI/R, R) �→ RI +� +�� +� +sum-zero subspace +: (S, T) ∈ 2I, S, T ̸= ∅ +� +. +In causal perturbation theory, the adjoint braid arrangement appears as the space of energy +components [Rue61, Section 2], and the hyperplanes correspond to subsets going ‘on-shell’. The +spherical representation of the adjoint braid arrangement is called the Steinmann sphere, or +Steinmann planet, e.g. [Eps16, Figure A.4]. The chambers of the adjoint braid arrangement are +indexed by combinatorial gadgets called cells S [EGS75, Definition 6], also known as maximal +unbalanced families [BMM+12] and positive sum systems [Bjo15]. +The primitive part Lie algebra Zie = P(Σ) (together with its dual Lie coalgebra Zie∗) has a +natural geometric realization over the adjoint braid arrangement [Rue61, Section 6], [Ocn18, Lecture +33], [LNO19], [NO19], which results in cells S corresponding to certain special primitive elements +DS ∈ Zie[I], see Section 1.5. +The special elements were named Dynkin elements by Aguiar- +Mahajan [AM17, Section 14.1 and 14.9.8]. It is shown in [NO19] that the Dynkin elements span Zie, +but they are not linearly independent. The relations which are satisfied by the Dynkin elements +are known as the Steinmann relations [Ste60b, Equation 44], see Section 1.6, first studied by +Steinmann in settings where Σ is represented as operator-valued distributions. More recently, they +have been studied in the context scattering amplitudes, where they appear to be related to cluster +algebras [DFG18], [CHDD+19], [CHDD+20]. +If we restrict a curried system of interacting generalized time-ordered products to the primitive +part Zie, then we obtain a Lie algebra homomorphism +Zie → H(EFloc[[ℏ]], UF[[ℏ,g]]), +DS �→ �RS. +6 we have used generalized function notation; GI is not a single function, but can be represented by a sequence of +functions + +6 +WILLIAM NORLEDGE +spanning set +operator-valued distributions +vacuum expectation values +E∗ +universal series +GI +time-ordered product +T(I) +time-ordered n-point +function +L +H-basis linear orders +Hℓ +T(i1) . . . T(in) +Wightman n-point +functions +Σ +H-basis set compositions +HF +generalized time-ordered products +T(S1) . . . T(Sk) +generalized time-ordered +functions +Zie +Dynkin elements +DS +generalized retarded products +RS +generalized retarded +functions +Figure 1. Dictionary between products/vacuum expectation values and elements +of the Hopf algebra Σ. +The operator-valued distributions �RS which are the images of the Dynkin elements DS are the +interacting generalized retarded products of the system, see e.g. [Ste60b], [Ara61], [EG73, Equation +79]. In this paper, we give an exposition of the Steinman algebra and Steinmann relations in +Section 1.4, Section 1.5 and Section 1.6. +Let L �→ Σ be the Hopf subalgebra of linear orders (=compositions with singleton lumps), and +let E∗ �→ Σ be the subcoalgebra of compositions with one lump. Then we have the dictionary in +Figure 1 between products/vacuum expectation values and elements of Σ. In the commutative +setting before Moyal deformation quantization, the species X and E are similarly related to the +smeared field and polynomial observables, see Section 6. +In Section 4.1 and Section 8, we formalize the perturbation of time-ordered products in casual +perturbation theory as follows. Our starting point is a fully normalized system of generalized +time-ordered products, that is a homomorphism of algebras +T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) +satisfying causal factorization, and such that the singleton components T{i} are the natural inclusion +Floc[[ℏ]] �→ F((ℏ)), +A �→ :A : . +The corresponding operator-valued distributions are determined everywhere on X I by causal +factorization, apart from on the fat diagonal (=coinciding interaction points). In particular, off +the fat diagonal, the time-ordered products T(I) are given by the Moyal star product ⋆F with +respect to the Feynman propagator ∆F for the Klein-Gordon field. The terms of the product ⋆F +may be encoded in finite multigraphs, i.e. Feynman graphs. The remaining inherent ambiguity +means one has to make choices when extending the T(I) to the fat diagonal, and these choices form +a torsor of the Stückelberg-Petermann renormalization group. This is Stora’s elaboration [PS16], +[Sto93a], [BF00] on Stückelberg-Bogoliubov-Epstein-Glaser normalization [EG73], which constructs +the T(I) inductively in n = |I|. We leave species-theoretic aspects of renormalization, and possible +connections to Connes-Kreimer theory [Pin00], [GBL00], [BK05], [DFKR14], to future work. +In the original formulation by Tomonaga, Schwinger, Feynman and Dyson, would-be time-ordered +products are obtained by informally multiplying Wick algebra products by step functions, which is +in general ill-defined by Hörmander’s criterion. This leads to the divergence of individual terms of +the formal power series, called UV-divergences. Then informal methods are used to obtain finite +values from these infinite terms [Sch95, Preface and Section 4.3]. + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +7 +The exponential species E, given by E[I] = C and 1C ∈ E[I] denoted HI, has the structure of an +algebra in species by linearizing taking unions of sets, +µS,T : E[S] ⊗ E[T] → E[I], +HS ⊗ HT �→ HI. +An E-module m = (m, ρ) is an associative and unital morphism +ρ : E • m → m +for m a species. Moreover, taking the inverse of µS,T as the comultiplication turns E into a connected +(co)commutative bialgebra, and so the category of E-modules Rep(E) is a symmetric monoidal +category with monoidal product the Cauchy product of E-modules. In particular, we may consider +Hopf/Lie algebras internal to Rep(E), which we call Hopf/Lie E-algebras. +The retarded Y ↓ (−) and advanced Y ↑ (−) Steinmann arrows are (we formalize as) raising +operators on Σ, whose precise definition is due to Epstein-Glaser-Stora [EGS75, p.82-83]. They +define two E-module structures on Σ, +E • Σ → Σ, +HY ⊗ HF �→ Y ↓ HF +and +E • Σ → Σ, +HY ⊗ HF �→ Y ↑ HF . +See Section 2.2. In particular, the retarded arrow is generated by putting {∗} ↓ H(I) = −H(∗,I)+H(∗I).7 +Then +Y ↓ H(I) = +� +Y1⊔Y2=Y +µY1,Y2⊔I +�s(H(Y1)) ⊗ H(Y2⊔I) +� +� +�� +� +denoted R(Y ;I) +where s : Σ → Σ is the antipode of Σ. The Steinmann arrows were first studied by Steinmann [Ste60b, +Section 3], where Σ is represented as operator-valued distributions. Here, the operator-valued +distribution which corresponds to R(Y ;I) ∈ Σ[Y ⊔ I] is called the retarded product R(Y ; I).8 +Since {∗} ↓ (−) is a commutative biderivation of Σ (Theorem 2.1), the retarded Steinmann arrow +gives Σ the structure of a Hopf E-algebra, and Zie the structure of a Lie E-algebra (similarly for +the advanced arrow). There is an interesting description of these Lie E-algebras in terms of the +adjoint braid arrangement, see Section 2.4. The Steinmann arrows are “two halves” of the restricted +adjoint representation L • Σ → Σ of Σ, which is reflected in [Ste60b, Equation 13]. This directly +corresponds to how the retarded ∆− and advanced ∆+ propagators are two halves of the causal +propagator ∆S = ∆+ − ∆−. +Let H +•(−, −) denote the internal hom for the Cauchy product of species, and let +(−)E = H +•(E, −). +See Section 2.3 for a more explicit definition. See also [Nor20, Section 2] for more details here +regarding the differentiation between the j -colored sets I (physically, the source field) and the +g-colored sets Y (physically, the coupling constant). Then (−)E is an endofunctor on species, which +is lax monoidal with respect to the Cauchy product. Therefore ΣE is naturally an algebra, with +multiplication inherited from Σ. Then, by currying the retarded Steinmann action E • Σ → Σ, we +obtain a homomorphism Σ → ΣE. Similarly for the setting with decorations, given a choice of +7 (∗I) denotes the composition of {∗} ⊔ I which has a single lump +8 note that some authors, e.g. [Düt19], call R(Y ; i) the retarded product, and then call R(Y ; I) the generalized +retarded product + +8 +WILLIAM NORLEDGE +adiabatically switched interaction action functional Sint ∈ Floc[[ℏ]], after acting with the retarded +Steinmann arrows and currying, we obtain the homomorphism +Σ ⊗ EFloc[[ℏ]] → (Σ ⊗ EFloc[[ℏ]])E +HF ⊗ Ai1 ⊗ · · · ⊗ Ain �→ +∞ +� +r=0 +↓ . . . ↓ +� �� � +r times +HF ⊗ Sint ⊗ · · · ⊗ Sint +� +�� +� +r times +⊗ Ai1 ⊗ · · · ⊗ Ain. +Compare this with the formalism for creation-annihilation operators in [AM10, Chapter 19]. Then, +finally, the corresponding system of perturbed interacting time-ordered products �T is given by +composing this homomorphism with the image of T under the endofunctor (−)E, +�T : Σ ⊗ EFloc[[ℏ]] → (Σ ⊗ EFloc[[ℏ]])E TE +−−→ (UF((ℏ)))E ∼= UF((ℏ))[[g]]. +See Section 4.1. It is a theorem of pAQFT that this does indeed land in UF[[ℏ,g]]. +Finally, in Section 3.3 and Section 7, we formalize S-matrices, or time-ordered exponentials, as +follows. Let Hom(−, −) denote the external hom for species, which lands in vector spaces Vec. We +let +S (−) = Hom(E, −). +This is lax monoidal with respect to the Cauchy product. In the presence of a generic system of +products on an algebra a, +ϕ : a ⊗ EV → UA, +series s ∈ S (a) of a +s : E → a, +HI �→ sI +induce S (UA) ∼= A[[j ]]-valued functions Ss on V as follows, +Ss : V → A[[j ]], +A �→ Ss(jA) := +∞ +� +n=0 +j n +n! ϕn(sn ⊗ A ⊗ · · · ⊗ A +� +�� +� +n times +). +If ϕ is a homomorphism of algebras, then +S(−) : S (a) → Func(V, A[[j ]]) +is a homomorphism of C-algebras. As a basic example, if we put a = E, A = C∞(V ∗), and set +j = 1 at the end, then one can recover the classical exponential function in this way. +For c ∈ C, the so-called (scaled) universal series G(c) of Σ is given by sending each finite set to +the (scaled) composition with one lump, +G(c) : E → Σ, +HI �→ G(c)I := cn H(I). +If we set c = 1/iℏ, then the function S = SG(1/iℏ) above for a fully normalized system of generalized +time-ordered products T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) recovers the usual perturbative S-matrix scheme +of pAQFT, +S : Floc[[ℏ]] → F((ℏ))[[j ]], +A �→ S(jA) = +∞ +� +n=0 +� 1 +iℏ +�n j n +n! Tn(H(n) ⊗ A ⊗ · · · ⊗ A +� +�� +� +n times +). +The image of S(jA) after applying perturbation by the retarded Steinmann arrow and a choice of +interaction Sint ∈ Floc[[ℏ]] is +ZgSint(jA) = +∞ +� +n=0 +∞ +� +r=0 +� 1 +iℏ +�r+n grj n +r! n! Rr;n(Sint ⊗ · · · ⊗ Sint +� +�� +� +r times +; A ⊗ · · · ⊗ A +� +�� +� +n times +) + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +9 +where, by our previous expression for R(Y ;I) = Y ↓ H(I) (and letting T denote the precomposition of +T with the antipode of Σ ⊗ EFloc[[ℏ]]), we have +RY ;I(S Y +int; AI) = TY ⊔I(Y ↓ H(I) ⊗ S Y +int ⊗ AI) = +� +Y1⊔Y2=Y +TY1(S Y1 +int) ⋆H TY2⊔I(S Y2 +int ⊗ AI). +Then, since +S(−) : S (Σ) → Func +�Floc[[ℏ]], F((ℏ))[[g]] +� +is a homomorphism of C-algebras, it follows that ZgSint is given by +ZgSint(jA) = S−1(gSint) ⋆H S(gSint + jA). +This is the generating function, or partition function, for time-ordered products of interacting field +observables, see e.g. [EG73, Section 8.1], [DF01, Section 6.2], going back to Bogoliubov [BS59, +Chapter 4]. In this paper, we arrive at the generating function ZgSint through purely Hopf-theoretic +considerations. However, it was originally motivated by attempts to make sense of the path integral +synthetically. For some recent developments, see [Col16], [HR20]. +Structure. This paper is divided into two parts. In part one, we focus on developing theory for +the Hopf algebra of compositions Σ and its primitive part Zie. In part two, we specialize to pAQFT +for the case of a real scalar field on Minkowski spacetime. +Acknowledgments. We thank Adrian Ocneanu for his support and useful discussions. This paper +would not have been written without Nick Early’s discovery that certain relations appearing in +Ocneanu’s work were known in quantum field theory as the Steinmann relations. We thank Yiannis +Loizides and Maria Teresa Chiri for helpful discussions during an early stage of this project. We +thank Arthur Jaffe for his support, useful suggestions, and encouragement to pursue this topic. We +thank Penn State maths department for their continued support. +Part 1. Hopf Monoids +1. The Algebras +We recall the Hopf algebra of compositions Σ, together with its Lie algebra of primitive elements +Zie �→ Σ. We show that Σ and Zie are naturally algebras over the exponential species E. This will +be a species-theoretic formalization of mathematical structure discovered by Steinmann [Ste60b] and +Epstein-Glaser-Stora [EGS75], which, combined with a certain ‘perturbation of systems of products’ +construction using the E-action, will recover the perturbative construction of interacting fields in +pAQFT, as in [EG73, Section 8.1], [DF01, Section 6.2], going back to Bogoliubov [BS59, Chapter 4]. +1.1. Compositions. Let I be a finite set of cardinality n. We think of I as having ‘color’ j +(physically, the source field). As a particular example of the set I, we have the set of integers +[n] := {1, . . . , n} (formally, we have picked a section of the decategorification functor I �→ n). For +k ∈ N, let +(k) := {1, . . . , k} +equipped with the ordering 1 > · · · > k. A composition F of I of length l(F) = k is a surjective +function F : I → (k). The set of all compositions of I is denoted Σ[I], +Σ[I] := +� +k∈N +�surjective functions F : I → (k) +�. + +10 +WILLIAM NORLEDGE +We often denote compositions by k-tuples +F = (S1, . . . , Sk) +where Sj := F −1(j), 1 ≤ j ≤ k. The Sj are called the lumps of F. In particular, we have the length +one composition (I) for I ̸= ∅, and the length zero composition ( ) which is the unique composition +of the empty set. The opposite ¯F of F is defined by +¯F := (Sk, . . . , S1), +i.e. +¯F −1(j) = F −1(k + 1 − j). +Given a decomposition I = S ⊔ T of I (S, T can be empty), for F = (S1, . . . , Sk) a composition of +S and G = (T1, . . . , Tl) a composition of T, their concatenation FG is the composition of I given by +FG := (S1, . . . , Sk, T1, . . . , Tl). +For S ⊆ I and F = (S1, . . . , Sk) ∈ Σ[I], the restriction F|S of F to S is the composition of S given +by +F|S := (S1 ∩ S, . . . , Sk ∩ S)+ +where (−)+ means we delete any sets from the list which are the empty set. +For compositions F, G ∈ Σ[I], we write G ≤ F if G can be obtained from F by iteratively merging +contiguous lumps. Given compositions G ≤ F with G = (T1, . . . , Tl), we let +l(F/G) := +k +� +j=1 +l(F|Tj) +and +(F/G)! := +k +� +j=1 +l(F|Tj)! . +1.2. The Cocommutative Hopf Monoid of Compositions. Let +Σ[I] := +�formal C-linear combinations of compositions of I +�. +The vector space Σ[I] is naturally a right module over the symmetric group on I, and these actions +extend to a contravariant functor from the category S of finite sets and bijections into the category +Vec of vector spaces over C, +Σ : Sop → Vec, +I �→ Σ[I]. +For F a composition of I, let HF ∈ Σ[I] denote the basis element corresponding to F. The sets +{HF : F ∈ Σ[I]} form the H-basis of Σ. +In general, functors p : Sop → Vec are called (complex) vector species, going back to Joyal [Joy81], +[Joy86]. Morphisms of vector species η : p → q are natural transformations; they consist of a linear +map ηI : p[I] → q[I] for each finite set I which commutes with the action of the bijections. When +I = [n] := {1, . . . , n}, we abbreviate ηn := η[n]. +We equip vector species with the tensor product p • q known as the Cauchy product [AM10, +Definition 8.5], given by +(1) +p • q[I] := +� +I=S⊔T +p[S] ⊗ q[T]. +This is the Day convolution with respect to disjoint union of sets and tensor product of vector +spaces. In this paper, we consider algebraic structures on species which are constructed using this +tensor product. In particular, a multiplication on a species p consists of linear maps +µS,T : p[S] ⊗ p[T] → p[I] +and a comultiplication on p consists of linear maps +∆S,T : p[I] → p[S] ⊗ p[T], + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +11 +where we have a map for each choice of decomposition I = S ⊔ T (S, T can be empty). We can then +impose conditions like (co)associativity, see e.g. [Nor20, Section 1.3]. +Following [AM13, Section 11], Σ is a connected9 bialgebra, meaning it is naturally equipped +with an associative, unital multiplication and a coassociative, counital comultiplication, which are +compatible in the sense they satisfy the bimonoid axiom. See [AM10, Section 8.3.1] for details. The +multiplication and comultiplication are given in terms of the H-basis by +µS,T (HF ⊗ HG) := HFG +and +∆S,T (HF ) := HF|S ⊗ HF|T . +We sometimes abbreviate HF HG := µS,T (HF ⊗ HG). The unit and counit are given by +1Σ := H( ) +and +ϵ∅(H( )) := 1C. +Let +(2) +HF := +� +G≥ ¯F +(−1)l(G) HG. +Then [AM10, Theorem 11.38] (in the case q = E∗ ++ and q = 1) shows that +(3) +� +S⊔T=I +HF|SHF|T = 0 +and +� +S⊔T=I +HF|SHF|T = 0. +In general, connected bialgebras are automatically Hopf algebras, and it follows from (3) that the +antipode s : Σ → Σ is given by +sI(HF ) = HF . +The Hopf algebra Σ is the free cocommutative Hopf algebra on the positive coalgebra E∗ ++ [AM10, +Section 11.2.5], and so Σ ∼= L ◦ E∗ ++ where ‘◦’ is plethysm of species and L �→ Σ is the subspecies of +singleton lump compositions (=linear orders). +There is a second important basis of Σ, called the Q-basis. The Q-basis is also indexed by +compositions, and is given by +QF := +� +G≥F +(−1)l(G)−l(F) +1 +l(G/F)HG +or equivalently +HF =: +� +G≥F +1 +(G/F)!QG. +For S ⊆ I and F ∈ Σ[I], we have deshuffling +F ∥S := +� +F|S +if S is a union of lumps of F 10 +0 ∈ Σ[S] +otherwise. +The multiplication and comultiplication of Σ is given in terms of the Q-basis by +µS,T (QF ⊗ QG) = QFG +and +∆S,T (QF ) = QF∥S ⊗ QF∥T . +1.3. Decorations. Given a complex vector space V , we can use V to ‘decorate’ Σ in order to +obtain an enlarged Hopf algebra Σ ⊗ EV . This goes as follows. +We have the species denoted EV , given by +EV [I] := V ⊗I = V ⊗ · · · ⊗ V +� +�� +� +a copy of V for each i ∈ I +. +The action of bijections is given by relabeling tensor factors. +Remark 1.1. Notice species of the form EV are exactly the monoidal functors EV : Sop → Vec. +9 a species p is connected if p[∅] = C +10 not necessarily contiguous + +12 +WILLIAM NORLEDGE +We denote vectors by A, S ∈ V , and we denote simple tensors of V ⊗I by +AI = Ai1 ⊗ · · · ⊗ Ain ∈ V ⊗I +where I = {i1, . . . , in}. If Ai = A for all i ∈ I, then we write +(4) +AI := A ⊗ · · · ⊗ A ∈ V ⊗I +and +An := A[n] ∈ V ⊗[n] +where [n] = {1, . . . , n} as usual. +We let ‘⊗’ denote the Hadamard product of species, which is given by componentwise tensoring, +see e.g. [Nor20, Section 1.2]. Then the species of V -decorated compositions Σ ⊗ EV is given by +Σ ⊗ EV [I] = Σ[I] ⊗ EV [I] = Σ[I] ⊗ V ⊗I. +Following [AM10, Section 8.13.4], Σ ⊗ EV is a connected bialgebra, with multiplication given by +µS,T +�(HF ⊗ AS) ⊗ (HG ⊗ AT ) +� := HF HG ⊗ AS ⊗ AT +and comultiplication given by +∆S,T (HF ⊗ AI) := (HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ). +The unit and counit are given by +1Σ⊗EV := H( ) ⊗ 1C +and +ϵ∅(H( ) ⊗ 1C) := 1C. +For HF ⊗ AI ∈ Σ ⊗ EV [I], we have +� +S⊔T=I +µS,T +�(HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ) +� = +� +S⊔T=I +HF|SHF|T +� +�� +� += 0 by (3) +⊗ AI = 0 +and +� +S⊔T=I +µS,T +�(HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ) +� = +� +S⊔T=I +HF|SHF|T +� +�� +� += 0 by (3) +⊗ AI = 0. +It follows that the antipode of Σ ⊗ EV is given by +(5) +sI(HF ⊗ AI) = HF ⊗ AI. +1.4. The Steinmann Algebra. The Hopf algebra Σ is connected and cocommutative, and so the +CMM Theorem applies, see [Nor20, Section 1.4]. We now describe the positive11 Lie algebra of +primitive elements +P(Σ) ⊂ Σ. +For I ∈ S a finite set, let a tree T over I be a planar12 full binary tree whose leaves are labeled +bijectively with the blocks of a partition of I (a partition P of I is a set of disjoint nonempty subsets +of I, called blocks, whose union is I). The blocks of this partition, called the lumps of T, form a +composition called the debracketing FT of T, by listing them in order of appearance from left to +right. We denote trees by nested products [ · , · ] of subsets or trees, see Figure 2. We make the +convention that no trees exist over the empty set ∅. +11 a species p is positive if p[∅] = 0 +12 i.e. a choice of left and right child is made at every node + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +13 +1 +9 +678 +24 +1 +23 +2 +3 +5 +4 +Figure 2. Let I be various subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9}. The trees [4], [1, 23] +(̸= [23, 1]), [[2, 3], 5], [[24, [1, 9]], 678] are shown. The debracketing of [[24, [1, 9]], 678] +is the composition (24, 1, 9, 678). If we put T1 = [24, [1, 9]] and T2 = [678], then +[T1, T2] would also denote this tree. +We define the positive species Zie by letting Zie[I] denote the vector space of formal C-linear +combinations of trees over I, modulo the relations of antisymmetry and the Jacobi identity as +interpreted on trees in the usual way. Explicitly, +(1) (antisymmetry) for all trees of the form [. . . [T1, T2] . . . ] (writing a tree in this form is +equivalent to picking a node) we have +[. . . [T1, T2] . . . ] + [. . . [T2, T1] . . . ] = 0. +(2) (Jacobi Identity) for all trees of the form [. . . [[T1, T2], T3] . . . ] we have +[. . . [[T1, T2], T3] . . . ] + [. . . [[T3, T1], T2] . . . ] + [. . . [[T2, T3], T1] . . . ] = 0. +Then Zie is a positive Lie algebra in species, with Lie bracket ∂∗ given by +∂∗ +S,T (T1 ⊗ T2) := [T1, T2]. +Remark 1.2. We have that Zie is the free Lie algebra on the positive exponential species E∗ ++, and +so the species Zie is also given by +Zie[I] = Lie ◦ E∗ ++[I] = +� +P +Lie[P] +where Lie is the species of the Lie operad, and the direct sum is over all partitions P of I. +The Lie algebra in species Zie is closely related to the Steinmann algebra from the physics +literature [BL75, Section III.1], [Rue61, Section 6]. Precisely, the Steinmann algebra is an ordinary +graded Lie algebra based on the structure map for the adjoint braid arrangement realization of Zie. +The adjoint braid arrangement realization of Zie is the topic of [LNO19], and the fact that the Lie +algebra there is indeed Zie was shown in [NO19]. +Via the commutator bracket, Σ is a Lie algebra in species, given by +[HF , HG] = HF HG − HGHF . +Let +[I; 2] := +�surjective functions I → {1, 2} +� +denote the set of compositions of I with two lumps. Since Σ is connected, its positive Lie subalgebra +of primitive elements P(Σ) ⊂ Σ is given on nonempty I by +P(Σ)[I] = +� +(S,T)∈[I;2] +ker +�∆S,T : Σ[I] → Σ[S] ⊗ Σ[T] +�. +In particular, Q(I) ∈ P(Σ)[I] for I nonempty. Since Zie is freely generated by stick trees [I], we can +define a homomorphism of Lie algebras by +Zie → P(Σ), +[I] �→ Q(I). + +14 +WILLIAM NORLEDGE +To describe this explicitly, given a tree T, let antisym(T) denote the set of 2l(FT)−1 many trees +which are obtained by switching left and right branches at nodes of T. For T′ ∈ antisym(T), let +(T, T′) ∈ Z/2Z denote the parity of the number of node switches required to bring T to T′. Then +the homomorphism is given in full by +Zie → P(Σ), +T �→ QT := +� +T′∈antisym(T) +(−1)(T,T′)QFT′. +By [AM10, Corollary 11.46], this is an isomorphism. From now on, we make the identification +Zie = P(Σ) +and retire the notation P(Σ). +1.5. Type A Dynkin Elements. Recall that the set of minuscule weights of (the root datum of) +SLI(C) is in natural bijection with [I; 2]. We denote the minuscule weight corresponding to (S, T) +by λST . See [NO19, Section 3.1] for more details. +A cell13 [EGS75, Definition 6] over I is (equivalent to) a subset S ⊆ [I; 2] such that for all +(S, T) ∈ [I; 2], exactly one of +(S, T) ∈ S +and +(T, S) ∈ S +is true, and whose corresponding set of minuscule weights is closed under conical combinations, that +is +λUV ∈ coni +�λST : (S, T) ∈ S +� +=⇒ +(U, V ) ∈ S. +By dualizing conical spaces generated by minuscule weights, cells are in natural bijection with +chambers of the adjoint of the braid arrangement, see [NO19, Section 3.3], [Eps16, Definition +2.5]. Their number is sequence A034997 in the OEIS. We denote the species of formal C-linear +combinations of cells by L∨. +Associated to each composition F of I is the subset FF ⊆ [I; 2] consisting of those compositions +(S, T) which are obtained by merging contiguous lumps of F, +FF := +�(S, T) ∈ [I; 2] : (S, T) ≤ F +�. +More geometrically, FF is the subset corresponding to the set of minuscule weights which are +contained in the closed braid arrangement face of F. Let us write F ⊆ S as abbreviation for +FF ⊆ S. +Consider the morphism of species given by +(6) +L∨ → Σ, +S �→ DS := − +� +¯F⊆S +(−1)l(F)HF . +The element DS is called the Dynkin element associated to the cell S. These special elements +were defined by Epstein-Glaser-Stora in [EGS75, Equation 1, p.26], and the name is due to +Aguiar-Mahajan [AM17, Equation 14.1] (see Remark 1.3). In fact, DS is a primitive element [AM17, +Proposition 14.1], and so we actually have a morphism L∨ → Zie. +For i ∈ I, let Si denote the cell given by +Si := +�(S, T) ∈ [I, 2] : i ∈ S +�. +13 also known as maximal unbalanced families [BMM+12] and positive sum systems [Bjo15] + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +15 +-1 ++1 +-1 ++1 ++1 +-1 +13 ∞ future +2 ∞ past +3 ∞ future +12 ∞ past +23 ∞ future +1 ∞ past +1 +2 +3 +DS +rS +S +Figure 3. A cell S over {1, 2, 3} (on the adjoint braid arrangement) and +its Dynkin element DS (on the tropical geometric realization of Σ, where the +multiplication embeds facets and the comultiplication projects onto facets, see [NO19, +Introduction])). In the presence of causal factorization, the time component of +the corresponding generalized retarded function rS is a C[[ℏ, g]]-valued generalized +function on the braid arrangement with support the gray cone. The Dynkin element +shown is DS = D3 = R(12;3). Its support consists of those configurations such that the +event labeled by 3 can be causally influenced by the events labeled by 1 and 2. +This is the cell corresponding to the adjoint braid arrangement chamber which contains the projection +of the basis element ei ∈ RI onto the sum-zero hyperplane. Let the total retarded Dynkin element +Di associated to i be given by +Di := DSi = − +� +F∈Σ[I] +i∈Sk +(−1)l(F)HF . +These Dynkin elements are considered in [AM13, Section 14.5]. For i ∈ I, let +¯Si := +�(S, T) ∈ [I, 2] : i ∈ T +�. +This is the cell corresponding to the adjoint braid arrangement chamber which is opposite to the +chamber of Si. Let the total advanced Dynkin element D¯i associated to i be given by +D¯i := D ¯Si = − +� +F∈Σ[I] +i∈S1 +(−1)l(F)HF . +Remark 1.3. More generally, Dynkin elements are certain Zie elements of generic real hyperplane +arrangements, which are indexed by chambers of the corresponding adjoint arrangement. They were +introduced by Aguiar-Mahajan in [AM17, Equation 14.1]. Specializing to the braid arrangement, +one recovers the type A Dynkin elements DS. +In [NO19], the following perspective on the Dynkin elements is given. The Hopf algebra Σ∗ which +is dual to Σ is realized as an algebra ˆΣ +∗ of piecewise-constant functions on the braid arrangement. +Then its dual, in the sense of polyhedral algebras [BP99, Theorem 2.7], is an algebra ˇΣ +∗ of certain +functionals of piecewise-constant functions on the adjoint braid arrangement, i.e. those coming from +evaluating on permutohedral cones. We have the morphism of species +ˇΣ +∗ → (L∨)∗ + +16 +WILLIAM NORLEDGE +defined by sending functionals to their restrictions to piecewise-constant functions on the complement +of the hyperplanes. Since the multiplication of ˇΣ +∗ corresponds to embedding hyperplanes, this +morphism is the indecomposable quotient of ˇΣ +∗ [NO19, Theorem 4.5]. Then, in [NO19, Proposition +5.1], we see that taking the linear dual of this morphism recovers the Dynkin elements map, +L∨ → Σ, +S �→ DS. +(Here we have identified Σ∗ = ˇΣ +∗.) Therefore we obtain the following. +Theorem 1.1 ([NO19]). The morphism of species L∨ → Zie is surjective. Therefore the Dynkin +elements {DS : S is a cell over I} span Zie. +1.6. The Steinmann Relations. The Dynkin elements span Zie, but they are not linearly +independent. The relations which are satisfied by the Dynkin elements are generated by relations +known in physics as the Steinmann relations, introduced in [Ste60a], [Ste60b]. +Let a pair of overlapping channels over I be a pair (S, T), (U, V ) ∈ [I; 2] of two-lump compositions +of I such that +S ∩ U ̸= ∅ +and +T ∩ U ̸= ∅. +Let S1, S2, S3, S4 be four cells over I with (S, T), (U, V ) ∈ S1, and such that S2, S3, S4 are obtained +from S1 by replacing, respectively, +(S, T), (U, V ) �→ (T, S), (U, V ) +(S, T), (U, V ) �→ (T, S), (V, U) +(S, T), (U, V ) �→ (S, T), (V, U). +Then, by inspecting the definition of the Dynkin elements (6), we see that14 +DS1 − DS2 + DS3 − DS4 = 0. +In general, a Steinmann relation is any relation between Dynkin elements obtained in this way, +i.e. an alternating sum of four Dynkin elements which are obtained from each other by switching +overlapping channels only. This definition of the Steinmann relations can be found in [EGS75, Seciton +4.3] (it is given slightly more generally there for paracells). +An alternative characterization of the Steinmann relations in terms of the Lie cobracket of the +dual Lie coalgebra Zie∗ is [LNO19, Definition 4.2]. Here, the Steinmann relations appear in the +same way one can arrive at generalized permutohedra, i.e. by insisting on type A ‘factorization’ in +the sense of species-theoretic coalgebra structure. See [NO19, Theorem 4.2 and Remark 4.2]. +Thus, Dynkin elements satisfy the Steinmann relations. Moreover, they are sufficient. +Theorem 1.2. The relations which are satisfied by the Dynkin elements are generated by the +Steinmann relations. That is, if +Stein[I] := +�DS1 − DS2 + DS3 − DS4 : DS1 − DS2 + DS3 − DS4 = 0 is a Steinmann relation +�15 +then +Zie ∼= L∨ /Stein. +Proof. This follows by combining [LNO19, Theorem 4.3] with [NO19, Theorems 4.2 and 4.5]. +□ +14 we go through the argument for the basic 4-point case in Example 1.1, which is sufficient to exhibit the general +phenomenon +15 angled brackets denote C-linear span + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +17 +Example 1.1. Let us give the basic 4-point example I = {1, 2, 3, 4}, which takes place on a square +facet of the type A coroot solid [LNO19, Figure 1]. Consider the following four cells over I (we +have marked where they differ, the names ‘s-channel’ and ‘u-channel’ are from physics and refer to +Mandelstam variables), +S1 = +� (23, 14) +� +�� +� +u-channel +, (12, 34), (1, 234), (13, 24), (13, 24), (134, 2), (3, 124)} +S2 = +�(23, 14), (34, 12) +� +�� +� +s-channel +, (1, 234), (13, 24), (13, 24), (134, 2), (3, 124) +� +S3 = +� (14, 23) +� +�� +� +u-channel +, (34, 12), (1, 234), (13, 24), (13, 24), (134, 2), (3, 124) +� +S4 = +�(14, 23), (12, 34) +� +�� +� +s-channel +, (1, 234), (13, 24), (13, 24), (134, 2), (3, 124) +�. +The s-channel and the u-channel overlap, and so we should now have +DS1 − DS2 + DS3 − DS4 = 0. +To see this, let us assume throughout that HF appears in the H-basis expansion (6) of DS1, i.e. +¯F ⊆ S1. Then we have +(♠) +¯F ⊆ S1 \ {(12, 34), (23, 14)} +=⇒ +¯F ⊆ S1, S2, S3, S4. +If ¯F ⊈ S1 \ {(12, 34), (23, 14)}, then either (12, 34) ∈ ¯F or (23, 14) ∈ ¯F but not both, since the +channels overlap. We then have +(♥) +(12, 34) ∈ ¯F =⇒ ¯F ⊆ S1, ¯F ⊈ S2, ¯F ⊈ S3, ¯F ⊆ S4. +We also have +(♦) +(23, 14) ∈ ¯F =⇒ ¯F ⊆ S1, ¯F ⊆ S2, ¯F ⊈ S3, ¯F ⊈ S4. +Notice that in all three cases (♠), (♥), (♦), the prefactors of HF sum to zero in the four term +alternating sum of the Steinmann relation. +Remark 1.4. In [NO19], the Steinmann condition is seen to be equivalent to the restriction to +generalized permutohedra in a certain local (or spherical) sense. Ocneanu [Ocn18] and Early [Ear19] +have studied an affine version of the Steinmann condition, in the context of higher structures and +matroid subdivisions. Here, one observes that the (translated) hyperplanes of the adjoint braid +arrangement for the Mandelstam variables give three subdivisions of the hypersimplex ∆(2, 4) +(octahedron). +12 +34 +14 +23 +24 +13 +s-channel +t-channel +u-channel +See [BC19], [CGUZ19] for the closely related study of generalized Feynman diagrams in generalized +biadjoint Φ3-theory. + +18 +WILLIAM NORLEDGE +1.7. Ruelle’s Identity. Since the Dynkin elements span Zie, we can ask what is the description +of the Lie bracket of Zie in terms of the Dynkin elements. The answer is known in the physics +literature as Ruelle’s identity. +In order to state Ruelle’s identity, we need to notice the following. For S ⊔ T = I, if S1 is a cell +over S and S2 is a cell over T, then S1 ⊔ S2 describes a collection of codimension one faces of the +adjoint braid arrangement which are supported by the hyperplane orthogonal to λST (in [LNO19], +such faces were called Steinmann equivalent). A cell S[S,T] over I which satisfies +S[S,T] ⊇ S1 ⊔ S2 +and +(S, T) ∈ S[S,T] +corresponds to a chamber arrived at by moving (by an arbitrarily small amount) from an interior +point of a face of S1 ⊔ S2 in the λST direction. In particular, such cells always exist, but they are +not unique (the Steinmann relations exactly quotient out this ambiguity). The chamber obtained +by moving in the opposite direction corresponds to the cell obtained by replacing (S, T) with (T, S) +in S[S,T]. +Proposition 1.3 (Ruelle’s Identity [Rue61, Equation 6.6]). For S ⊔ T = I, let S1 be a cell over S +and let S2 be a cell over T. Let S[S,T] be a cell over I which satisfies +S[S,T] ⊇ S1 ⊔ S2 +and +(S, T) ∈ S[S,T]. +Let S[T,S] denote the cell obtained by replacing (S, T) with (T, S) in S[S,T]. Then the Lie bracket of +Zie is given by +(7) +[DS1, DS2] = DS[S,T ] − DS[T,S]. +Proof. This result is clear from [LNO19, Section 5.2]; the Lie bracket which was given to the adjoint +braid arrangement realization of Zie (denoted there by Γ) coincides with (7). Alternatively, we can +just explicitly check, as in [EGS75, Section 4.3]. +□ +2. Σ as a Hopf E-Algebra +We now recall the Steinmann arrows, which are (or we interpret as) actions of the exponential +species E on Σ. We show that they give Σ the structure of a Hopf E-algebra (=Hopf monoid +internal to E-modules) in two ways, and thus the primitive part Zie = P(Σ) the structure of a Lie +E-algebra in two ways. +2.1. Derivations and Coderivations of Σ. Let Y = {y1, . . . , yr} be a finite set with cardinality +r ∈ N. We think of Y as having ‘color’ g (physically, the coupling constant). Given a species p, we +have the Y -derivative p[Y ] of p, which is the species given by +p[Y ][I] := p[Y ⊔ I] +and +p[Y ][σ] := p[idY ⊔ σ]. +A raising operator u on p is a morphism of species of the form16 +u : p → p[Y ], +a �→ u(a). +Remark 2.1. Moreover, there is an endomorphism algebra of raising operators [Nor20, Section +2.4], which features when considering modules internal to species, see [Nor20, Section 5.1]. +16 for raising operators, we often abbreviate u(a) := uI(a) + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +19 +As a particular example of the set Y , we have the set of formal symbols [r] := {∗1, . . . , ∗r} +(formally, we have picked a section of the decategorification functor Y �→ r). We often abbreviate +∗ = ∗1, also ∗ = {∗} and ∗I = {∗} ⊔ I. The derivative p′ of p is the Y -derivative in the singleton +case Y = {∗}, thus +p′[I] := p[∗][I] = p[∗I]. +Following [AM10, Section 8.12.1], an up operator u on p is a raising operator of the form u : p → p′. +Writing u∗(a) = u(a) in order to specify the name of the adjoined singleton, we call an up operator +commutative if +u∗2(u∗1(a)) = u∗1(u∗2(a)). +Raising operators can be obtained by iteratively applying commutative up operators, see [Nor20, +Section 5.4]. Following [AM10, Section 8.12.4], an up operator on an algebra a is called an up +derivation if +(8) +u +�µS,T (a ⊗ b) +� = µ∗S,T +�u(a) ⊗ b +� + µS,∗T +�a ⊗ u(b) +� +(it follows that u(1a) = 0 if a is unital) and an up operator on a coalgebra c is called an up +coderivation if +(9) +�u ⊗ id + id ⊗ u +� ◦ ∆S,T (a) = ∆∗S,T +�u(a) +� + ∆S,∗T +�u(a) +�. +An up biderivation on a bialgebra h is an up operator which is both an up derivation and an +up coderivation. The data of an up (co/bi)derivation on a connected species h is equivalent to +giving h the structure of an L-(co/Hopf)algebra (= an (co/Hopf)monoid internal to L-modules). +The data of a commutative up (co/bi)derivation on h is equivalent to giving h the structure of an +E-(co/Hopf)algebra. See [Nor20, Section 5] for more details and proofs. +Thus, an up derivation u of Σ is a morphism of species +u : Σ → Σ′, +HF �→ u(HF ) +such that +u(HF HG) = u(HF )HG + HF u(HG). +An up derivation of Σ is determined by its values on the elements H(I), I ∈ S, since then +u(HF ) = u(H(S1))H(S2) . . . H(Sk) + · · · + H(S1) . . . H(Sk−1)u(H(Sk)). +An up derivation must have u(H( )) = 0, since 1Σ = H( ). An up coderivation u of Σ is a morphism +of species +u : Σ → Σ′, +HF �→ u(HF ) +such that +∆∗S,T +�u(HF ) +� = u(HF|S) ⊗ HF|T . +In particular, an up coderivation must have +∆∗S,T +�u(H(I)) +� = u(H(S)) ⊗ H(T). +Therefore, an up biderivation u of Σ must have +u(H(i)) = a1H(∗,i) + a2H(∗i) + a3H(i,∗) +where +a1 + a2 + a3 = 0 ∈ C. +Motivated by this, given a, b ∈ C, we define an up derivation ua,b of Σ by +(10) +ua,b : Σ → Σ′, +ua,b(H(I)) := −aH(∗,I) + (a + b)H(∗I) − bH(I,∗). +Towards an explicit description, consider the following example for I = {1, 2, 3}, +ua,b(H(12,3)) = ua,b(H(12))H(3) + H(12)ua,b(H(3)) += (−aH(∗,12) + (a + b)H(∗12) − bH(12,∗))H(3) + H(12)(−aH(∗,3) + (a + b)H(∗3) − bH(3,∗)) += −aH(∗,12,3) + (a + b)H(∗12,3) − bH(12,∗,3)) − aH(12,∗,3) + (a + b)H(12,∗3) − bH(12,3,∗). + +20 +WILLIAM NORLEDGE +From this, we see that in general +ua,b(HF ) = +� +1≤m≤k +−aH(S1,...,∗,Sm,...,Sk) + (a + b)H(S1,...,∗Sm,...,Sk) − bH(S1,...,Sm,∗,...,Sk). +Theorem 2.1. Given a, b ∈ C, the morphism of species +Σ → Σ′, +HF �→ ua,b(HF ) +is an up biderivation of Σ (it follows this gives Σ the structure of a Hopf L-algebra). +Proof. In the following, for F = (S1, . . . , Sk) a composition of I and S ⊆ I, we write +(U1, . . . , Uk) := (S1 ∩ S, . . . , Sk ∩ S). +In general, (U1, . . . , Uk) is a decomposition of I. +First, ua,b defines a derivation of Σ by construction. To see that ua,b also defines a coderivation, +we have +∆∗S,T +�ua,b(HF ) +� = +∆∗S,T +� +� +1≤m≤k +−aH(S1,...,∗,Sm,...,Sk) + (a + b)H(S1,...,∗Sm,...,Sk) − bH(S1,...,Sm,∗,...,Sk) +� += +� +� +1≤m≤k +−aH(U1,...,∗,Um,...,Uk)+ + (a + b)H(U1,...,∗Um,...,Uk)+ − bH(U1,...,Um,∗,...,Uk)+ +� +⊗ HF|T += +� +� +1≤m≤k +Um̸=∅ +−aH(U1,...,∗,Um,...,Uk)+ + (a + b)H(U1,...,∗Um,...,Uk)+ − bH(U1,...,Um,∗,...,Uk)+ +� +⊗ HF|T ++ +� +� +1≤m≤k +Um=∅ +� − a + (a + b) − b +� H(U1,...,Um−1,∗,Um+1,...,Uk)+ +� +� +�� +� +=0 +⊗ HF|T += +u(HF|S) ⊗ HF|T . +Therefore ua,b is a biderivation of Σ. +□ +2.2. The Steinmann Arrows. We now recall the Steinmann arrows for Σ, whose precise definition +is due to Epstein-Glaser-Stora [EGS75, p.82-83]. The Steinmann arrows were first considered by +Steinmann in settings where Σ is represented as operator-valued distributions [Ste60b, Section 3]. +Let the retarded Steinmann arrow be the up biderivation of Σ given by +(11) +∗ ↓ (−) : Σ → Σ′, +∗ ↓ HF := u1,0(HF ) = +� +1≤m≤k +−H(S1,...,∗,Sm,...,Sk) + H(S1,...,∗Sm,...,Sk). +Let the advanced Steinmann arrow be the up biderivation of Σ given by +(12) +∗ ↑ (−) : Σ → Σ′, +∗ ↑ HF := u0,1(HF ) = +� +1≤m≤k +H(S1,...,∗Sm,...,Sk) − H(S1,...,Sm,∗,...,Sk). +We use this arrow notation from now on instead of ‘u’ in order to match the physics literature. In +particular +∗ ↓ H(I) = −H(∗,I) + H(∗I) +and +∗ ↑ H(I) = H(∗I) − H(I,∗). + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +21 +We have +∗ ↑ HF − ∗ ↓ HF = u−1,1(HF ) = [H(∗), HF ]. +This identity appears often in the physics literature for operator-valued distributions, e.g. [Ste60b, +Equation 13], [EG73, Equation 83]. The biderivation u−1,1 gives Σ the structure of a Hopf L-algebra. +This L-action is the restriction of the adjoint representation of Σ. Notice the Steinmann arrows +are commutative up operators. By [Nor20, Proposition 5.4], we can restrict them to obtain up +derivations of Zie, +∗ ↓ (−) : Zie → Zie′, +DS �→ ∗ ↓ DS +and +∗ ↑ (−) : Zie → Zie′, +DS �→ ∗ ↑ DS. +Following [Nor20, Section 5], the Steinmann arrows equip Σ with the structure of a Hopf E-algebra +(and Zie with the structure of a Lie E-algebra) in two ways. The details are as follows. First, E is +the exponential species, given by +E[I] := C +for all +I ∈ S. +We denote HI := 1C ∈ E[I]. The exponential species is an algebra in species when equipped with +the trivial multiplication +µS,T : E[S] ⊗ E[T] = C ⊗ C ∼ +−→ C = E[I], +HS ⊗ HT �→ HI. +We have the following E-modules induced by the Steinmann arrows, as defined in [Nor20, Equation +23], +E • Σ → Σ, +HY ⊗ a �→ Y ↓ a := yr ↓ ◦ · · · ◦ y1 ↓ +� +�� +� +invariant of the order +(a) +and +E • Σ → Σ, +HY ⊗ a �→ Y ↑ a := yr ↑ ◦ · · · ◦ y1 ↑ +� +�� +� +invariant of the order +(a) +where Y = {y1, . . . , yr} as usual. In particular, Y ↓ (−) and Y ↑ (−) are the Steinmann arrow +raising operators obtained from iterating the Steinmann arrow up operators ∗ ↓ (−) and ∗ ↓ (−), as +mentioned in Section 2.1. For example, the retarded arrow Y ↓ (−) consists of a linear map of the +form +Σ[I] → Σ[Y ⊔ I] +for each choice of finite set I. For Y = [r] := {∗1, . . . , ∗r}, we abbreviate +↓ (−) := ∗ ↓ (−), +↓↓ (−) := {∗1, ∗2} ↓ (−), +. . . +and similarly for the advanced arrow. Since the arrows are derivations, they respect the multiplication +of Σ, and since the arrows are coderivations, they respect the comultiplication of Σ. It follows that +these E-actions give Σ the structure of a Hopf monoid constructed internal to E-modules. +By inspecting the definitions, we see that +(13) Y ↓ H(I) = R(Y ;I) := +� +Y1⊔Y2=Y +H(Y1)H(Y2⊔I) +and +Y ↑ H(I) = A(Y ;I) := +� +Y1⊔Y2=Y +H(Y1⊔I)H(Y2). +It follows that +Y ↓ HF = +� +Y1⊔···⊔Yk=Y +R(Y1;S1) . . . R(Yk;Sk) +and +Y ↑ HF = +� +Y1⊔···⊔Yk=Y +A(Y1;S1) . . . A(Yk;Sk). + +22 +WILLIAM NORLEDGE +The +sums +are +over +all +decompositions +(Y1, . . . , Yk) +of +Y +of +length +l(F). +We +call +R(Y ;I), A(Y ;I) ∈ Σ[Y ⊔ I] the retarded and advanced elements respectively. The total retarded and +total advanced elements are given by +Y ↓ H(i) = R(Y ;i) = +� +Y1⊔Y2=Y +H(Y1) H(Y2i) +and +Y ↑ H(i) = A(Y ;i) = +� +Y1⊔Y2=Y +H(Y2i) H(Y1) +respectively. +Remark 2.2. If we put I = J ⊔ {i}, then we have +R(J;i) = +� +S⊔T=I +i∈T +H(S) H(T) = − +� +F∈Σ[I] +i∈Sk +(−1)l(F)HF = Di +and +A(J;i) = +� +S⊔T=I +i∈T +H(T) H(S) = − +� +F∈Σ[I] +i∈S1 +(−1)l(F)HF = D¯i. +2.3. Currying the Steinmann Arrows. Given a species p, we let pE denote the species given +by +pE[I] := +∞ +� +r=0 +�p[r][I] +�Sr. +Here, p[r] is the Y -derivative of p for Y = [r], and (−)Sr denotes the subspace of Sr-invariants, +where Sr is the symmetric group on [r]. We denote elements of pE[I] using formal power series +notation +∞ +� +r=0 +xr, +xr ∈ p[r][I]. +Explicitly, xr is an element of the vector space p[{∗1, . . . , ∗r} ⊔ I] which is invariant under the action +of permuting {∗1, . . . , ∗r} and leaving I fixed. +The mapping p �→ pE extends to an endofunctor on species. In particular, given a morphism of +species η : p → q, we have the morphism ηE given by +(14) +ηE : pE → qE, +∞ +� +r=0 +xr �→ +∞ +� +r=0 +η[r]⊔I(xr). +A series of a species p is a morphism of species of the form s : E → p. Notice the elements of pE[I] +are naturally series of the species Y �→ p[Y ][I]. See [Nor20, Section 3.2] for more details. For the +connection between pE and the internal hom for the Cauchy product, see [Nor20, Section 2.3]. +If a is an algebra in species, then so is aE, see [Nor20, Equation 12]. In particular, ΣE is an +algebra, with multiplication given by +∞ +� +r=0 +xr ⊗ +∞ +� +r=0 +yr �→ +∞ +� +r=0 +� +r1+r2=r +r! +r1! r2!µ[r1]⊔S,[r2]⊔T (xr1 ⊗ yr2). +Theorem 2.2. We have the following homomorphisms of algebras in species, +Σ → ΣE, +HF �→ +∞ +� +r=0 +� +Y1⊔···⊔Yk=[r] +R(Y1;S1) . . . R(Yk;Sk) +and +Σ → ΣE, +HF �→ +∞ +� +r=0 +� +Y1⊔···⊔Yk=[r] +A(Y1;S1) . . . A(Yk;Sk). + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +23 +1 +2 +3 +* +Figure 4. Schematic for the action of the retarded Steinmann arrow ∗ ↓ for +I = {1, 2, 3} on the Steinmann sphere (left) and the tropical geometric realization of +Σ (right, see [NO19, Introduction]). +Proof. The Steinmann arrows are commutative up biderivations of Σ, and so give Σ the structure +of a Hopf E-algebra. This result is then a special case of [Nor20, Theorem 5.1]. +□ +The homomorphisms of Theorem 2.2 are the unique extensions of the maps +H(I) �→ +∞ +� +r=0 +R(r;I) +and +H(I) �→ +∞ +� +r=0 +A(r;I) +to homomorphisms. In the application to causal perturbation theory, we shall be interested in the +decorated analog of these homomorphisms, see Section 4.1. +Remark 2.3. These homomorphisms Σ → ΣE come from currying the E-actions of the Steinmann +arrows. See [Nor20, Section 5.1] for details. +2.4. The Steinmann Arrows and Dynkin Elements. We now show that the restriction of the +Steinmann arrows to Zie, which are derivations for its Lie bracket, have an interesting description +in terms of cells, i.e. chambers of the adjoint braid arrangement. +Following [Eps16, Section 2], we define the commutative up operators +∗ ↓ (−) : L∨ → L∨′, +∗ ↓ S := +�(∗S, T), (S, ∗T), (I, ∗) : (S, T) ∈ S +� +and +∗ ↑ (−) : L∨ → L∨′, +∗ ↑ S := +�(∗S, T), (S, ∗T), (∗, I) : (S, T) ∈ S +�. +These are indeed well-defined; ∗ ↓ S corresponds to the adjoint braid arrangement chamber on +the I side of the hyperplane λ∗,I = 0 which has the face of S as a facet, and ∗ ↑ S corresponds +to the chamber on the ∗ side of the hyperplane λ∗,I = 0 which has the face of S as a facet. See +around [LNO19, Remark 2.2] for more details. Thus, it follows from Proposition 1.3 (Ruelle’s +identity) that +[H(∗), DS] = D∗↑S − D∗↓S. +The induced E-modules are given by +E • L∨ → L∨, +HY ⊗ S �→ Y ↓ S := +�(Y1 ⊔ S, Y2 ⊔ T) ∈ [Y ⊔ I; 2] : (S, T) ∈ S or S = I +� +and +E • L∨ → L∨, +HY ⊗ S �→ Y ↑ S := +�(Y1 ⊔ S, Y2 ⊔ T) ∈ [Y ⊔ I; 2] : (S, T) ∈ S or T = I +�. + +24 +WILLIAM NORLEDGE +Proposition 2.3. Given a cell S over I, we have +Y ↓ DS = DY ↓S +and +Y ↑ DS = DY ↑S. +Proof. We consider the retarded case Y ↓ DS = DY ↓S only, since the advanced case then follows +similarly. It is sufficient to consider the case Y = {∗}. We have +↓ DS = − +� +¯F⊆S +(−1)l(F) ↓ HF +and +D↓S = − +� +¯F⊆ ↓S +(−1)l(F) HF . +So, the result follows if we have the following equality +� +¯F⊆S +(−1)l(F) +� +1≤m≤k +−H(S1,...,∗,Sm,...,Sk) + H(S1,...,∗Sm,...,Sk) +?= +� +¯G⊆ ↓S +(−1)l(G) HG. +Indeed, notice that the H-basis elements HG ∈ Σ[∗I] which appear on the LHS are exactly those +such that +¯G ⊆↓ S. +Notice also that each HG appears with total sign (−1)l(G), since when ∗ is inserted as a singleton +lump, thus increasing l(G) by one, it appears also with a negative sign. +□ +Remark 2.4. This interpretation of the E-module structure of Σ restricted to the primitive part +Zie = P(Σ) in terms of the adjoint braid arrangement suggests obvious generalizations of the +Steinmann arrows in the direction of [AM17], [AM20], since the generalization of Hopf monoids +there is via hyperplane arrangements. +Corollary 2.3.1. We have the following homomorphisms of Lie algebras in species, +Zie → ZieE, +DS �→ D(−)↓S = +∞ +� +r=0 +D[r]↓S = DS + D↓S + D↓↓S + · · · +and +Zie → ZieE, +DS �→ D(−)↑S = +∞ +� +r=0 +D[r]↑S = DS + D↑S + D↑↑S + · · · . +Proof. The Steinmann arrows are commutative up biderivations of Zie, and so give Zie the structure +of a Lie E-algebra. This result is then a special case of [Nor20, Theorem 5.1]. +□ +3. Products and Series +We now recall several basic constructions of casual perturbation theory in the current, clean, +abstract setting. We do this without yet imposing causal factorization/causal additivity. We say +e.g. ‘T-product’ and ‘R-product’ for now, and then change to ‘time-ordered product’ and ‘retarded +product’ in the presence of causal factorization. +3.1. T-Products, Generalized T-Products, and Generalized R-Products. Let V be a +vector space over C. Let A be a C-algebra with multiplication denoted by ⋆. Let UA be the +algebra in species given by +UA[I] := A. +The action of bijections is trivial, and the multiplication is the multiplication of A. +The positive exponential species E∗ ++ is given by +E∗ ++[I] := C +if +I ̸= ∅ +and +E∗ ++[∅] = 0. + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +25 +Let a system of T-products T be a system of products for the positive exponential species E∗ ++, as +defined in [Nor20, Section 6.2]. This means T is a morphism of species of the form17 +T : E∗ ++ ⊗ EV → UA, +H(I) ⊗ AI �→ TI(H(I) ⊗ AI) +where recall E∗ ++ ⊗ EV is the Hadamard product of species, given by +E∗ ++ ⊗ EV [I] := E∗ ++[I] ⊗ EV [I]. +Thus, if I ̸= 0, we have +E∗ ++ ⊗ EV [I] ∼= V ⊗I. +We abbreviate +(15) +TI(AI) := TI(H(I) ⊗ AI). +Let H(EV , UA) denote the species of linear maps between components, given by +H(EV , UA)[I] := HomVec +�EV [I], UA[I]) = HomVec +�V ⊗I, A). +We have that H(−, −) is the hom for the Hadamard product. Therefore we can curry T to give the +morphism of species +E∗ ++ → H(EV , UA), +H(I) �→ T(I) +where T(I) is the linear map +T(I) : V ⊗I → A, +AI �→ TI(AI). +The linear maps T(I) are called T-products. Notice that T-products are commutative in the sense +that +TI +�EV [σ](AI) +� = TI(AI) +for all bijections +σ : I → I. +This property holds because the system T is a morphism of species, and bijections act trivially for +UA. This commutativity exists despite the fact that the algebra A is noncommutative in general. +Remark 3.1. In applications to QFT, we shall also have a causal structure on V . Then T is meant +to first order the vectors of AI according to the causal structure, and then multiply in A, giving rise +to this commutativity. +Let the system of generalized T-products associated to a system of T-products be the unique +extension to a system of products for Σ = L ◦ E∗ ++ which is a homomorphism, as defined in [Nor20, +Section 6.2]. Thus +T : Σ ⊗ EV → UA, +HF ⊗ AI �→ TI(HF ⊗ AI) := TS1(AS1) ⋆ · · · ⋆ TSk(ASk). +The currying of T is denoted by +Σ → H(EV , UA), +HF �→ T(S1) . . . T(Sk). +The linear maps +T(S1) . . . T(Sk) : V ⊗I → A, +AI �→ TI(HF ⊗ AI) +are called generalized T-products. Let the system of generalized R-products associated to a system +of T-products be the restriction to the Lie algebra of primitive elements Zie, +R : Zie ⊗ EV → UA, +DS ⊗ AI �→ RI(DS ⊗ AI) := TI(DS ⊗ AI). +17 recall the definition and notation for EV from Section 1.3 + +26 +WILLIAM NORLEDGE +This is a morphism of Lie algebras, where UA is equipped with the commutator bracket. The +currying of R is denoted by +Zie → H(EV , UA), +DS �→ RS. +The linear maps +RS : EV [I] → A, +AI �→ RI(DS ⊗ AI) +are called generalized R-products. From the expansion (6) of Dynkin elements DS in terms of the +H-basis, we recover [EG73, Equation 79], +RS = − +� +FF ⊆ ¯S +(−1)k T(S1) . . . T(Sk). +Remark 3.2. Consider a system of products of the form +Z : E∗ ++ ⊗ EV → UV , +H(I) ⊗ AI �→ ZI(AI). +Then we obtain a new T-product T′, given by +T′ : E∗ ++ ⊗ EV → UA, +T′ +I(AI) := +� +P +TP +�ZS1(AS1) . . . ZSk(ASk) +�. +The sum is over all partitions P = {S1, . . . , Sk} of I. This construction underlies renormalization +in pAQFT [Düt19, Section 3.6.2], which deals with the remaining ambiguity of T-products after +imposing causal factorization, and perhaps other renormalization conditions. +3.2. Reverse T-Products. The system of reverse generalized T-products T of a system of +generalized T-products is given by precomposing T with the antipode (5) of Σ ⊗ EV , thus +T : Σ ⊗ EV → UAop, +TI(HF ⊗ AI) := TI +�HF ⊗ AI +�. +Since the antipode is a homomorphism Σ ⊗ EV → (Σ ⊗ EV )op,cop [AM10, Proposition 1.22 (iii)], +this is a system of generalized T-products into the opposite algebra UAop. The image of H(I) under +the currying of T is called the reverse T-product +T(I) : EV [I] → Aop. +From (2), we obtain +T(I) = +� +F∈Σ[I] +(−1)k T(S1) . . . T(Sk). +Note that reverse T-products in [EG73, Equation 11] are defined to be (−1)n T(I). Our definition +agrees with [Sch20, Definition 15.35]. +3.3. T-Exponentials. For details on series in species, see [AM10, Section 12]. +The (scaled) +universal series G(c) is the group-like series of Σ given by +G(c) : E → Σ, +HI �→ G(c)I := cn H(I) +for +c ∈ C. +The fundamental nature of this series is described in [AM13, Section 13.6]. The series s ◦ G(c) which +is the composition of G(c) with the antipode s of Σ is given by +(16) +s ◦ G(c) : E → Σ, +HI �→ +�s ◦ G(c) +� +I = cn H(I). +Let A[[j ]] denote the C-algebra of formal power series in the formal symbol j with coefficients in +A. Given a system of generalized T-products +T : Σ ⊗ EV → UA + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +27 +let the T-exponential S := SG(c) of this system be the A[[j ]]-valued function on the vector space V +associated to the series G(c), as constructed in [Nor20, Section 6.3]. Thus, we have18 +(17) +S : V → A[[j ]], +A �→ S(jA) = +∞ +� +n=0 +cn +n!Tn +�jA ⊗ · · · ⊗ jA +� +� +�� +� +n times +:= +∞ +� +n=0 +j ncn +n! Tn(An). +By [Nor20, Equation 34] and (16), the T-exponential for the system of reverse T-products is the +inverse of S as an element of the C-algebra of functions Func(V, A[[j ]]), given by +S−1 : V → A[[j ]], +A �→ S−1(jA) := +∞ +� +n=0 +j ncn +n! Tn(An) = +∞ +� +n=0 +j ncn +n! Tn(H(n) ⊗ An). +Therefore +S(jA) ⋆ S−1(jA) = S−1(jA) ⋆ S(jA) = 1A +for all A ∈ V . This appears in e.g. [EG73, Equation 2]. +4. Perturbation of T-Products +For the perturbation of T-products by a certain up coderivation of E which gives the S-matrix +scheme SgS(jA) = S(gS + jA), see [Nor20, Section 10.1]. +4.1. Perturbation of T-Products by Steinmann Arrows. Suppose we have a system of +generalized T-products +T : Σ ⊗ EV → UA, +HF ⊗ AI �→ TI(HF ⊗ AI). +Following [Nor20, Section 6.4], given a choice of decorations vector S ∈ V , we can use the retarded +Steinmann arrow (11) to perturb T as follows. +Recall the decorated Hopf algebra Σ ⊗ EV from Section 1.3. Recall also the derivative (Σ ⊗ EV )′ +of Σ ⊗ EV from Section 2.1, given by +(Σ ⊗ EV )′[I] = Σ[∗I] ⊗ V ⊗ V ⊗I. +We have the up derivation of Σ ⊗ EV which is the decorated analog of the retarded Steinmann +arrow, given by +Σ ⊗ EV → (Σ ⊗ EV )′, +HF ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain �→ ∗ ↓ HF ⊗ S ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain. +This is indeed still an up derivation by [Nor20, Proposition 6.4]. Analogous to the setting without +decorations, we have the induced raising operators and associated E-action by iterating, which, +after currying, give us the homomorphism +Σ ⊗ EV → (Σ ⊗ EV )E +HF ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain �→ +∞ +� +r=0 +↓ . . . ↓ +� �� � +r times +HF ⊗ S ⊗ · · · ⊗ S +� +�� +� +r times +⊗Ai1 ⊗ · · · ⊗ ⊗Ain. +This is a homomorphism by [Nor20, Theorem 5.1]. Then, a new ‘perturbed’ system of generalized +T-products is given by composing this homomorphism with TE (defined in (14)), +�T : Σ ⊗ EV → (Σ ⊗ EV )E TE +−−→ (UA)E ∼= UA[[g]]. +For the result that (UA)E ∼= UA[[g]], see [Nor20, Section 4]. +18 we use the abbreviations (4) and (15), and also Tn := T[n] + +28 +WILLIAM NORLEDGE +Remark 4.1. The fact �T is still a homomorphism, and is thus still a generalized system of products, +depends crucially on the fact the Steinmann arrow is a derivation [Nor20, Theorem 5.1], and that +(−)E is a monoidal functor [Nor20, Section 2.5]. We can similarly perturb a system of generalized +R-products, which uses the fact the Steinmann arrow is a biderivation. +We now unpack all this formalism to give a fully explicit description of the new perturbed system +of products. Let us abbreviate +SY AI = Sy1 ⊗ · · · ⊗ Syr ⊗ Ai1 ⊗ · · · ⊗ Ain ∈ EV [Y ⊔ I]. +Let +(18) +RY ;I(SY ; AI) := TY ⊔I(R(Y ;I) ⊗ SY AI) = +� +Y1⊔Y2=Y +TY1⊔∅(SY1) ⋆ TY2⊔I(SY2AI) +� +�� +� +by (13) +Then the new perturbed system is given by19 +�T : Σ ⊗ EV → UA[[g]], +HF ⊗ AI �→ +∞ +� +r=0 +� +r1+ ··· +rk=r +gr +r! Rr1;S1(S r1; AS1) ⋆ · · · ⋆ Rrk;Sk(S rk; ASk). +In particular, the restriction to E∗ ++ ⊗ EV , i.e. the new perturbed T-product, is given by +�TI(AI) = +∞ +� +r=0 +gr +r! Rr;I(S r; AI) += TI(AI) + g T∗1I(↓ H(I) ⊗ SAI) + g2 +2! T∗2∗1I(↓↓ H(I) ⊗ SSAI) + · · · +� +�� +� +perturbation +. +Similar, we can perturb a system of generalized T-products using the advanced Steinmann arrow. +We let VgS, respectively WgS, denote the T-exponential (as defined in (17)) for the new perturbed +system of generalized T-products using the retarded, respectively advanced, Steinmann arrows. +Thus +VgS : V → A[[g,j ]], +VgS(jA) := +∞ +� +n=0 +j ncn +n! +�Tn(An) = +∞ +� +n=0 +∞ +� +r=0 +grj ncr+n +r! n! +Rr;n(S r; An) +and +WgS : V → A[[g,j ]], +WgS(jA) := +∞ +� +n=0 +j ncn +n! +�Tn(An) = +∞ +� +n=0 +∞ +� +r=0 +grj ncr+n +r! n! +Ar;n(S r; An) +where +AY ;I(SY ; AI) := TY ⊔I(A(Y ;I) ⊗ SY AI) = +� +Y1⊔Y2=Y +TY1⊔I(SY1AI) ⋆ TY2⊔∅(SY2) +� +�� +� +by (13) +. +Theorem 4.1. We have +VgS(jA) = S−1(gS) ⋆ S(gS + jA) +and +WgS(jA) = S(gS + jA) ⋆ S−1(gS). +19 we abbreviate Rr;I(S r; AI) := R[r];I(S [r]; AI) = R[r];I(S ⊗ · · · ⊗ S +� +�� +� +r times +; AI) + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +29 +Proof. We have +Rr;I(S r; AI) = +� +Y1⊔Y2=[r] +TY1⊔∅(SY1) ⋆ TY2⊔I(SY2AI). +Then +VgS(jA) = +∞ +� +n=0 +∞ +� +r=0 +grj ncr+n +r! n! +Rr;n(S r; An) += +∞ +� +n=0 +∞ +� +r=0 +grj ncr+n +r! n! +� +Y1⊔Y2=[r] +TY1⊔∅(SY1) ⋆ TY2⊔[n](SY2An) += +∞ +� +r=0 +grcr +r! Tr+0(S r) ⋆ +∞ +� +n=0 +∞ +� +r=0 +cn +n!Tr+n(S rAn) += S−1(gS) ⋆ S(gS + jA) +The proof for WgS(jA) is similar. +□ +Corollary 4.1.1 (Bogoliubov’s Formula [BS59, Chapter 4]). We have +(19) +�Ti(A) = 1 +c +d +dj +��� +j =0VgS(jA). +Proof. We have +d +dj VgS(jA) = d +dj +∞ +� +n=0 +j ncn +n! +�Tn(An) = +∞ +� +n=1 +j n−1cn +(n − 1)! +�Tn(An). +Then, putting j = 0, we obtain +d +dj +��� +j =0VgS(jA) = c �T1(A). +□ +This formula was originally motivated by the path integral heuristic, see e.g. [Sch20, Remark +15.16]. +4.2. R-Products and A-Products. The linear maps R(Y ; I) which are given by +R(Y ; I) : E[Y ] +V [I] → A, +SY AI �→ RY ;I(SY AI) +are called R-products. In the case of singletons I = {i}, the maps R(Y ; i) are called total R-products. +By (13), R-products are given in terms of T-products and reverse T-products by +R(Y ; I) = +� +Y1⊔Y2=Y +T(Y1) ⋆ T(Y2 ⊔ I). +Then +�T(I) = +∞ +� +r=0 +cr +r! R(r; I). +In a similar way, we can define the A-products A(Y ; I), so that +A(Y ; I) = +� +Y1⊔Y2=Y +T(Y1 ⊔ I) ⋆ T(Y2). +The total R-products are both R-products and generalized R-products, which is due to the double +description appearing in Remark 2.2. A related result is [AM13, Proposition 109]. + +30 +WILLIAM NORLEDGE +Remark 4.2. In the literature, the total retarded products in our sense are sometimes called +retarded products, and the retarded products in our sense are then called generalized retarded +products, e.g. [Pol58], [Düt19, Exercise 3.3.16]. +Part 2. Perturbative Algebraic Quantum Field Theory +We now apply the theory we have developed to the case of a real scalar quantum field on a Minkowski +spacetime, as described by pAQFT.20 Mathematically, the important extra property is a causal +structure on the vector space of decorations V , which allows one to impose causal factorization. +Connections between QFT and species have been previously studied in [Abd04], [Far11], [GK18]. +Our references for pAQFT are [DF01], [Rej16], [Düt19], [Sch20]. We mainly adopt the notation +and presentation of [Sch20]. Key features of pAQFT are its local, i.e. sheaf-theoretic, approach, +the (closely related) use of adiabatic switching of interaction terms to avoid IR-divergences, and +the interpretation of renormalization as the extension of distributions to the fat diagonal to avoid +UV-divergences. The Wilsonian cutoff, sometimes called heuristic quantum field theory, may be +rigorously formulated within pAQFT [BDF09], [Düt12], [Düt19, Section 3.8], [Sch20, Section 16]. +5. Spacetime and Field Configurations +Let X ∼= R1,p denote a (p + 1)-dimensional Minkowski spacetime, for p ∈ N. Thus, X is a real vector +space equipped with a metric tensor which is a symmetric nondegenerate bilinear form X × X → R +with signature (1, p). The bilinear form gives rise to a volume form on X, which we denote by +dvolX ∈ Ωp+1(X). For regions of spacetime X1, X2 ⊂ X, we write +X1∨∧X2 +if one cannot travel from X1 to X2 on a future-directed timelike or lightlike curve. We have the set +valued species X (−) given by +I �→ X I := +�functions I → X +�. +For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on X. Therefore, let +E → X be a smooth real vector bundle over X with one-dimensional fibers. An (off-shell) field +configuration Φ is a smooth section of the bundle E → X, +Φ : X �→ E, +x �→ Φ(x). +The space of all field configurations, denoted Γ(E), has the structure of a Fréchet topological (real) +vector space. +Remark 5.1. We can always pick an isomorphism (E → X) ∼= (X × R → X), which induces +an isomorphism Γ(E) ∼= C∞(X, R), so that field configurations are modeled as smooth functions +X → R. +Let E∗ → X denote the dual vector bundle of E, and let the canonical pairing be denoted by +⟨−, −⟩ : E∗ ⊗ E → R. +Let a compactly supported distributional section α be a distribution of field configurations +α : Γ(E) → R, +20 although pAQFT deals more generally with perturbative Yang-Mills gauge theory on curved spacetimes + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +31 +i.e. an element of the topological dual vector space of Γ(E), which is modeled as a sequence (αj)j∈N +of smooth compactly supported sections of the dual bundle E∗ → X, +αj : X �→ E∗, +j ∈ N, +where the modeled distribution is recovered as the following limit of integrals, +Γ(E) → R, +Φ �→ +� +x∈X +�α(x), Φ(x) +�dvolX := lim +j→∞ +� +x∈X +⟨αj(x), Φ(x)⟩dvolX +� +�� +� +sometimes called generalized function notation +. +The space of all compactly supported distributional sections is denoted Γ′ +cp(E∗). By e.g. [Bär15, +Lemma 2.15], all distributions Γ(E) → R may be obtained as compactly supported distributional +sections in this way. +We can pullback the vector bundle E∗ to X I along each canonical projection +X I → X {i} ∼= X, +i ∈ I. +The tensor product of these n many pullback bundles is the exterior tensor product bundle (E∗)⊠I. +This defines a presheaf of smooth vector bundles on S, +Sop → Diff/X , +I �→ (E∗)⊠I. +By taking compexified compactly supported distributional sections Γ′C +cp(−) := Γ′ +cp(−) ⊗R C, we +obtain the complex vector species Γ′C +cp(E∗), given by +Γ′C +cp(E∗)[I] := Γ′C +cp +� +(E∗)⊠I� +. +Of course, Γ′C +cp(E∗) does not ‘factorize’ in the sense that it is not a monoidal functor, +(20) +Γ′C +cp(E∗)[I] ≇ Γ′C +cp(E∗)[i1] ⊗ · · · ⊗ Γ′C +cp(E∗)[in] +where I = {i1, . . . , in}. There are more distributional sections then just those coming from the +tensor product. +6. Observables +An off-shell observable O is a smooth functional of field configurations into the complex numbers, +O : Γ(E) → C, +Φ �→ O(Φ). +The space of all observables is denoted Obs. We can pointwise multiply observables, sometimes +called the normal ordered product, so that observables form a commutative C-algebra, +Obs ⊗ Obs → Obs, +O1 ⊗ O2 �→ O1 · O2 +where +O1 · O2(Φ) := O1(Φ)O2(Φ) +� +�� +� +multiplication in C +. +Thus, we may form the commutative algebra in species UObs, given by UObs[I] = Obs. +A linear observable O ∈ Obs is an observable which is additionally a linear functional, that is +O(Φ1 + Φ2) = O(Φ1) + O(Φ2) +and +O(cΦ) = cO(Φ) +for +c ∈ C. +The space of linear observables is denoted LinObs ⊂ Obs. In particular, for each spacetime event +x ∈ X, we have the field observable Φ(x) ∈ LinObs, given by +Φ(x) : Γ(E) → C, +Φ �→ Φ(x). + +32 +WILLIAM NORLEDGE +We now show how linear observables and so-called polynomial observables arise species-theoretically, +via (generalized) systems of products for the species E and X = P(E). +Let X denote the species given by X +�{i} +� := C for singletons and X[I] := 0 otherwise. We denote +Hi := 1 ∈ X +�{i} +�. We have the following morphism of species, +X ⊗ Γ′C +cp(E∗) → UObs, +Hi ⊗ α �→ +� +Φ �→ +� +x∈X +�α(x), Φ(x) +�dvolX +� +. +This is like a system of products for X, however Γ′C +cp(E∗) does not factorize (20), and so cannot +be written in the form EV . It follows from [Bär15, Lemma 2.15] that the colimit (as defined +in [AM10, Remark 15.7]) of the species which is the image of this morphism is the space of linear +observables LinObs. The currying of this map is given by +X → H +�Γ′C +cp(E∗), UObs +�, +Hi �→ Φi = Φ +where +Φ(α) := +� +Φ �→ +� +x∈X +�α(x), Φ(x) +�dvolX +� +. +If we restrict Φ to bump functions b ∈ Γcp(E∗) ⊗R C, also called ‘smearing functions’, then one +might call the linear map +Φ : Γcp(E∗) ⊗R C → Obs, +b �→ Φ(b) +an ‘observable-valued distribution’, and this is sometimes referred to as ‘the (smeared) field’. The +field observable Φ(x) is recovered by evaluating Φ on the Dirac delta function δx localized at x. +One views b as the smearing of a Dirac delta function, hence smearing functions and smeared field. +We extend the smeared field by replacing X with E to define the following morphism of species, +E ⊗ Γ′C +cp(E∗) → UObs, +HI ⊗ αI �→ +� +Φ �→ +� +X I +�αI(xi1, . . . , xin), Φ(xi1) . . . Φ(xin) +�dvolX I +� +. +This is like a system of products for E, but again without factorization. The colimit of the species +which is the image of this morphism is the vector space of polynomial observables, as defined in +e.g. [Sch20, Definition 7.13], denoted +PolyObs ⊂ Obs. +(Alternatively, if we restrict the limit of this map S (Γ′C +cp(E∗)) → Obs[[j ]] to finite series and set +j = 1, then we recover [Düt19, Definition 1.2.1].) The space of microcausal polynomial observables +F is the subspace +F ⊂ PolyObs +consisting of those polynomial observables which satisfy a certain microlocal-theoretic condition +called microcausality, see [Düt19, Definition 1.2.1 (ii)]. Following [Düt19, Definition 1.3.4], the space +of local observables +Floc ⊂ Obs +consists of those observables obtained by integrating a polynomial with real coefficients in the field +and its derivatives (‘field polynomials’) against a bump function b ∈ Γcp(E∗) ⊗R C. Importantly, we +have a natural inclusion +Floc �→ F, +A �→ : A : . +Let Floc[[ℏ]] and F[[ℏ]] denote the spaces of formal power series in ℏ with coefficients in Floc and F +respectively, and let F((ℏ)) denote the space of Laurent series in ℏ with coefficients in F. + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +33 +Applying Moyal deformation quantization with formal Planck’s constant ℏ, F[[ℏ]] is a formal +power series ∗-algebra, called the (abstract, off-shell) Wick algebra, with multiplication the Moyal +star product [Düt19, Definition 2.1.1] defined with respect to the Wightman propagator ∆H for the +Klein-Gordan field [Düt19, Section 2.2], +F[[ℏ]] ⊗ F[[ℏ]] → F[[ℏ]], +O1 ⊗ O2 �→ O1 ⋆H O2. +We may form the algebra in species UF[[ℏ]], or, allowing negative powers of ℏ, UF((ℏ)). +7. Time-Ordered Products and S-Matrix Schemes +For A ∈ Floc[[ℏ]], let supp(A) denote the spacetime support of A. Given a composition G of I, we +say that AI ∈ EFloc[[ℏ]][I] respects G if +supp(Ai1) ∨∧ supp(Ai2) +for all +(i1, i2) +such that +G|{i1,i2} = (i1, i2).21 +Consider a system of T-products (as defined in Section 3.1) of the form +T : E∗ ++ ⊗ EFloc[[ℏ]] → UF((ℏ)), +H(I) ⊗ AI �→ TI(H(I) ⊗ AI) = TI(AI). +Since Σ is the free algebra on E∗ ++, we have the unique extension to a system of generalized T-products +T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)), +TI(HF ⊗ AI) := TS1(AS1) ⋆H · · · ⋆H TSk(ASk). +Then: +1. (perturbation) we say that T satisfies perturbation if the singleton components Ti are +isomorphic to the inclusion Floc[[ℏ]] �→ F((ℏ)), that is +Ti(A) = :A: +2. (causal factorization) we say that T satisfies causal factorization if for all compositions (S, T) +of I with two lumps, if AI ∈ EFloc[[ℏ]][I] respects (S, T)22 then +(21) +TI(H(I) ⊗ AI) = TI(H(S,T) ⊗ AI).23 +Let a (fully normalized) system of time-ordered products be a system of T-products which satisfies +perturbation and causal factorization. The corresponding unique extension of T to Σ is called the +associated system of generalized time-ordered products. After currying +Σ → H(EFloc[[ℏ]], UF((ℏ))), +HF �→ T(S1) . . . T(Sk) +the linear maps +T(S1) . . . T(Sk) : Floc[[ℏ]]⊗I → F((ℏ)), +AI �→ TI(HF ⊗ AI) +are called generalized time-ordered products. The linear maps T(I) are called time-ordered products. +After fixing a field polynomial, so that each Aij of AI is determined by a bump function bij, they +are usually presented in generalized function notation as follows, +TI(Ai1 ⊗ · · · ⊗ Ain) = +� +X I T(xi1, . . . , xin)bi1(xi1) . . . bin(xi1)dxi1 . . . dxin +where (xi1, . . . , xin) �→ T(xi1, . . . , xin) is an ‘operator-valued’ generalized function. See e.g. [EG73, +Section 1.2]. +21 G|{i1,i2} = (i1, i2) means that i1 and i2 are in different lumps, with the lump containing i1 appearing to the left +of the lump containing i2 +22 explicitly, supp(Ai1) ∨∧ supp(Ai2) for all i1 ∈ S and i2 ∈ T +23 or equivalently TI(AI) = TS(AS) ⋆H TT (AT ) + +34 +WILLIAM NORLEDGE +Given compositions F = (S1, . . . , Sk) and G = (U1, . . . , Ul) of I, let +HF ▷ HG := H(U1∩S1,...,Ul∩S1,......,U1∩Sk,...,Ul∩Sk)+. +This is called the Tits product, going back to Tits [Tit74]. See [AM13, Section 13] for more on the +structure of the Tits product, where it is shown it is given by the action of Σ on itself by Hopf +powers. See also [AB08, Section 1.4.6] for the context of other Coxeter systems and Dynkin types. +Proposition 7.1. Let +T : E∗ ++ ⊗ EFloc[[ℏ]] → UF((ℏ)) +be a system of T-products which satisfies causal factorization. Given a composition G = (U1, . . . , Uk) +of I, and AI ∈ EFloc[[ℏ]][I] which respects G, then +TI(a ⊗ AI) = TI(a ▷ HG ⊗ AI) +for all +a ∈ Σ[I]. +Proof. We have +TI(HG ⊗ AI) = TU1(AU1) ⋆H · · · ⋆H TUk(AUk) = TI(AI) +� +�� +� +by repeated applications of causal factorization +. +Observe that the action HF �→ HF ▷ HG, for F ∈ Σ[I], replaces the lumps of F with their intersections +with G. But we just saw that TI(AI) = TI(HG ⊗ AI), and so it follows that +TI(HF ⊗ AI) = TI(HF ▷ HG ⊗ AI). +Since the claim is true for the H-basis, it is true for all a ∈ Σ[I]. +□ +Corollary 7.1.1. If a ▷ HG = 0, then +TI(a ⊗ AI) = 0 +for all AI ∈ EFloc[[ℏ]][I] which respect G. +The restriction of T to the primitive part Lie algebra is called the associated system of generalized +retarded products, +R : Zie ⊗ EFloc[[ℏ]] → UF((ℏ)). +The image of the Dynkin elements DS under the currying of R are the generalized retarded products +RS, see e.g. [EG73, Equation 79]. It follows from Corollary 7.1.1 and the structure of Dynkin +elements under the Tits product that generalized retarded products have nice support properties. +This is described in [EGS75]. +Given a system of generalized time-ordered products +T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) +the T-exponential S = SG(1/iℏ) (defined in (17)) for the group-like series +G(1/iℏ) : E → Σ, +HI �→ 1 +iℏH(I) +is called the associated perturbative S-matrix scheme. Thus, S is the function +S : Floc[[ℏ]] → F((ℏ))[[j ]], +A �→ S(jA) := +∞ +� +n=0 +� 1 +iℏ +�n j n +n! Tn(An). + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +35 +8. Interactions +Given a choice of adiabatically switched interaction Sint ∈ Floc[[ℏ]], and a system of fully normalized +generalized time-ordered products +T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)), +we have the new system of interacting generalized time-ordered products which is obtained by the +construction of Section 4.1, +�T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ))[[g]]. +The associated generating function scheme ZgSint for interacting field observables, and more generally +for time-ordered products of interacting field observables, is the new T-exponential for the group-like +series G(1/iℏ), denoted VgSint in Section 4.1. Thus, ZgSint is the function +ZgSint : Floc[[ℏ]] → F((ℏ))[[g,j ]], +A �→ ZgSint(jA) +where +ZgSint(jA):= +∞ +� +n=0 +� 1 +iℏ +�nj n +n! +�Tn(An) = +∞ +� +n=0 +∞ +� +r=0 +� 1 +iℏ +�r+ngrj n +r! n! Rr;n(S r +int; An) = S−1(gSint)⋆H S(gSint +jA). +Then +Aint := �Ti(A) = +∞ +� +r=0 +� 1 +iℏ +�r gr +r! Rr+1(S r +int; A) ∈ F((ℏ))[[g]] +is the local interacting field observable of A. Bogoliubov’s formula (19) now reads +Aint = iℏ d +dj +��� +j =0ZgSint(jA). +One views Aint as the deformation of the local observable A due to the interaction Sint. One can +show that �T does indeed land in UF[[ℏ,g]] [DF01, Proposition 2 (ii)]. The perturbative interacting +quantum field theory then has a classical limit [Col16], [HR20]. +9. Scattering Amplitudes +We finish with a translation of a standard result in pAQFT (see [Sch20, Example 15.12]) into our +notation, which relates S-matrix schemes as presented in Section 7 to S-matrices used to compute +scattering amplitudes, which are predictions of pAQFT that are tested with scattering experiments +at particle accelerators. +Following [Düt19, Definition 2.5.2], the Hadamard vacuum state ⟨−⟩0 is the linear map given by +⟨−⟩0 : F[[ℏ, g]] → C[[ℏ, g]], +O �→ ⟨O ⟩0 := O (Φ = 0). +Let Sint ∈ Floc[[ℏ]]. We say that the Hadamard vacuum state ⟨−⟩0 is stable with respect to the +interaction Sint if for all O ∈ F[[ℏ, g]], we have +(22) +�O ⋆H S(gSint) +� +0 = +�O +� +0 +�S(gSint) +� +0 +and +�S−1(gSint) ⋆H O +� +0 = +1 +�S(gSint) +� +0 +�O +� +0. +In situations where +Sint ⊗ AI ∈ E′ +Floc[[ℏ]][I] +respects the composition +(S, ∗, T) + +36 +WILLIAM NORLEDGE +we can interpret free particles/wave packets labeled by T coming in from the far past, interacting in +a compact region according to the adiabatically switched interaction Sint, and then emerging into +the far future, labeled by S. For AI ∈ EFloc[[ℏ]][I], let +GI(AI) := +��T(AI) +� +0. +If we fix the field polynomial of local observables to be P(Φ) = Φ, then AI �→ GI(AI) is the +time-ordered n-point correlation function, or Green’s function. They are usually presented in +generalized function notation as follows, +GI(bi1 ⊗ · · · ⊗ bin) = +� +X I +� +T +�Φ(xi1) . . . Φ(xin) +�� +0bi1(xi1) . . . bin(xin)dxi1 . . . dxin. +Note that to obtain the realistic Green’s functions, we still have to take the adiabatic limit. +Proposition 9.1. If the Hadamard vacuum state ⟨−⟩0 is stable with respect to Sint ∈ Floc[[ℏ]], and +if Sint ⊗ AI ∈ E′ +Floc[[ℏ]][I] respects the composition (S, ∗, T), then +GI(AI) = +1 +� +S(gSint) +� +0 +� +TS(AS) ⋆H S(gSint) ⋆H TT (AT ) +� +0.24 +Proof. We have +GI(AI) = +��T(AI) +� +0 += +� ∞ +� +r=0 +gr +r! Rr;I(S r +int; AI) +� +0 += +� ∞ +� +r=0 +� +r1+r2=r +gr +r1! r2!T[r1]⊔∅(S r1 +int) ⋆H T[r2]⊔I(S r2 +intAI) +� +0 +. +To obtain the final line, we expanded the retarded products according to (18). Then, by causal +factorization (21), we have +T[r2]⊔I(S r2 +intAI) = TS(AS) ⋆H T[r2]⊔∅(S r2 +int) ⋆H TT (AT ). +Therefore +GI(AI) = +� ∞ +� +r=0 +� +r1+r2=r +gr +r1! r2!T[r1]⊔∅(S r1 +int) ⋆H TS(AS) ⋆H T[r2]⊔∅(S r2 +int) ⋆H TT (AT ) +� +0 +. += +� ∞ +� +r=0 +gr +r! T[r]⊔∅(S r +int) ⋆H TS(AS) ⋆H +∞ +� +r=0 +gr +r! T[r]⊔∅(S r +int) ⋆H TT (AT ) +� +0 +. += +� +S−1(gSint) ⋆H TS(AS) ⋆H S(gSint) ⋆H TT (AT ) +� +0 += +1 +� +S(gSint) +� +0 +� +TS(AS) ⋆H S(gSint) ⋆H TT (AT ) +� +0. +For the final step, we used vacuum stability (22). +□ +24 the element S(gSint) ∈ F((ℏ))[[g]] is called the perturbative S-matrix + +HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY +37 +References +[AB08] +Peter Abramenko and Kenneth S. Brown. Buildings, volume 248 of Graduate Texts in Mathematics. +Springer, New York, 2008. Theory and applications. 34 +[Abd04] +Abdelmalek Abdesselam. Feynman diagrams in algebraic combinatorics. Sém. Lothar. Combin., 49:Art. +B49c, 45, 2002/04. 30 +[AM10] +Marcelo Aguiar and Swapneel Mahajan. Monoidal functors, species and Hopf algebras, volume 29 of CRM +Monograph Series. American Mathematical Society, Providence, RI, 2010. With forewords by Kenneth +Brown, Stephen Chase and André Joyal. 2, 3, 4, 8, 10, 11, 12, 14, 19, 26, 32 +[AM13] +Marcelo Aguiar and Swapneel Mahajan. 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Lecture Notes in Mathematics, Vol. 386. +Springer-Verlag, Berlin-New York, 1974. 3, 34 +Pennsylvania State University +Email address: wxn39@psu.edu + diff --git a/ptAyT4oBgHgl3EQfzflz/content/tmp_files/load_file.txt b/ptAyT4oBgHgl3EQfzflz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f1db1afd72019c0d103a31ea60c50f25a6d331c --- /dev/null +++ b/ptAyT4oBgHgl3EQfzflz/content/tmp_files/load_file.txt @@ -0,0 +1,1884 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf,len=1883 +page_content='HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY WILLIAM NORLEDGE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We develop an algebraic formalism for perturbative quantum field theory (pQFT) which is based on Joyal’s combinatorial species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We show that certain basic structures of pQFT are correctly viewed as algebraic structures internal to species, constructed with respect to the Cauchy monoidal product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Aspects of this formalism have appeared in the physics literature, particularly in the work of Bogoliubov-Shirkov, Steinmann, Ruelle, and Epstein-Glaser-Stora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we give a fully explicit account in terms of modern theory developed by Aguiar-Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We describe the central construction of causal perturbation theory as a homomorphism from the Hopf monoid of set compositions, decorated with local observables, into the Wick algebra of microcausal polynomial observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The operator-valued distributions called (generalized) time-ordered products and (generalized) retarded products are obtained as images of fundamental elements of this Hopf monoid under the curried homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The perturbative S-matrix scheme corresponds to the so-called universal series, and the property of causal factorization is naturally expressed in terms of the action of the Hopf monoid on itself by Hopf powers, called the Tits product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a system of fully renormalized time-ordered products, the perturbative construction of the corresponding interacting products is via an up biderivation of the Hopf monoid, which recovers Bogoliubov’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Contents Introduction 1 Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Hopf Monoids 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Algebras 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Σ as a Hopf E-Algebra 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Products and Series 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbation of T-Products 27 Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbative Algebraic Quantum Field Theory 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Spacetime and Field Configurations 30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Observables 31 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Time-Ordered Products and S-Matrix Schemes 33 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Interactions 35 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Scattering Amplitudes 35 References 37 Introduction The theory of species is a richer, categorified version of analyzing combinatorial structures in terms of generating functions, going back to André Joyal [Joy81], [Joy86], [BLL98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this approach, This paper is an abridged version of ‘Species-theoretic foundations of perturbative quantum field theory’, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='09969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='00702v1 [math-ph] 2 Jan 2023 2 WILLIAM NORLEDGE one sees additional structure by encoding processes of relabeling combinatorial objects, that is by modeling combinatorial objects as presheaves on the category S of finite sets I (the labels) and bijections σ (relabelings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we are concerned with species p valued in complex vector spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' functors of the form p : Sop → Vec, I �→ p[I], σ �→ p[σ] where Vec is the category of complex vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Explicitly, p consists of a complex vector space p[I] for each finite set I, and a bijective linear map p[σ] : p[I] → p[J] for each bijection σ : J → I such that composition of bijections is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A highly structured theory of gebras1 internal to vector species has been developed by Aguiar-Mahajan [AM10], [AM13], building on the work of Barratt [Bar78], Joyal [Joy86], Schmitt [Sch93], Stover [Sto93b], and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For the internalization, one uses the Day convolution monoidal product p • q with respect to disjoint union and tensor product, given by p • q[I] = p ⊗Day q[I] = � S⊔T=I p[S] ⊗ q[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This may be viewed as a categorification of the Cauchy product of formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2 Various decategorifications of Aguiar-Mahajan’s theory recovers the plethora of graded combinatorial Hopf algebras which have been studied [AM10, Chapter 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' On the other hand, quantum field theory (QFT) may be viewed as a kind of modern infinite dimensional calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbative quantum field theory (pQFT) is the part of QFT which considers Taylor series approximations of smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By an argument of Dyson [Dys52], Taylor series of realistic pQFTs are expected to have vanishing radius of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Nevertheless, if an actual smooth function of a non-perturbative quantum field theory is being approximated, then they are asymptotic series, and so one might expect their truncations to agree to reasonable precision with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is indeed the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There are two main synthetic approaches to (non-perturbative) QFT, which grew out of the failure to make sense of the path integral analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There is functorial quantum field theory (FQFT), which formalizes the Schrödinger picture by assigning time evolution operators to cobordisms between spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There is also algebraic quantum field theory (AQFT), going back to [HK64], which formalizes the Heisenberg picture by assigning C∗-algebras of observables to regions of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Low dimension examples of AQFTs/Wightman field theories were rigorously constructed in seminal work of Glimm-Jaffe and others [GJ68], [CJ70], [GJS74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbative algebraic quantum field theory (pAQFT) [Rej16], [Düt19], [Sch20, nLab], due to Brunetti, Dütsch, Fredenhagen, Hollands, Rejzner, Wald, and others, is (mathematically precise, realistic) pQFT based on causal perturbation theory [Ste71], [EG73], [Sch95], due to Stückelberg, Bogoliubov, Steinmann, Epstein, Glaser, Stora, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [Düt19, Foreword] for an account of the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [IS78], [BF00], [DF01], in which one takes the algebraic adiabatic limit to handle IR-divergences, pAQFT satisfies the Haag-Kastler axioms of AQFT, but with C∗-algebras replaced by formal power series ∗-algebras, reflecting the fact that pQFT deals with Taylor series approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we show that the construction and structure of these formal power series algebras is naturally described in terms of gebra theory internal to species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1 meaning (co/bi/Hopf)algebras and Lie (co)algebras 2 from the perspective of S-colored (co)operads, as defined in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Pet13, Section 3], there is an equivalent description of these gebras as (co)algebras over the left (co)action (co)monads of the (co)operads Com(∗), Ass(∗), Lie(∗) [AM10, Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5], which relates the gebras of this paper to structures such as cyclic operads, which already appear in mathematical physics HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 3 For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on Minkowski spacetime X ∼= Rp,1, p ∈ N (pAQFT may be applied in more general settings, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Hol08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore for us, an off-shell field configuration Φ is a smooth function Φ : X → R, x �→ Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, we do not impose conditions on the asymptotic behaviour of Φ at infinite times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Floc denote the space of local observables A ∈ Floc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' these are functionals of field configurations which are obtained by integrating polynomials in Φ and its derivatives against bump functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let F denote the commutative ∗-algebra of microcausal polynomial observables O ∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' these are polynomial functionals of field configurations satisfying a microlocal-theoretic condition known as microcausality, with multiplication the pointwise multiplication of functionals, sometimes called the normal-ordered product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then F[[ℏ]] is a formal power series ∗-algebra in formal Planck’s constant ℏ, called the (abstract, off-shell) Wick algebra, with multiplication the Moyal star product for the Wightman propagator ∆H of the Klein-Gordan field F[[ℏ]] ⊗ F[[ℏ]] → F[[ℏ]], O1 ⊗ O2 �→ O1 ⋆HO2, sometimes called the operator product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perhaps the most fundamental Hopf monoid of Aguiar-Mahajan’s theory is the cocommutative Hopf algebra3 of compositions Σ, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2, which is a Hopf monoid internal to vector species defined with respect to the Day convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (More familiar is perhaps a certain decategorification of Σ, which is the graded Hopf algebra of noncommutative symmetric functions NSym, see [AM10, Section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=') A composition F of I is a surjective function of the form F : I → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , k}, for some k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The ordering 1 > · · · > k is understood, so that F models the kth ordinal with I-marked points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We let Sj = F −1(j), called the lumps of F, and write F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Each component Σ[I] is the space of formal linear combinations of compositions F of I, Σ[I] = � a = � F cF HF �� cF ∈ C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The multiplication µS,T : Σ[S] ⊗ Σ[T] → Σ[I], HF ⊗ HG �→ HFG is the linearization of concatenating compositions (‘gluing’ via ordinal sum), and the comultiplication ∆S,T : Σ[I] → Σ[S] ⊗ Σ[T], HF �→ HF|S ⊗ HF|T is the linearization of restricting compositions to subsets (‘forgetting marked points’), where S⊔T = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Aspects of Σ have appeared in the physics literature as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Firstly, Epstein-Glaser-Stora’s algebra of proper sequences [EGS75, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] is the action of Σ on itself by Hopf powers, called the Tits product [AM13, Section 13], going back to Tits [Tit74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Secondly, the primitive part Zie = P(Σ)4, which is a Lie algebra internal to species, is essentially the Steinmann algebra from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Rue61, Section 6], [BL75, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' More precisely, the Steinmann algebra is a graded Lie algebra based on the structure map of the adjoint realization of Zie, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thirdly and fourthly, and outside the scope of this paper, see below regarding work of Losev-Manin and Feynman integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 3 we say ‘algebra’ and not ‘monoid’ since vector species form a linear category 4 the name ‘Zie’ comes from [AM17] 4 WILLIAM NORLEDGE The central idea of this paper is to formalize the construction of a system of interacting time-ordered products in causal perturbation theory as the construction of a homomorphism �T of algebras internal to species of the form �T : Σ ⊗ EFloc[[ℏ]] → UF[[ℏ,g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We describe this construction in a clean abstract setting in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1, and then specialize to QFT in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Here, ⊗ is the Hadamard monoidal product (=componentwise tensoring), EFloc[[ℏ]] is the species given by I �→ (Floc[[ℏ]])⊗I, and UF[[ℏ,g]] is the algebra in species which has the Wick algebra, with formal coupling constant g adjoined, in each I-component, EFloc[[ℏ]][I] = (Floc[[ℏ]])⊗I, UF[[ℏ,g]][I] = F[[ℏ, g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows that the data of a system of products �T is equivalently a homomorphism of C-algebras ˆΣ(Floc[[ℏ]]) → F[[ℏ, g]] where ˆΣ(−) : Vec → Vec is the analytic endofunctor, or Schur functor, on vector spaces associated to Σ [AM10, Section 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5 Decategorified versions of this formalization appear in graded Hopf algebra approaches to pQFT [Bro09], [Bor11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 635].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, there is an interpretation of the Moyal deformation quantization in terms of Laplace pairings (=coquasitriangular structures) [Fau01], [Bro09, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Also related is the notion of a Losev-Manin cohomological field theory [LM00, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], [SZ11, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3], where finite ordinals are replaced by strings of Riemann spheres glued at the poles, giving a Hopf monoid structure on the toric variety of the permutohedron, and Σ is replaced by the ordinary homology of this toric variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Hopf monoid structure of this toric variety is also central to modern approaches to Feynman integrals [Bro17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6], [Sch18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We shall study this Hopf monoid in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Explicitly, the homomorphism �T consists of component linear maps �TI : Σ[I] ⊗ (Floc[[ℏ]])⊗I → F[[ℏ, g]], HF ⊗ Ai1 ⊗ · · · ⊗ Ain �→ �TI(HF ⊗ Ai1 ⊗ · · · ⊗ Ain) for each finite set I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , in}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This homomorphism should also satisfy causal factorization, which says �TI(a ⊗ Ai1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Ain) = �TI( a ▷ HG � �� � Tits product ⊗Ai1 ⊗ · · · ⊗ Ain) for all a ∈ Σ[I] whenever the local observables Ai1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Ain respect the ordering of I induced by the composition G, see Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Additional properties are often included, such as translation equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can curry �T with respect to the internal hom H(−, −) for the Hadamard product, giving a homomorphism of algebras Σ → H(EFloc[[ℏ]], UF[[ℏ,g]]), HF = H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) �→ �T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �T(Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The resulting linear maps �T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �T(Sk) : (Floc[[ℏ]])⊗I → F[[ℏ, g]] are called interacting generalized time-ordered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For each choice of a field polynomial, the curried homomorphism is a ‘representation’ of Σ as F[[ℏ, g]]-valued generalized functions on X I, 5 the hat ˆΣ is meant to suggest a kind of categorified Fourier transform HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 5 called operator-valued distributions since the Wick algebra is often represented on a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The composition of the time-ordered products �T(I) with the Hadamard vacuum state ⟨−⟩0 : F[[ℏ, g]] → C[[ℏ, g]], O �→ O(Φ = 0) are then translation invariant C[[ℏ, g]]-valued generalized functions GI : X I → C[[ℏ, g]], (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin) �→ GI(xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin)6 called time-ordered n-point correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' After taking the adiabatic limit, and in the presence of vacuum stability, these functions may be interpreted as the probabilistic predictions made by the pQFT of the outcomes of scattering experiments, called scattering amplitudes, see Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' However, their values are formal power series in ℏ and g, and so have to be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Central to Aguiar-Mahajan’s work is the interpretation of Σ (and other Hopf monoids) in terms of the geometry of the type A reflection hyperplane arrangement, called the (essentialized) braid arrangement Br[I] = �{xi1 − xi2 = 0} ⊆ RI/R ↞ RI � �� � quotient by translations : (i1, i2) ∈ I2, i1 ̸= i2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In causal perturbation theory, the braid arrangement appears as the space of time components of configurations X I modulo translational symmetry [Rue61, Section 2], and the reflection hyperplanes are the coinciding interaction points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Every real hyperplane arrangement A has a corresponding adjoint hyperplane arrangement A∨ [AM17, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The free vector space RI on I is naturally Hom(RI, R), and so the adjoint of the braid arrangement is given by Br∨[I] = �� � i∈S xi = � i∈T xi = 0 � ⊆ Hom(RI/R, R) �→ RI � �� � sum-zero subspace : (S, T) ∈ 2I, S, T ̸= ∅ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In causal perturbation theory, the adjoint braid arrangement appears as the space of energy components [Rue61, Section 2], and the hyperplanes correspond to subsets going ‘on-shell’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The spherical representation of the adjoint braid arrangement is called the Steinmann sphere, or Steinmann planet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Eps16, Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The chambers of the adjoint braid arrangement are indexed by combinatorial gadgets called cells S [EGS75, Definition 6], also known as maximal unbalanced families [BMM+12] and positive sum systems [Bjo15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The primitive part Lie algebra Zie = P(Σ) (together with its dual Lie coalgebra Zie∗) has a natural geometric realization over the adjoint braid arrangement [Rue61, Section 6], [Ocn18, Lecture 33], [LNO19], [NO19], which results in cells S corresponding to certain special primitive elements DS ∈ Zie[I], see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The special elements were named Dynkin elements by Aguiar- Mahajan [AM17, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It is shown in [NO19] that the Dynkin elements span Zie, but they are not linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The relations which are satisfied by the Dynkin elements are known as the Steinmann relations [Ste60b, Equation 44], see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6, first studied by Steinmann in settings where Σ is represented as operator-valued distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' More recently, they have been studied in the context scattering amplitudes, where they appear to be related to cluster algebras [DFG18], [CHDD+19], [CHDD+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we restrict a curried system of interacting generalized time-ordered products to the primitive part Zie, then we obtain a Lie algebra homomorphism Zie → H(EFloc[[ℏ]], UF[[ℏ,g]]), DS �→ �RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 6 we have used generalized function notation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' GI is not a single function, but can be represented by a sequence of functions 6 WILLIAM NORLEDGE spanning set operator-valued distributions vacuum expectation values E∗ universal series GI time-ordered product T(I) time-ordered n-point function L H-basis linear orders Hℓ T(i1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(in) Wightman n-point functions Σ H-basis set compositions HF generalized time-ordered products T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk) generalized time-ordered functions Zie Dynkin elements DS generalized retarded products RS generalized retarded functions Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Dictionary between products/vacuum expectation values and elements of the Hopf algebra Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The operator-valued distributions �RS which are the images of the Dynkin elements DS are the interacting generalized retarded products of the system, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Ste60b], [Ara61], [EG73, Equation 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we give an exposition of the Steinman algebra and Steinmann relations in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5 and Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let L �→ Σ be the Hopf subalgebra of linear orders (=compositions with singleton lumps), and let E∗ �→ Σ be the subcoalgebra of compositions with one lump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then we have the dictionary in Figure 1 between products/vacuum expectation values and elements of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the commutative setting before Moyal deformation quantization, the species X and E are similarly related to the smeared field and polynomial observables, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 and Section 8, we formalize the perturbation of time-ordered products in casual perturbation theory as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Our starting point is a fully normalized system of generalized time-ordered products, that is a homomorphism of algebras T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) satisfying causal factorization, and such that the singleton components T{i} are the natural inclusion Floc[[ℏ]] �→ F((ℏ)), A �→ :A : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The corresponding operator-valued distributions are determined everywhere on X I by causal factorization, apart from on the fat diagonal (=coinciding interaction points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, off the fat diagonal, the time-ordered products T(I) are given by the Moyal star product ⋆F with respect to the Feynman propagator ∆F for the Klein-Gordon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The terms of the product ⋆F may be encoded in finite multigraphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Feynman graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The remaining inherent ambiguity means one has to make choices when extending the T(I) to the fat diagonal, and these choices form a torsor of the Stückelberg-Petermann renormalization group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is Stora’s elaboration [PS16], [Sto93a], [BF00] on Stückelberg-Bogoliubov-Epstein-Glaser normalization [EG73], which constructs the T(I) inductively in n = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We leave species-theoretic aspects of renormalization, and possible connections to Connes-Kreimer theory [Pin00], [GBL00], [BK05], [DFKR14], to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the original formulation by Tomonaga, Schwinger, Feynman and Dyson, would-be time-ordered products are obtained by informally multiplying Wick algebra products by step functions, which is in general ill-defined by Hörmander’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This leads to the divergence of individual terms of the formal power series, called UV-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then informal methods are used to obtain finite values from these infinite terms [Sch95, Preface and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 7 The exponential species E, given by E[I] = C and 1C ∈ E[I] denoted HI, has the structure of an algebra in species by linearizing taking unions of sets, µS,T : E[S] ⊗ E[T] → E[I], HS ⊗ HT �→ HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An E-module m = (m, ρ) is an associative and unital morphism ρ : E • m → m for m a species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Moreover, taking the inverse of µS,T as the comultiplication turns E into a connected (co)commutative bialgebra, and so the category of E-modules Rep(E) is a symmetric monoidal category with monoidal product the Cauchy product of E-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, we may consider Hopf/Lie algebras internal to Rep(E), which we call Hopf/Lie E-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The retarded Y ↓ (−) and advanced Y ↑ (−) Steinmann arrows are (we formalize as) raising operators on Σ, whose precise definition is due to Epstein-Glaser-Stora [EGS75, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='82-83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' They define two E-module structures on Σ, E • Σ → Σ, HY ⊗ HF �→ Y ↓ HF and E • Σ → Σ, HY ⊗ HF �→ Y ↑ HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, the retarded arrow is generated by putting {∗} ↓ H(I) = −H(∗,I)+H(∗I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='7 Then Y ↓ H(I) = � Y1⊔Y2=Y µY1,Y2⊔I �s(H(Y1)) ⊗ H(Y2⊔I) � � �� � denoted R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) where s : Σ → Σ is the antipode of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann arrows were first studied by Steinmann [Ste60b, Section 3], where Σ is represented as operator-valued distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Here, the operator-valued distribution which corresponds to R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) ∈ Σ[Y ⊔ I] is called the retarded product R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='8 Since {∗} ↓ (−) is a commutative biderivation of Σ (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1), the retarded Steinmann arrow gives Σ the structure of a Hopf E-algebra, and Zie the structure of a Lie E-algebra (similarly for the advanced arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There is an interesting description of these Lie E-algebras in terms of the adjoint braid arrangement, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann arrows are “two halves” of the restricted adjoint representation L • Σ → Σ of Σ, which is reflected in [Ste60b, Equation 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This directly corresponds to how the retarded ∆− and advanced ∆+ propagators are two halves of the causal propagator ∆S = ∆+ − ∆−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let H (−, −) denote the internal hom for the Cauchy product of species, and let (−)E = H (E, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3 for a more explicit definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See also [Nor20, Section 2] for more details here regarding the differentiation between the j -colored sets I (physically, the source field) and the g-colored sets Y (physically, the coupling constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then (−)E is an endofunctor on species, which is lax monoidal with respect to the Cauchy product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore ΣE is naturally an algebra, with multiplication inherited from Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, by currying the retarded Steinmann action E • Σ → Σ, we obtain a homomorphism Σ → ΣE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Similarly for the setting with decorations, given a choice of 7 (∗I) denotes the composition of {∗} ⊔ I which has a single lump 8 note that some authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Düt19], call R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' i) the retarded product, and then call R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I) the generalized retarded product 8 WILLIAM NORLEDGE adiabatically switched interaction action functional Sint ∈ Floc[[ℏ]], after acting with the retarded Steinmann arrows and currying, we obtain the homomorphism Σ ⊗ EFloc[[ℏ]] → (Σ ⊗ EFloc[[ℏ]])E HF ⊗ Ai1 ⊗ · · · ⊗ Ain �→ ∞ � r=0 ↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ↓ � �� � r times HF ⊗ Sint ⊗ · · · ⊗ Sint � �� � r times ⊗ Ai1 ⊗ · · · ⊗ Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Compare this with the formalism for creation-annihilation operators in [AM10, Chapter 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, finally, the corresponding system of perturbed interacting time-ordered products �T is given by composing this homomorphism with the image of T under the endofunctor (−)E, �T : Σ ⊗ EFloc[[ℏ]] → (Σ ⊗ EFloc[[ℏ]])E TE −−→ (UF((ℏ)))E ∼= UF((ℏ))[[g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It is a theorem of pAQFT that this does indeed land in UF[[ℏ,g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Finally, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3 and Section 7, we formalize S-matrices, or time-ordered exponentials, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Hom(−, −) denote the external hom for species, which lands in vector spaces Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We let S (−) = Hom(E, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is lax monoidal with respect to the Cauchy product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the presence of a generic system of products on an algebra a, ϕ : a ⊗ EV → UA, series s ∈ S (a) of a s : E → a, HI �→ sI induce S (UA) ∼= A[[j ]]-valued functions Ss on V as follows, Ss : V → A[[j ]], A �→ Ss(jA) := ∞ � n=0 j n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ϕn(sn ⊗ A ⊗ · · · ⊗ A � �� � n times ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If ϕ is a homomorphism of algebras, then S(−) : S (a) → Func(V, A[[j ]]) is a homomorphism of C-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' As a basic example, if we put a = E, A = C∞(V ∗), and set j = 1 at the end, then one can recover the classical exponential function in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For c ∈ C, the so-called (scaled) universal series G(c) of Σ is given by sending each finite set to the (scaled) composition with one lump, G(c) : E → Σ, HI �→ G(c)I := cn H(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we set c = 1/iℏ, then the function S = SG(1/iℏ) above for a fully normalized system of generalized time-ordered products T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) recovers the usual perturbative S-matrix scheme of pAQFT, S : Floc[[ℏ]] → F((ℏ))[[j ]], A �→ S(jA) = ∞ � n=0 � 1 iℏ �n j n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tn(H(n) ⊗ A ⊗ · · · ⊗ A � �� � n times ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The image of S(jA) after applying perturbation by the retarded Steinmann arrow and a choice of interaction Sint ∈ Floc[[ℏ]] is ZgSint(jA) = ∞ � n=0 ∞ � r=0 � 1 iℏ �r+n grj n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='n(Sint ⊗ · · · ⊗ Sint � �� � r times ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A ⊗ · · · ⊗ A � �� � n times ) HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 9 where, by our previous expression for R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) = Y ↓ H(I) (and letting T denote the precomposition of T with the antipode of Σ ⊗ EFloc[[ℏ]]), we have RY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S Y int;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) = TY ⊔I(Y ↓ H(I) ⊗ S Y int ⊗ AI) = � Y1⊔Y2=Y TY1(S Y1 int) ⋆H TY2⊔I(S Y2 int ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, since S(−) : S (Σ) → Func �Floc[[ℏ]], F((ℏ))[[g]] � is a homomorphism of C-algebras, it follows that ZgSint is given by ZgSint(jA) = S−1(gSint) ⋆H S(gSint + jA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is the generating function, or partition function, for time-ordered products of interacting field observables, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [EG73, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], [DF01, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2], going back to Bogoliubov [BS59, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we arrive at the generating function ZgSint through purely Hopf-theoretic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' However, it was originally motivated by attempts to make sense of the path integral synthetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For some recent developments, see [Col16], [HR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This paper is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In part one, we focus on developing theory for the Hopf algebra of compositions Σ and its primitive part Zie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In part two, we specialize to pAQFT for the case of a real scalar field on Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We thank Adrian Ocneanu for his support and useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This paper would not have been written without Nick Early’s discovery that certain relations appearing in Ocneanu’s work were known in quantum field theory as the Steinmann relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We thank Yiannis Loizides and Maria Teresa Chiri for helpful discussions during an early stage of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We thank Arthur Jaffe for his support, useful suggestions, and encouragement to pursue this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We thank Penn State maths department for their continued support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Hopf Monoids 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Algebras We recall the Hopf algebra of compositions Σ, together with its Lie algebra of primitive elements Zie �→ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We show that Σ and Zie are naturally algebras over the exponential species E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This will be a species-theoretic formalization of mathematical structure discovered by Steinmann [Ste60b] and Epstein-Glaser-Stora [EGS75], which, combined with a certain ‘perturbation of systems of products’ construction using the E-action, will recover the perturbative construction of interacting fields in pAQFT, as in [EG73, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], [DF01, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2], going back to Bogoliubov [BS59, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let I be a finite set of cardinality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We think of I as having ‘color’ j (physically, the source field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' As a particular example of the set I, we have the set of integers [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , n} (formally, we have picked a section of the decategorification functor I �→ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For k ∈ N, let (k) := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , k} equipped with the ordering 1 > · · · > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A composition F of I of length l(F) = k is a surjective function F : I → (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The set of all compositions of I is denoted Σ[I], Σ[I] := � k∈N �surjective functions F : I → (k) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 10 WILLIAM NORLEDGE We often denote compositions by k-tuples F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk) where Sj := F −1(j), 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Sj are called the lumps of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, we have the length one composition (I) for I ̸= ∅, and the length zero composition ( ) which is the unique composition of the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The opposite ¯F of F is defined by ¯F := (Sk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , S1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ¯F −1(j) = F −1(k + 1 − j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a decomposition I = S ⊔ T of I (S, T can be empty), for F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk) a composition of S and G = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Tl) a composition of T, their concatenation FG is the composition of I given by FG := (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Tl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For S ⊆ I and F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk) ∈ Σ[I], the restriction F|S of F to S is the composition of S given by F|S := (S1 ∩ S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk ∩ S)+ where (−)+ means we delete any sets from the list which are the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For compositions F, G ∈ Σ[I], we write G ≤ F if G can be obtained from F by iteratively merging contiguous lumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given compositions G ≤ F with G = (T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Tl), we let l(F/G) := k � j=1 l(F|Tj) and (F/G)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' := k � j=1 l(F|Tj)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Cocommutative Hopf Monoid of Compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Σ[I] := �formal C-linear combinations of compositions of I �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The vector space Σ[I] is naturally a right module over the symmetric group on I, and these actions extend to a contravariant functor from the category S of finite sets and bijections into the category Vec of vector spaces over C, Σ : Sop → Vec, I �→ Σ[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For F a composition of I, let HF ∈ Σ[I] denote the basis element corresponding to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The sets {HF : F ∈ Σ[I]} form the H-basis of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In general, functors p : Sop → Vec are called (complex) vector species, going back to Joyal [Joy81], [Joy86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Morphisms of vector species η : p → q are natural transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' they consist of a linear map ηI : p[I] → q[I] for each finite set I which commutes with the action of the bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' When I = [n] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , n}, we abbreviate ηn := η[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We equip vector species with the tensor product p • q known as the Cauchy product [AM10, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5], given by (1) p • q[I] := � I=S⊔T p[S] ⊗ q[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is the Day convolution with respect to disjoint union of sets and tensor product of vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In this paper, we consider algebraic structures on species which are constructed using this tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, a multiplication on a species p consists of linear maps µS,T : p[S] ⊗ p[T] → p[I] and a comultiplication on p consists of linear maps ∆S,T : p[I] → p[S] ⊗ p[T], HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 11 where we have a map for each choice of decomposition I = S ⊔ T (S, T can be empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can then impose conditions like (co)associativity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Nor20, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [AM13, Section 11], Σ is a connected9 bialgebra, meaning it is naturally equipped with an associative, unital multiplication and a coassociative, counital comultiplication, which are compatible in the sense they satisfy the bimonoid axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [AM10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The multiplication and comultiplication are given in terms of the H-basis by µS,T (HF ⊗ HG) := HFG and ∆S,T (HF ) := HF|S ⊗ HF|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We sometimes abbreviate HF HG := µS,T (HF ⊗ HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The unit and counit are given by 1Σ := H( ) and ϵ∅(H( )) := 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let (2) HF := � G≥ ¯F (−1)l(G) HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then [AM10, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='38] (in the case q = E∗ + and q = 1) shows that (3) � S⊔T=I HF|SHF|T = 0 and � S⊔T=I HF|SHF|T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In general, connected bialgebras are automatically Hopf algebras, and it follows from (3) that the antipode s : Σ → Σ is given by sI(HF ) = HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Hopf algebra Σ is the free cocommutative Hopf algebra on the positive coalgebra E∗ + [AM10, Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5], and so Σ ∼= L ◦ E∗ + where ‘◦’ is plethysm of species and L �→ Σ is the subspecies of singleton lump compositions (=linear orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There is a second important basis of Σ, called the Q-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Q-basis is also indexed by compositions, and is given by QF := � G≥F (−1)l(G)−l(F) 1 l(G/F)HG or equivalently HF =: � G≥F 1 (G/F)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='QG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For S ⊆ I and F ∈ Σ[I], we have deshuffling F ∥S := � F|S if S is a union of lumps of F 10 0 ∈ Σ[S] otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The multiplication and comultiplication of Σ is given in terms of the Q-basis by µS,T (QF ⊗ QG) = QFG and ∆S,T (QF ) = QF∥S ⊗ QF∥T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Decorations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a complex vector space V , we can use V to ‘decorate’ Σ in order to obtain an enlarged Hopf algebra Σ ⊗ EV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the species denoted EV , given by EV [I] := V ⊗I = V ⊗ · · · ⊗ V � �� � a copy of V for each i ∈ I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The action of bijections is given by relabeling tensor factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice species of the form EV are exactly the monoidal functors EV : Sop → Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 9 a species p is connected if p[∅] = C 10 not necessarily contiguous 12 WILLIAM NORLEDGE We denote vectors by A, S ∈ V , and we denote simple tensors of V ⊗I by AI = Ai1 ⊗ · · · ⊗ Ain ∈ V ⊗I where I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , in}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If Ai = A for all i ∈ I, then we write (4) AI := A ⊗ · · · ⊗ A ∈ V ⊗I and An := A[n] ∈ V ⊗[n] where [n] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , n} as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We let ‘⊗’ denote the Hadamard product of species, which is given by componentwise tensoring, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Nor20, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then the species of V -decorated compositions Σ ⊗ EV is given by Σ ⊗ EV [I] = Σ[I] ⊗ EV [I] = Σ[I] ⊗ V ⊗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [AM10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], Σ ⊗ EV is a connected bialgebra, with multiplication given by µS,T �(HF ⊗ AS) ⊗ (HG ⊗ AT ) � := HF HG ⊗ AS ⊗ AT and comultiplication given by ∆S,T (HF ⊗ AI) := (HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The unit and counit are given by 1Σ⊗EV := H( ) ⊗ 1C and ϵ∅(H( ) ⊗ 1C) := 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For HF ⊗ AI ∈ Σ ⊗ EV [I], we have � S⊔T=I µS,T �(HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ) � = � S⊔T=I HF|SHF|T � �� � = 0 by (3) ⊗ AI = 0 and � S⊔T=I µS,T �(HF|S ⊗ AI|S) ⊗ (HF|T ⊗ AI|T ) � = � S⊔T=I HF|SHF|T � �� � = 0 by (3) ⊗ AI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows that the antipode of Σ ⊗ EV is given by (5) sI(HF ⊗ AI) = HF ⊗ AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Hopf algebra Σ is connected and cocommutative, and so the CMM Theorem applies, see [Nor20, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We now describe the positive11 Lie algebra of primitive elements P(Σ) ⊂ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For I ∈ S a finite set, let a tree T over I be a planar12 full binary tree whose leaves are labeled bijectively with the blocks of a partition of I (a partition P of I is a set of disjoint nonempty subsets of I, called blocks, whose union is I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The blocks of this partition, called the lumps of T, form a composition called the debracketing FT of T, by listing them in order of appearance from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote trees by nested products [ · , · ] of subsets or trees, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We make the convention that no trees exist over the empty set ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 11 a species p is positive if p[∅] = 0 12 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' a choice of left and right child is made at every node HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 13 1 9 678 24 1 23 2 3 5 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let I be various subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The trees [4], [1, 23] (̸= [23, 1]), [[2, 3], 5], [[24, [1, 9]], 678] are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The debracketing of [[24, [1, 9]], 678] is the composition (24, 1, 9, 678).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we put T1 = [24, [1, 9]] and T2 = [678], then [T1, T2] would also denote this tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We define the positive species Zie by letting Zie[I] denote the vector space of formal C-linear combinations of trees over I, modulo the relations of antisymmetry and the Jacobi identity as interpreted on trees in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Explicitly, (1) (antisymmetry) for all trees of the form [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [T1, T2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] (writing a tree in this form is equivalent to picking a node) we have [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [T1, T2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] + [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [T2, T1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (2) (Jacobi Identity) for all trees of the form [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [[T1, T2], T3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] we have [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [[T1, T2], T3] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] + [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [[T3, T1], T2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] + [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [[T2, T3], T1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then Zie is a positive Lie algebra in species, with Lie bracket ∂∗ given by ∂∗ S,T (T1 ⊗ T2) := [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have that Zie is the free Lie algebra on the positive exponential species E∗ +, and so the species Zie is also given by Zie[I] = Lie ◦ E∗ +[I] = � P Lie[P] where Lie is the species of the Lie operad, and the direct sum is over all partitions P of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Lie algebra in species Zie is closely related to the Steinmann algebra from the physics literature [BL75, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], [Rue61, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Precisely, the Steinmann algebra is an ordinary graded Lie algebra based on the structure map for the adjoint braid arrangement realization of Zie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The adjoint braid arrangement realization of Zie is the topic of [LNO19], and the fact that the Lie algebra there is indeed Zie was shown in [NO19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Via the commutator bracket, Σ is a Lie algebra in species, given by [HF , HG] = HF HG − HGHF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] := �surjective functions I → {1, 2} � denote the set of compositions of I with two lumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since Σ is connected, its positive Lie subalgebra of primitive elements P(Σ) ⊂ Σ is given on nonempty I by P(Σ)[I] = � (S,T)∈[I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2] ker �∆S,T : Σ[I] → Σ[S] ⊗ Σ[T] �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, Q(I) ∈ P(Σ)[I] for I nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since Zie is freely generated by stick trees [I], we can define a homomorphism of Lie algebras by Zie → P(Σ), [I] �→ Q(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 14 WILLIAM NORLEDGE To describe this explicitly, given a tree T, let antisym(T) denote the set of 2l(FT)−1 many trees which are obtained by switching left and right branches at nodes of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For T′ ∈ antisym(T), let (T, T′) ∈ Z/2Z denote the parity of the number of node switches required to bring T to T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then the homomorphism is given in full by Zie → P(Σ), T �→ QT := � T′∈antisym(T) (−1)(T,T′)QFT′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By [AM10, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='46], this is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' From now on, we make the identification Zie = P(Σ) and retire the notation P(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Type A Dynkin Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Recall that the set of minuscule weights of (the root datum of) SLI(C) is in natural bijection with [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote the minuscule weight corresponding to (S, T) by λST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [NO19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A cell13 [EGS75, Definition 6] over I is (equivalent to) a subset S ⊆ [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] such that for all (S, T) ∈ [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2], exactly one of (S, T) ∈ S and (T, S) ∈ S is true, and whose corresponding set of minuscule weights is closed under conical combinations, that is λUV ∈ coni �λST : (S, T) ∈ S � =⇒ (U, V ) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By dualizing conical spaces generated by minuscule weights, cells are in natural bijection with chambers of the adjoint of the braid arrangement, see [NO19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3], [Eps16, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Their number is sequence A034997 in the OEIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote the species of formal C-linear combinations of cells by L∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Associated to each composition F of I is the subset FF ⊆ [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] consisting of those compositions (S, T) which are obtained by merging contiguous lumps of F, FF := �(S, T) ∈ [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] : (S, T) ≤ F �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' More geometrically, FF is the subset corresponding to the set of minuscule weights which are contained in the closed braid arrangement face of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let us write F ⊆ S as abbreviation for FF ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Consider the morphism of species given by (6) L∨ → Σ, S �→ DS := − � ¯F⊆S (−1)l(F)HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The element DS is called the Dynkin element associated to the cell S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' These special elements were defined by Epstein-Glaser-Stora in [EGS75, Equation 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='26], and the name is due to Aguiar-Mahajan [AM17, Equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In fact, DS is a primitive element [AM17, Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], and so we actually have a morphism L∨ → Zie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For i ∈ I, let Si denote the cell given by Si := �(S, T) ∈ [I, 2] : i ∈ S �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 13 also known as maximal unbalanced families [BMM+12] and positive sum systems [Bjo15] HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 15 1 +1 1 +1 +1 1 13 ∞ future 2 ∞ past 3 ∞ future 12 ∞ past 23 ∞ future 1 ∞ past 1 2 3 DS rS S Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A cell S over {1, 2, 3} (on the adjoint braid arrangement) and its Dynkin element DS (on the tropical geometric realization of Σ, where the multiplication embeds facets and the comultiplication projects onto facets, see [NO19, Introduction])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the presence of causal factorization, the time component of the corresponding generalized retarded function rS is a C[[ℏ, g]]-valued generalized function on the braid arrangement with support the gray cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Dynkin element shown is DS = D3 = R(12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Its support consists of those configurations such that the event labeled by 3 can be causally influenced by the events labeled by 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is the cell corresponding to the adjoint braid arrangement chamber which contains the projection of the basis element ei ∈ RI onto the sum-zero hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the total retarded Dynkin element Di associated to i be given by Di := DSi = − � F∈Σ[I] i∈Sk (−1)l(F)HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' These Dynkin elements are considered in [AM13, Section 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For i ∈ I, let ¯Si := �(S, T) ∈ [I, 2] : i ∈ T �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is the cell corresponding to the adjoint braid arrangement chamber which is opposite to the chamber of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the total advanced Dynkin element D¯i associated to i be given by D¯i := D ¯Si = − � F∈Σ[I] i∈S1 (−1)l(F)HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' More generally, Dynkin elements are certain Zie elements of generic real hyperplane arrangements, which are indexed by chambers of the corresponding adjoint arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' They were introduced by Aguiar-Mahajan in [AM17, Equation 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Specializing to the braid arrangement, one recovers the type A Dynkin elements DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In [NO19], the following perspective on the Dynkin elements is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Hopf algebra Σ∗ which is dual to Σ is realized as an algebra ˆΣ ∗ of piecewise-constant functions on the braid arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then its dual, in the sense of polyhedral algebras [BP99, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='7], is an algebra ˇΣ ∗ of certain functionals of piecewise-constant functions on the adjoint braid arrangement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' those coming from evaluating on permutohedral cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the morphism of species ˇΣ ∗ → (L∨)∗ 16 WILLIAM NORLEDGE defined by sending functionals to their restrictions to piecewise-constant functions on the complement of the hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since the multiplication of ˇΣ ∗ corresponds to embedding hyperplanes, this morphism is the indecomposable quotient of ˇΣ ∗ [NO19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, in [NO19, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], we see that taking the linear dual of this morphism recovers the Dynkin elements map, L∨ → Σ, S �→ DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (Here we have identified Σ∗ = ˇΣ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=') Therefore we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 ([NO19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The morphism of species L∨ → Zie is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore the Dynkin elements {DS : S is a cell over I} span Zie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann Relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Dynkin elements span Zie, but they are not linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The relations which are satisfied by the Dynkin elements are generated by relations known in physics as the Steinmann relations, introduced in [Ste60a], [Ste60b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let a pair of overlapping channels over I be a pair (S, T), (U, V ) ∈ [I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] of two-lump compositions of I such that S ∩ U ̸= ∅ and T ∩ U ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let S1, S2, S3, S4 be four cells over I with (S, T), (U, V ) ∈ S1, and such that S2, S3, S4 are obtained from S1 by replacing, respectively, (S, T), (U, V ) �→ (T, S), (U, V ) (S, T), (U, V ) �→ (T, S), (V, U) (S, T), (U, V ) �→ (S, T), (V, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, by inspecting the definition of the Dynkin elements (6), we see that14 DS1 − DS2 + DS3 − DS4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In general, a Steinmann relation is any relation between Dynkin elements obtained in this way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' an alternating sum of four Dynkin elements which are obtained from each other by switching overlapping channels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This definition of the Steinmann relations can be found in [EGS75, Seciton 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3] (it is given slightly more generally there for paracells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An alternative characterization of the Steinmann relations in terms of the Lie cobracket of the dual Lie coalgebra Zie∗ is [LNO19, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Here, the Steinmann relations appear in the same way one can arrive at generalized permutohedra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' by insisting on type A ‘factorization’ in the sense of species-theoretic coalgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [NO19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, Dynkin elements satisfy the Steinmann relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Moreover, they are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The relations which are satisfied by the Dynkin elements are generated by the Steinmann relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' That is, if Stein[I] := �DS1 − DS2 + DS3 − DS4 : DS1 − DS2 + DS3 − DS4 = 0 is a Steinmann relation �15 then Zie ∼= L∨ /Stein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This follows by combining [LNO19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3] with [NO19, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ 14 we go through the argument for the basic 4-point case in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1, which is sufficient to exhibit the general phenomenon 15 angled brackets denote C-linear span HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 17 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let us give the basic 4-point example I = {1, 2, 3, 4}, which takes place on a square facet of the type A coroot solid [LNO19, Figure 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Consider the following four cells over I (we have marked where they differ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' the names ‘s-channel’ and ‘u-channel’ are from physics and refer to Mandelstam variables),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' S1 = � (23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 14) � �� � u-channel ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 34),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 234),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (134,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 124)} S2 = �(23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 14),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 12) � �� � s-channel ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 234),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (134,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 124) � S3 = � (14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 23) � �� � u-channel ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 234),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (134,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 124) � S4 = �(14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 34) � �� � s-channel ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 234),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (134,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 124) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The s-channel and the u-channel overlap, and so we should now have DS1 − DS2 + DS3 − DS4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' To see this, let us assume throughout that HF appears in the H-basis expansion (6) of DS1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ¯F ⊆ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then we have (♠) ¯F ⊆ S1 \\ {(12, 34), (23, 14)} =⇒ ¯F ⊆ S1, S2, S3, S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If ¯F ⊈ S1 \\ {(12, 34), (23, 14)}, then either (12, 34) ∈ ¯F or (23, 14) ∈ ¯F but not both, since the channels overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We then have (♥) (12, 34) ∈ ¯F =⇒ ¯F ⊆ S1, ¯F ⊈ S2, ¯F ⊈ S3, ¯F ⊆ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We also have (♦) (23, 14) ∈ ¯F =⇒ ¯F ⊆ S1, ¯F ⊆ S2, ¯F ⊈ S3, ¯F ⊈ S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice that in all three cases (♠), (♥), (♦), the prefactors of HF sum to zero in the four term alternating sum of the Steinmann relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In [NO19], the Steinmann condition is seen to be equivalent to the restriction to generalized permutohedra in a certain local (or spherical) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Ocneanu [Ocn18] and Early [Ear19] have studied an affine version of the Steinmann condition, in the context of higher structures and matroid subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Here, one observes that the (translated) hyperplanes of the adjoint braid arrangement for the Mandelstam variables give three subdivisions of the hypersimplex ∆(2, 4) (octahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 12 34 14 23 24 13 s-channel t-channel u-channel See [BC19], [CGUZ19] for the closely related study of generalized Feynman diagrams in generalized biadjoint Φ3-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 18 WILLIAM NORLEDGE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Ruelle’s Identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since the Dynkin elements span Zie, we can ask what is the description of the Lie bracket of Zie in terms of the Dynkin elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The answer is known in the physics literature as Ruelle’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In order to state Ruelle’s identity, we need to notice the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For S ⊔ T = I, if S1 is a cell over S and S2 is a cell over T, then S1 ⊔ S2 describes a collection of codimension one faces of the adjoint braid arrangement which are supported by the hyperplane orthogonal to λST (in [LNO19], such faces were called Steinmann equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A cell S[S,T] over I which satisfies S[S,T] ⊇ S1 ⊔ S2 and (S, T) ∈ S[S,T] corresponds to a chamber arrived at by moving (by an arbitrarily small amount) from an interior point of a face of S1 ⊔ S2 in the λST direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, such cells always exist, but they are not unique (the Steinmann relations exactly quotient out this ambiguity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The chamber obtained by moving in the opposite direction corresponds to the cell obtained by replacing (S, T) with (T, S) in S[S,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3 (Ruelle’s Identity [Rue61, Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For S ⊔ T = I, let S1 be a cell over S and let S2 be a cell over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let S[S,T] be a cell over I which satisfies S[S,T] ⊇ S1 ⊔ S2 and (S, T) ∈ S[S,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let S[T,S] denote the cell obtained by replacing (S, T) with (T, S) in S[S,T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then the Lie bracket of Zie is given by (7) [DS1, DS2] = DS[S,T ] − DS[T,S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This result is clear from [LNO19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' the Lie bracket which was given to the adjoint braid arrangement realization of Zie (denoted there by Γ) coincides with (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Alternatively, we can just explicitly check, as in [EGS75, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Σ as a Hopf E-Algebra We now recall the Steinmann arrows, which are (or we interpret as) actions of the exponential species E on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We show that they give Σ the structure of a Hopf E-algebra (=Hopf monoid internal to E-modules) in two ways, and thus the primitive part Zie = P(Σ) the structure of a Lie E-algebra in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Derivations and Coderivations of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , yr} be a finite set with cardinality r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We think of Y as having ‘color’ g (physically, the coupling constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a species p, we have the Y -derivative p[Y ] of p, which is the species given by p[Y ][I] := p[Y ⊔ I] and p[Y ][σ] := p[idY ⊔ σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A raising operator u on p is a morphism of species of the form16 u : p → p[Y ], a �→ u(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Moreover, there is an endomorphism algebra of raising operators [Nor20, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], which features when considering modules internal to species, see [Nor20, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 16 for raising operators, we often abbreviate u(a) := uI(a) HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 19 As a particular example of the set Y , we have the set of formal symbols [r] := {∗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , ∗r} (formally, we have picked a section of the decategorification functor Y �→ r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We often abbreviate ∗ = ∗1, also ∗ = {∗} and ∗I = {∗} ⊔ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The derivative p′ of p is the Y -derivative in the singleton case Y = {∗}, thus p′[I] := p[∗][I] = p[∗I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [AM10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], an up operator u on p is a raising operator of the form u : p → p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Writing u∗(a) = u(a) in order to specify the name of the adjoined singleton, we call an up operator commutative if u∗2(u∗1(a)) = u∗1(u∗2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Raising operators can be obtained by iteratively applying commutative up operators, see [Nor20, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [AM10, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], an up operator on an algebra a is called an up derivation if (8) u �µS,T (a ⊗ b) � = µ∗S,T �u(a) ⊗ b � + µS,∗T �a ⊗ u(b) � (it follows that u(1a) = 0 if a is unital) and an up operator on a coalgebra c is called an up coderivation if (9) �u ⊗ id + id ⊗ u � ◦ ∆S,T (a) = ∆∗S,T �u(a) � + ∆S,∗T �u(a) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An up biderivation on a bialgebra h is an up operator which is both an up derivation and an up coderivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The data of an up (co/bi)derivation on a connected species h is equivalent to giving h the structure of an L-(co/Hopf)algebra (= an (co/Hopf)monoid internal to L-modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The data of a commutative up (co/bi)derivation on h is equivalent to giving h the structure of an E-(co/Hopf)algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [Nor20, Section 5] for more details and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, an up derivation u of Σ is a morphism of species u : Σ → Σ′, HF �→ u(HF ) such that u(HF HG) = u(HF )HG + HF u(HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An up derivation of Σ is determined by its values on the elements H(I), I ∈ S, since then u(HF ) = u(H(S1))H(S2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' H(Sk) + · · · + H(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' H(Sk−1)u(H(Sk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An up derivation must have u(H( )) = 0, since 1Σ = H( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An up coderivation u of Σ is a morphism of species u : Σ → Σ′, HF �→ u(HF ) such that ∆∗S,T �u(HF ) � = u(HF|S) ⊗ HF|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, an up coderivation must have ∆∗S,T �u(H(I)) � = u(H(S)) ⊗ H(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore, an up biderivation u of Σ must have u(H(i)) = a1H(∗,i) + a2H(∗i) + a3H(i,∗) where a1 + a2 + a3 = 0 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Motivated by this, given a, b ∈ C, we define an up derivation ua,b of Σ by (10) ua,b : Σ → Σ′, ua,b(H(I)) := −aH(∗,I) + (a + b)H(∗I) − bH(I,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Towards an explicit description, consider the following example for I = {1, 2, 3}, ua,b(H(12,3)) = ua,b(H(12))H(3) + H(12)ua,b(H(3)) = (−aH(∗,12) + (a + b)H(∗12) − bH(12,∗))H(3) + H(12)(−aH(∗,3) + (a + b)H(∗3) − bH(3,∗)) = −aH(∗,12,3) + (a + b)H(∗12,3) − bH(12,∗,3)) − aH(12,∗,3) + (a + b)H(12,∗3) − bH(12,3,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 20 WILLIAM NORLEDGE From this, we see that in general ua,b(HF ) = � 1≤m≤k −aH(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) + (a + b)H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) − bH(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sm,∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a, b ∈ C, the morphism of species Σ → Σ′, HF �→ ua,b(HF ) is an up biderivation of Σ (it follows this gives Σ the structure of a Hopf L-algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the following, for F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk) a composition of I and S ⊆ I, we write (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Uk) := (S1 ∩ S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk ∩ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In general, (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Uk) is a decomposition of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' First, ua,b defines a derivation of Σ by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' To see that ua,b also defines a coderivation, we have ∆∗S,T �ua,b(HF ) � = ∆∗S,T � � 1≤m≤k −aH(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) + (a + b)H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) − bH(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sm,∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) � = � � 1≤m≤k −aH(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Um,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ + (a + b)H(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Um,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ − bH(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Um,∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ � ⊗ HF|T = � � 1≤m≤k Um̸=∅ −aH(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Um,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ + (a + b)H(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Um,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ − bH(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Um,∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ � ⊗ HF|T + � � 1≤m≤k Um=∅ � − a + (a + b) − b � H(U1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Um−1,∗,Um+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Uk)+ � � �� � =0 ⊗ HF|T = u(HF|S) ⊗ HF|T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore ua,b is a biderivation of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann Arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We now recall the Steinmann arrows for Σ, whose precise definition is due to Epstein-Glaser-Stora [EGS75, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='82-83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann arrows were first considered by Steinmann in settings where Σ is represented as operator-valued distributions [Ste60b, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the retarded Steinmann arrow be the up biderivation of Σ given by (11) ∗ ↓ (−) : Σ → Σ′, ∗ ↓ HF := u1,0(HF ) = � 1≤m≤k −H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) + H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the advanced Steinmann arrow be the up biderivation of Σ given by (12) ∗ ↑ (−) : Σ → Σ′, ∗ ↑ HF := u0,1(HF ) = � 1≤m≤k H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) − H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sm,∗,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We use this arrow notation from now on instead of ‘u’ in order to match the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular ∗ ↓ H(I) = −H(∗,I) + H(∗I) and ∗ ↑ H(I) = H(∗I) − H(I,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 21 We have ∗ ↑ HF − ∗ ↓ HF = u−1,1(HF ) = [H(∗), HF ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This identity appears often in the physics literature for operator-valued distributions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Ste60b, Equation 13], [EG73, Equation 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The biderivation u−1,1 gives Σ the structure of a Hopf L-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This L-action is the restriction of the adjoint representation of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice the Steinmann arrows are commutative up operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By [Nor20, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], we can restrict them to obtain up derivations of Zie, ∗ ↓ (−) : Zie → Zie′, DS �→ ∗ ↓ DS and ∗ ↑ (−) : Zie → Zie′, DS �→ ∗ ↑ DS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [Nor20, Section 5], the Steinmann arrows equip Σ with the structure of a Hopf E-algebra (and Zie with the structure of a Lie E-algebra) in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The details are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' First, E is the exponential species, given by E[I] := C for all I ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote HI := 1C ∈ E[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The exponential species is an algebra in species when equipped with the trivial multiplication µS,T : E[S] ⊗ E[T] = C ⊗ C ∼ −→ C = E[I], HS ⊗ HT �→ HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the following E-modules induced by the Steinmann arrows, as defined in [Nor20, Equation 23], E • Σ → Σ, HY ⊗ a �→ Y ↓ a := yr ↓ ◦ · · · ◦ y1 ↓ � �� � invariant of the order (a) and E • Σ → Σ, HY ⊗ a �→ Y ↑ a := yr ↑ ◦ · · · ◦ y1 ↑ � �� � invariant of the order (a) where Y = {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , yr} as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, Y ↓ (−) and Y ↑ (−) are the Steinmann arrow raising operators obtained from iterating the Steinmann arrow up operators ∗ ↓ (−) and ∗ ↓ (−), as mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For example, the retarded arrow Y ↓ (−) consists of a linear map of the form Σ[I] → Σ[Y ⊔ I] for each choice of finite set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For Y = [r] := {∗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , ∗r}, we abbreviate ↓ (−) := ∗ ↓ (−), ↓↓ (−) := {∗1, ∗2} ↓ (−), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' and similarly for the advanced arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since the arrows are derivations, they respect the multiplication of Σ, and since the arrows are coderivations, they respect the comultiplication of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows that these E-actions give Σ the structure of a Hopf monoid constructed internal to E-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By inspecting the definitions, we see that (13) Y ↓ H(I) = R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) := � Y1⊔Y2=Y H(Y1)H(Y2⊔I) and Y ↑ H(I) = A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) := � Y1⊔Y2=Y H(Y1⊔I)H(Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows that Y ↓ HF = � Y1⊔···⊔Yk=Y R(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' R(Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Sk) and Y ↑ HF = � Y1⊔···⊔Yk=Y A(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A(Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 22 WILLIAM NORLEDGE The sums are over all decompositions (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Yk) of Y of length l(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We call R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I), A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) ∈ Σ[Y ⊔ I] the retarded and advanced elements respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The total retarded and total advanced elements are given by Y ↓ H(i) = R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='i) = � Y1⊔Y2=Y H(Y1) H(Y2i) and Y ↑ H(i) = A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='i) = � Y1⊔Y2=Y H(Y2i) H(Y1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we put I = J ⊔ {i}, then we have R(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='i) = � S⊔T=I i∈T H(S) H(T) = − � F∈Σ[I] i∈Sk (−1)l(F)HF = Di and A(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='i) = � S⊔T=I i∈T H(T) H(S) = − � F∈Σ[I] i∈S1 (−1)l(F)HF = D¯i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Currying the Steinmann Arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a species p, we let pE denote the species given by pE[I] := ∞ � r=0 �p[r][I] �Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Here, p[r] is the Y -derivative of p for Y = [r], and (−)Sr denotes the subspace of Sr-invariants, where Sr is the symmetric group on [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote elements of pE[I] using formal power series notation ∞ � r=0 xr, xr ∈ p[r][I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Explicitly, xr is an element of the vector space p[{∗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , ∗r} ⊔ I] which is invariant under the action of permuting {∗1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , ∗r} and leaving I fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The mapping p �→ pE extends to an endofunctor on species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, given a morphism of species η : p → q, we have the morphism ηE given by (14) ηE : pE → qE, ∞ � r=0 xr �→ ∞ � r=0 η[r]⊔I(xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A series of a species p is a morphism of species of the form s : E → p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice the elements of pE[I] are naturally series of the species Y �→ p[Y ][I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [Nor20, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For the connection between pE and the internal hom for the Cauchy product, see [Nor20, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If a is an algebra in species, then so is aE, see [Nor20, Equation 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, ΣE is an algebra, with multiplication given by ∞ � r=0 xr ⊗ ∞ � r=0 yr �→ ∞ � r=0 � r1+r2=r r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' r1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' r2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='µ[r1]⊔S,[r2]⊔T (xr1 ⊗ yr2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the following homomorphisms of algebras in species, Σ → ΣE, HF �→ ∞ � r=0 � Y1⊔···⊔Yk=[r] R(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' R(Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Sk) and Σ → ΣE, HF �→ ∞ � r=0 � Y1⊔···⊔Yk=[r] A(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A(Yk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 23 1 2 3 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Schematic for the action of the retarded Steinmann arrow ∗ ↓ for I = {1, 2, 3} on the Steinmann sphere (left) and the tropical geometric realization of Σ (right, see [NO19, Introduction]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann arrows are commutative up biderivations of Σ, and so give Σ the structure of a Hopf E-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This result is then a special case of [Nor20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ The homomorphisms of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2 are the unique extensions of the maps H(I) �→ ∞ � r=0 R(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) and H(I) �→ ∞ � r=0 A(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) to homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the application to causal perturbation theory, we shall be interested in the decorated analog of these homomorphisms, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' These homomorphisms Σ → ΣE come from currying the E-actions of the Steinmann arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [Nor20, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann Arrows and Dynkin Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We now show that the restriction of the Steinmann arrows to Zie, which are derivations for its Lie bracket, have an interesting description in terms of cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' chambers of the adjoint braid arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [Eps16, Section 2], we define the commutative up operators ∗ ↓ (−) : L∨ → L∨′, ∗ ↓ S := �(∗S, T), (S, ∗T), (I, ∗) : (S, T) ∈ S � and ∗ ↑ (−) : L∨ → L∨′, ∗ ↑ S := �(∗S, T), (S, ∗T), (∗, I) : (S, T) ∈ S �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' These are indeed well-defined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ∗ ↓ S corresponds to the adjoint braid arrangement chamber on the I side of the hyperplane λ∗,I = 0 which has the face of S as a facet, and ∗ ↑ S corresponds to the chamber on the ∗ side of the hyperplane λ∗,I = 0 which has the face of S as a facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See around [LNO19, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, it follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3 (Ruelle’s identity) that [H(∗), DS] = D∗↑S − D∗↓S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The induced E-modules are given by E • L∨ → L∨, HY ⊗ S �→ Y ↓ S := �(Y1 ⊔ S, Y2 ⊔ T) ∈ [Y ⊔ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] : (S, T) ∈ S or S = I � and E • L∨ → L∨, HY ⊗ S �→ Y ↑ S := �(Y1 ⊔ S, Y2 ⊔ T) ∈ [Y ⊔ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2] : (S, T) ∈ S or T = I �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24 WILLIAM NORLEDGE Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a cell S over I, we have Y ↓ DS = DY ↓S and Y ↑ DS = DY ↑S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We consider the retarded case Y ↓ DS = DY ↓S only, since the advanced case then follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It is sufficient to consider the case Y = {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have ↓ DS = − � ¯F⊆S (−1)l(F) ↓ HF and D↓S = − � ¯F⊆ ↓S (−1)l(F) HF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' So, the result follows if we have the following equality � ¯F⊆S (−1)l(F) � 1≤m≤k −H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗,Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) + H(S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',∗Sm,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Sk) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='= � ¯G⊆ ↓S (−1)l(G) HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Indeed, notice that the H-basis elements HG ∈ Σ[∗I] which appear on the LHS are exactly those such that ¯G ⊆↓ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice also that each HG appears with total sign (−1)l(G), since when ∗ is inserted as a singleton lump, thus increasing l(G) by one, it appears also with a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This interpretation of the E-module structure of Σ restricted to the primitive part Zie = P(Σ) in terms of the adjoint braid arrangement suggests obvious generalizations of the Steinmann arrows in the direction of [AM17], [AM20], since the generalization of Hopf monoids there is via hyperplane arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the following homomorphisms of Lie algebras in species, Zie → ZieE, DS �→ D(−)↓S = ∞ � r=0 D[r]↓S = DS + D↓S + D↓↓S + · · · and Zie → ZieE, DS �→ D(−)↑S = ∞ � r=0 D[r]↑S = DS + D↑S + D↑↑S + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Steinmann arrows are commutative up biderivations of Zie, and so give Zie the structure of a Lie E-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This result is then a special case of [Nor20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Products and Series We now recall several basic constructions of casual perturbation theory in the current, clean, abstract setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We do this without yet imposing causal factorization/causal additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We say e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ‘T-product’ and ‘R-product’ for now, and then change to ‘time-ordered product’ and ‘retarded product’ in the presence of causal factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T-Products, Generalized T-Products, and Generalized R-Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let V be a vector space over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let A be a C-algebra with multiplication denoted by ⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let UA be the algebra in species given by UA[I] := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The action of bijections is trivial, and the multiplication is the multiplication of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The positive exponential species E∗ + is given by E∗ +[I] := C if I ̸= ∅ and E∗ +[∅] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 25 Let a system of T-products T be a system of products for the positive exponential species E∗ +, as defined in [Nor20, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This means T is a morphism of species of the form17 T : E∗ + ⊗ EV → UA, H(I) ⊗ AI �→ TI(H(I) ⊗ AI) where recall E∗ + ⊗ EV is the Hadamard product of species, given by E∗ + ⊗ EV [I] := E∗ +[I] ⊗ EV [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, if I ̸= 0, we have E∗ + ⊗ EV [I] ∼= V ⊗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We abbreviate (15) TI(AI) := TI(H(I) ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let H(EV , UA) denote the species of linear maps between components, given by H(EV , UA)[I] := HomVec �EV [I], UA[I]) = HomVec �V ⊗I, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have that H(−, −) is the hom for the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore we can curry T to give the morphism of species E∗ + → H(EV , UA), H(I) �→ T(I) where T(I) is the linear map T(I) : V ⊗I → A, AI �→ TI(AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The linear maps T(I) are called T-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Notice that T-products are commutative in the sense that TI �EV [σ](AI) � = TI(AI) for all bijections σ : I → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This property holds because the system T is a morphism of species, and bijections act trivially for UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This commutativity exists despite the fact that the algebra A is noncommutative in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In applications to QFT, we shall also have a causal structure on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then T is meant to first order the vectors of AI according to the causal structure, and then multiply in A, giving rise to this commutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the system of generalized T-products associated to a system of T-products be the unique extension to a system of products for Σ = L ◦ E∗ + which is a homomorphism, as defined in [Nor20, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus T : Σ ⊗ EV → UA, HF ⊗ AI �→ TI(HF ⊗ AI) := TS1(AS1) ⋆ · · · ⋆ TSk(ASk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The currying of T is denoted by Σ → H(EV , UA), HF �→ T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The linear maps T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk) : V ⊗I → A, AI �→ TI(HF ⊗ AI) are called generalized T-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let the system of generalized R-products associated to a system of T-products be the restriction to the Lie algebra of primitive elements Zie, R : Zie ⊗ EV → UA, DS ⊗ AI �→ RI(DS ⊗ AI) := TI(DS ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 17 recall the definition and notation for EV from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3 26 WILLIAM NORLEDGE This is a morphism of Lie algebras, where UA is equipped with the commutator bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The currying of R is denoted by Zie → H(EV , UA), DS �→ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The linear maps RS : EV [I] → A, AI �→ RI(DS ⊗ AI) are called generalized R-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' From the expansion (6) of Dynkin elements DS in terms of the H-basis, we recover [EG73, Equation 79], RS = − � FF ⊆ ¯S (−1)k T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Consider a system of products of the form Z : E∗ + ⊗ EV → UV , H(I) ⊗ AI �→ ZI(AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then we obtain a new T-product T′, given by T′ : E∗ + ⊗ EV → UA, T′ I(AI) := � P TP �ZS1(AS1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ZSk(ASk) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The sum is over all partitions P = {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk} of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This construction underlies renormalization in pAQFT [Düt19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2], which deals with the remaining ambiguity of T-products after imposing causal factorization, and perhaps other renormalization conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Reverse T-Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The system of reverse generalized T-products T of a system of generalized T-products is given by precomposing T with the antipode (5) of Σ ⊗ EV , thus T : Σ ⊗ EV → UAop, TI(HF ⊗ AI) := TI �HF ⊗ AI �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since the antipode is a homomorphism Σ ⊗ EV → (Σ ⊗ EV )op,cop [AM10, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='22 (iii)], this is a system of generalized T-products into the opposite algebra UAop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The image of H(I) under the currying of T is called the reverse T-product T(I) : EV [I] → Aop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' From (2), we obtain T(I) = � F∈Σ[I] (−1)k T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Note that reverse T-products in [EG73, Equation 11] are defined to be (−1)n T(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Our definition agrees with [Sch20, Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T-Exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For details on series in species, see [AM10, Section 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The (scaled) universal series G(c) is the group-like series of Σ given by G(c) : E → Σ, HI �→ G(c)I := cn H(I) for c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The fundamental nature of this series is described in [AM13, Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The series s ◦ G(c) which is the composition of G(c) with the antipode s of Σ is given by (16) s ◦ G(c) : E → Σ, HI �→ �s ◦ G(c) � I = cn H(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let A[[j ]] denote the C-algebra of formal power series in the formal symbol j with coefficients in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a system of generalized T-products T : Σ ⊗ EV → UA HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 27 let the T-exponential S := SG(c) of this system be the A[[j ]]-valued function on the vector space V associated to the series G(c), as constructed in [Nor20, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, we have18 (17) S : V → A[[j ]], A �→ S(jA) = ∞ � n=0 cn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Tn �jA ⊗ · · · ⊗ jA � � �� � n times := ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tn(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By [Nor20, Equation 34] and (16), the T-exponential for the system of reverse T-products is the inverse of S as an element of the C-algebra of functions Func(V, A[[j ]]), given by S−1 : V → A[[j ]], A �→ S−1(jA) := ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tn(An) = ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tn(H(n) ⊗ An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore S(jA) ⋆ S−1(jA) = S−1(jA) ⋆ S(jA) = 1A for all A ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This appears in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [EG73, Equation 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbation of T-Products For the perturbation of T-products by a certain up coderivation of E which gives the S-matrix scheme SgS(jA) = S(gS + jA), see [Nor20, Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbation of T-Products by Steinmann Arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Suppose we have a system of generalized T-products T : Σ ⊗ EV → UA, HF ⊗ AI �→ TI(HF ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [Nor20, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], given a choice of decorations vector S ∈ V , we can use the retarded Steinmann arrow (11) to perturb T as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Recall the decorated Hopf algebra Σ ⊗ EV from Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Recall also the derivative (Σ ⊗ EV )′ of Σ ⊗ EV from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1, given by (Σ ⊗ EV )′[I] = Σ[∗I] ⊗ V ⊗ V ⊗I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the up derivation of Σ ⊗ EV which is the decorated analog of the retarded Steinmann arrow, given by Σ ⊗ EV → (Σ ⊗ EV )′, HF ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain �→ ∗ ↓ HF ⊗ S ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is indeed still an up derivation by [Nor20, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Analogous to the setting without decorations, we have the induced raising operators and associated E-action by iterating, which, after currying, give us the homomorphism Σ ⊗ EV → (Σ ⊗ EV )E HF ⊗ Ai1 ⊗ · · · ⊗ ⊗Ain �→ ∞ � r=0 ↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ↓ � �� � r times HF ⊗ S ⊗ · · · ⊗ S � �� � r times ⊗Ai1 ⊗ · · · ⊗ ⊗Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is a homomorphism by [Nor20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, a new ‘perturbed’ system of generalized T-products is given by composing this homomorphism with TE (defined in (14)), �T : Σ ⊗ EV → (Σ ⊗ EV )E TE −−→ (UA)E ∼= UA[[g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For the result that (UA)E ∼= UA[[g]], see [Nor20, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 18 we use the abbreviations (4) and (15), and also Tn := T[n] 28 WILLIAM NORLEDGE Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The fact �T is still a homomorphism, and is thus still a generalized system of products, depends crucially on the fact the Steinmann arrow is a derivation [Nor20, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1], and that (−)E is a monoidal functor [Nor20, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can similarly perturb a system of generalized R-products, which uses the fact the Steinmann arrow is a biderivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We now unpack all this formalism to give a fully explicit description of the new perturbed system of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let us abbreviate SY AI = Sy1 ⊗ · · · ⊗ Syr ⊗ Ai1 ⊗ · · · ⊗ Ain ∈ EV [Y ⊔ I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let (18) RY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(SY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) := TY ⊔I(R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) ⊗ SY AI) = � Y1⊔Y2=Y TY1⊔∅(SY1) ⋆ TY2⊔I(SY2AI) � �� � by (13) Then the new perturbed system is given by19 �T : Σ ⊗ EV → UA[[g]], HF ⊗ AI �→ ∞ � r=0 � r1+ ··· +rk=r gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='S1(S r1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AS1) ⋆ · · · ⋆ Rrk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Sk(S rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' ASk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, the restriction to E∗ + ⊗ EV , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' the new perturbed T-product, is given by �TI(AI) = ∞ � r=0 gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) = TI(AI) + g T∗1I(↓ H(I) ⊗ SAI) + g2 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T∗2∗1I(↓↓ H(I) ⊗ SSAI) + · · · � �� � perturbation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Similar, we can perturb a system of generalized T-products using the advanced Steinmann arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We let VgS, respectively WgS, denote the T-exponential (as defined in (17)) for the new perturbed system of generalized T-products using the retarded, respectively advanced, Steinmann arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus VgS : V → A[[g,j ]], VgS(jA) := ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �Tn(An) = ∞ � n=0 ∞ � r=0 grj ncr+n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='n(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An) and WgS : V → A[[g,j ]], WgS(jA) := ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �Tn(An) = ∞ � n=0 ∞ � r=0 grj ncr+n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Ar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='n(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An) where AY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(SY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) := TY ⊔I(A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I) ⊗ SY AI) = � Y1⊔Y2=Y TY1⊔I(SY1AI) ⋆ TY2⊔∅(SY2) � �� � by (13) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have VgS(jA) = S−1(gS) ⋆ S(gS + jA) and WgS(jA) = S(gS + jA) ⋆ S−1(gS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 19 we abbreviate Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) := R[r];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S [r];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) = R[r];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S ⊗ · · · ⊗ S � �� � r times ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) = � Y1⊔Y2=[r] TY1⊔∅(SY1) ⋆ TY2⊔I(SY2AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then VgS(jA) = ∞ � n=0 ∞ � r=0 grj ncr+n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='n(S r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An) = ∞ � n=0 ∞ � r=0 grj ncr+n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' � Y1⊔Y2=[r] TY1⊔∅(SY1) ⋆ TY2⊔[n](SY2An) = ∞ � r=0 grcr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tr+0(S r) ⋆ ∞ � n=0 ∞ � r=0 cn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='Tr+n(S rAn) = S−1(gS) ⋆ S(gS + jA) The proof for WgS(jA) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 (Bogoliubov’s Formula [BS59, Chapter 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have (19) �Ti(A) = 1 c d dj ��� j =0VgS(jA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have d dj VgS(jA) = d dj ∞ � n=0 j ncn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �Tn(An) = ∞ � n=1 j n−1cn (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �Tn(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, putting j = 0, we obtain d dj ��� j =0VgS(jA) = c �T1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ This formula was originally motivated by the path integral heuristic, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Sch20, Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' R-Products and A-Products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The linear maps R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I) which are given by R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I) : E[Y ] V [I] → A, SY AI �→ RY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(SY AI) are called R-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the case of singletons I = {i}, the maps R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' i) are called total R-products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By (13), R-products are given in terms of T-products and reverse T-products by R(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I) = � Y1⊔Y2=Y T(Y1) ⋆ T(Y2 ⊔ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then �T(I) = ∞ � r=0 cr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' R(r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In a similar way, we can define the A-products A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I), so that A(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' I) = � Y1⊔Y2=Y T(Y1 ⊔ I) ⋆ T(Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The total R-products are both R-products and generalized R-products, which is due to the double description appearing in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A related result is [AM13, Proposition 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 30 WILLIAM NORLEDGE Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In the literature, the total retarded products in our sense are sometimes called retarded products, and the retarded products in our sense are then called generalized retarded products, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Pol58], [Düt19, Exercise 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Perturbative Algebraic Quantum Field Theory We now apply the theory we have developed to the case of a real scalar quantum field on a Minkowski spacetime, as described by pAQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='20 Mathematically, the important extra property is a causal structure on the vector space of decorations V , which allows one to impose causal factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Connections between QFT and species have been previously studied in [Abd04], [Far11], [GK18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Our references for pAQFT are [DF01], [Rej16], [Düt19], [Sch20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We mainly adopt the notation and presentation of [Sch20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Key features of pAQFT are its local, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' sheaf-theoretic, approach, the (closely related) use of adiabatic switching of interaction terms to avoid IR-divergences, and the interpretation of renormalization as the extension of distributions to the fat diagonal to avoid UV-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The Wilsonian cutoff, sometimes called heuristic quantum field theory, may be rigorously formulated within pAQFT [BDF09], [Düt12], [Düt19, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='8], [Sch20, Section 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Spacetime and Field Configurations Let X ∼= R1,p denote a (p + 1)-dimensional Minkowski spacetime, for p ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, X is a real vector space equipped with a metric tensor which is a symmetric nondegenerate bilinear form X × X → R with signature (1, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The bilinear form gives rise to a volume form on X, which we denote by dvolX ∈ Ωp+1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For regions of spacetime X1, X2 ⊂ X, we write X1∨∧X2 if one cannot travel from X1 to X2 on a future-directed timelike or lightlike curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the set valued species X (−) given by I �→ X I := �functions I → X �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For simplicity, we restrict ourselves to the Klein-Gordan real scalar field on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore, let E → X be a smooth real vector bundle over X with one-dimensional fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An (off-shell) field configuration Φ is a smooth section of the bundle E → X, Φ : X �→ E, x �→ Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The space of all field configurations, denoted Γ(E), has the structure of a Fréchet topological (real) vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can always pick an isomorphism (E → X) ∼= (X × R → X), which induces an isomorphism Γ(E) ∼= C∞(X, R), so that field configurations are modeled as smooth functions X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let E∗ → X denote the dual vector bundle of E, and let the canonical pairing be denoted by ⟨−, −⟩ : E∗ ⊗ E → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let a compactly supported distributional section α be a distribution of field configurations α : Γ(E) → R, 20 although pAQFT deals more generally with perturbative Yang-Mills gauge theory on curved spacetimes HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 31 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' an element of the topological dual vector space of Γ(E), which is modeled as a sequence (αj)j∈N of smooth compactly supported sections of the dual bundle E∗ → X, αj : X �→ E∗, j ∈ N, where the modeled distribution is recovered as the following limit of integrals, Γ(E) → R, Φ �→ � x∈X �α(x), Φ(x) �dvolX := lim j→∞ � x∈X ⟨αj(x), Φ(x)⟩dvolX � �� � sometimes called generalized function notation .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The space of all compactly supported distributional sections is denoted Γ′ cp(E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Bär15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='15], all distributions Γ(E) → R may be obtained as compactly supported distributional sections in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can pullback the vector bundle E∗ to X I along each canonical projection X I → X {i} ∼= X, i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The tensor product of these n many pullback bundles is the exterior tensor product bundle (E∗)⊠I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This defines a presheaf of smooth vector bundles on S, Sop → Diff/X , I �→ (E∗)⊠I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' By taking compexified compactly supported distributional sections Γ′C cp(−) := Γ′ cp(−) ⊗R C, we obtain the complex vector species Γ′C cp(E∗), given by Γ′C cp(E∗)[I] := Γ′C cp � (E∗)⊠I� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Of course, Γ′C cp(E∗) does not ‘factorize’ in the sense that it is not a monoidal functor, (20) Γ′C cp(E∗)[I] ≇ Γ′C cp(E∗)[i1] ⊗ · · · ⊗ Γ′C cp(E∗)[in] where I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , in}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' There are more distributional sections then just those coming from the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Observables An off-shell observable O is a smooth functional of field configurations into the complex numbers, O : Γ(E) → C, Φ �→ O(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The space of all observables is denoted Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We can pointwise multiply observables, sometimes called the normal ordered product, so that observables form a commutative C-algebra, Obs ⊗ Obs → Obs, O1 ⊗ O2 �→ O1 · O2 where O1 · O2(Φ) := O1(Φ)O2(Φ) � �� � multiplication in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, we may form the commutative algebra in species UObs, given by UObs[I] = Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A linear observable O ∈ Obs is an observable which is additionally a linear functional, that is O(Φ1 + Φ2) = O(Φ1) + O(Φ2) and O(cΦ) = cO(Φ) for c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The space of linear observables is denoted LinObs ⊂ Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In particular, for each spacetime event x ∈ X, we have the field observable Φ(x) ∈ LinObs, given by Φ(x) : Γ(E) → C, Φ �→ Φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 32 WILLIAM NORLEDGE We now show how linear observables and so-called polynomial observables arise species-theoretically, via (generalized) systems of products for the species E and X = P(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let X denote the species given by X �{i} � := C for singletons and X[I] := 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We denote Hi := 1 ∈ X �{i} �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have the following morphism of species, X ⊗ Γ′C cp(E∗) → UObs, Hi ⊗ α �→ � Φ �→ � x∈X �α(x), Φ(x) �dvolX � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is like a system of products for X, however Γ′C cp(E∗) does not factorize (20), and so cannot be written in the form EV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows from [Bär15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='15] that the colimit (as defined in [AM10, Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='7]) of the species which is the image of this morphism is the space of linear observables LinObs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The currying of this map is given by X → H �Γ′C cp(E∗), UObs �, Hi �→ Φi = Φ where Φ(α) := � Φ �→ � x∈X �α(x), Φ(x) �dvolX � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we restrict Φ to bump functions b ∈ Γcp(E∗) ⊗R C, also called ‘smearing functions’, then one might call the linear map Φ : Γcp(E∗) ⊗R C → Obs, b �→ Φ(b) an ‘observable-valued distribution’, and this is sometimes referred to as ‘the (smeared) field’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The field observable Φ(x) is recovered by evaluating Φ on the Dirac delta function δx localized at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' One views b as the smearing of a Dirac delta function, hence smearing functions and smeared field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We extend the smeared field by replacing X with E to define the following morphism of species, E ⊗ Γ′C cp(E∗) → UObs, HI ⊗ αI �→ � Φ �→ � X I �αI(xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin), Φ(xi1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Φ(xin) �dvolX I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is like a system of products for E, but again without factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The colimit of the species which is the image of this morphism is the vector space of polynomial observables, as defined in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [Sch20, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='13], denoted PolyObs ⊂ Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (Alternatively, if we restrict the limit of this map S (Γ′C cp(E∗)) → Obs[[j ]] to finite series and set j = 1, then we recover [Düt19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=') The space of microcausal polynomial observables F is the subspace F ⊂ PolyObs consisting of those polynomial observables which satisfy a certain microlocal-theoretic condition called microcausality, see [Düt19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 (ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [Düt19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4], the space of local observables Floc ⊂ Obs consists of those observables obtained by integrating a polynomial with real coefficients in the field and its derivatives (‘field polynomials’) against a bump function b ∈ Γcp(E∗) ⊗R C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Importantly, we have a natural inclusion Floc �→ F, A �→ : A : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Floc[[ℏ]] and F[[ℏ]] denote the spaces of formal power series in ℏ with coefficients in Floc and F respectively, and let F((ℏ)) denote the space of Laurent series in ℏ with coefficients in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 33 Applying Moyal deformation quantization with formal Planck’s constant ℏ, F[[ℏ]] is a formal power series ∗-algebra, called the (abstract, off-shell) Wick algebra, with multiplication the Moyal star product [Düt19, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1] defined with respect to the Wightman propagator ∆H for the Klein-Gordan field [Düt19, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2], F[[ℏ]] ⊗ F[[ℏ]] → F[[ℏ]], O1 ⊗ O2 �→ O1 ⋆H O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We may form the algebra in species UF[[ℏ]], or, allowing negative powers of ℏ, UF((ℏ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Time-Ordered Products and S-Matrix Schemes For A ∈ Floc[[ℏ]], let supp(A) denote the spacetime support of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a composition G of I, we say that AI ∈ EFloc[[ℏ]][I] respects G if supp(Ai1) ∨∧ supp(Ai2) for all (i1, i2) such that G|{i1,i2} = (i1, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='21 Consider a system of T-products (as defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1) of the form T : E∗ + ⊗ EFloc[[ℏ]] → UF((ℏ)), H(I) ⊗ AI �→ TI(H(I) ⊗ AI) = TI(AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since Σ is the free algebra on E∗ +, we have the unique extension to a system of generalized T-products T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)), TI(HF ⊗ AI) := TS1(AS1) ⋆H · · · ⋆H TSk(ASk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (perturbation) we say that T satisfies perturbation if the singleton components Ti are isomorphic to the inclusion Floc[[ℏ]] �→ F((ℏ)), that is Ti(A) = :A: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' (causal factorization) we say that T satisfies causal factorization if for all compositions (S, T) of I with two lumps, if AI ∈ EFloc[[ℏ]][I] respects (S, T)22 then (21) TI(H(I) ⊗ AI) = TI(H(S,T) ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='23 Let a (fully normalized) system of time-ordered products be a system of T-products which satisfies perturbation and causal factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The corresponding unique extension of T to Σ is called the associated system of generalized time-ordered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' After currying Σ → H(EFloc[[ℏ]], UF((ℏ))), HF �→ T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk) the linear maps T(S1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T(Sk) : Floc[[ℏ]]⊗I → F((ℏ)), AI �→ TI(HF ⊗ AI) are called generalized time-ordered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The linear maps T(I) are called time-ordered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' After fixing a field polynomial, so that each Aij of AI is determined by a bump function bij, they are usually presented in generalized function notation as follows, TI(Ai1 ⊗ · · · ⊗ Ain) = � X I T(xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin)bi1(xi1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' bin(xi1)dxi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' dxin where (xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin) �→ T(xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , xin) is an ‘operator-valued’ generalized function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [EG73, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 21 G|{i1,i2} = (i1, i2) means that i1 and i2 are in different lumps, with the lump containing i1 appearing to the left of the lump containing i2 22 explicitly, supp(Ai1) ∨∧ supp(Ai2) for all i1 ∈ S and i2 ∈ T 23 or equivalently TI(AI) = TS(AS) ⋆H TT (AT ) 34 WILLIAM NORLEDGE Given compositions F = (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Sk) and G = (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Ul) of I, let HF ▷ HG := H(U1∩S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Ul∩S1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='.,U1∩Sk,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=',Ul∩Sk)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is called the Tits product, going back to Tits [Tit74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See [AM13, Section 13] for more on the structure of the Tits product, where it is shown it is given by the action of Σ on itself by Hopf powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' See also [AB08, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='6] for the context of other Coxeter systems and Dynkin types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let T : E∗ + ⊗ EFloc[[ℏ]] → UF((ℏ)) be a system of T-products which satisfies causal factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a composition G = (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' , Uk) of I, and AI ∈ EFloc[[ℏ]][I] which respects G, then TI(a ⊗ AI) = TI(a ▷ HG ⊗ AI) for all a ∈ Σ[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have TI(HG ⊗ AI) = TU1(AU1) ⋆H · · · ⋆H TUk(AUk) = TI(AI) � �� � by repeated applications of causal factorization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Observe that the action HF �→ HF ▷ HG, for F ∈ Σ[I], replaces the lumps of F with their intersections with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' But we just saw that TI(AI) = TI(HG ⊗ AI), and so it follows that TI(HF ⊗ AI) = TI(HF ▷ HG ⊗ AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Since the claim is true for the H-basis, it is true for all a ∈ Σ[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If a ▷ HG = 0, then TI(a ⊗ AI) = 0 for all AI ∈ EFloc[[ℏ]][I] which respect G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The restriction of T to the primitive part Lie algebra is called the associated system of generalized retarded products, R : Zie ⊗ EFloc[[ℏ]] → UF((ℏ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The image of the Dynkin elements DS under the currying of R are the generalized retarded products RS, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' [EG73, Equation 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' It follows from Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1 and the structure of Dynkin elements under the Tits product that generalized retarded products have nice support properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' This is described in [EGS75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Given a system of generalized time-ordered products T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)) the T-exponential S = SG(1/iℏ) (defined in (17)) for the group-like series G(1/iℏ) : E → Σ, HI �→ 1 iℏH(I) is called the associated perturbative S-matrix scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, S is the function S : Floc[[ℏ]] → F((ℏ))[[j ]], A �→ S(jA) := ∞ � n=0 � 1 iℏ �n j n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Tn(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 35 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Interactions Given a choice of adiabatically switched interaction Sint ∈ Floc[[ℏ]], and a system of fully normalized generalized time-ordered products T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ)), we have the new system of interacting generalized time-ordered products which is obtained by the construction of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1, �T : Σ ⊗ EFloc[[ℏ]] → UF((ℏ))[[g]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The associated generating function scheme ZgSint for interacting field observables, and more generally for time-ordered products of interacting field observables, is the new T-exponential for the group-like series G(1/iℏ), denoted VgSint in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Thus, ZgSint is the function ZgSint : Floc[[ℏ]] → F((ℏ))[[g,j ]], A �→ ZgSint(jA) where ZgSint(jA):= ∞ � n=0 � 1 iℏ �nj n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' �Tn(An) = ∞ � n=0 ∞ � r=0 � 1 iℏ �r+ngrj n r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='n(S r int;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' An) = S−1(gSint)⋆H S(gSint +jA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then Aint := �Ti(A) = ∞ � r=0 � 1 iℏ �r gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr+1(S r int;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' A) ∈ F((ℏ))[[g]] is the local interacting field observable of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Bogoliubov’s formula (19) now reads Aint = iℏ d dj ��� j =0ZgSint(jA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' One views Aint as the deformation of the local observable A due to the interaction Sint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' One can show that �T does indeed land in UF[[ℏ,g]] [DF01, Proposition 2 (ii)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' The perturbative interacting quantum field theory then has a classical limit [Col16], [HR20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Scattering Amplitudes We finish with a translation of a standard result in pAQFT (see [Sch20, Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='12]) into our notation, which relates S-matrix schemes as presented in Section 7 to S-matrices used to compute scattering amplitudes, which are predictions of pAQFT that are tested with scattering experiments at particle accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Following [Düt19, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='2], the Hadamard vacuum state ⟨−⟩0 is the linear map given by ⟨−⟩0 : F[[ℏ, g]] → C[[ℏ, g]], O �→ ⟨O ⟩0 := O (Φ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Let Sint ∈ Floc[[ℏ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We say that the Hadamard vacuum state ⟨−⟩0 is stable with respect to the interaction Sint if for all O ∈ F[[ℏ, g]], we have (22) �O ⋆H S(gSint) � 0 = �O � 0 �S(gSint) � 0 and �S−1(gSint) ⋆H O � 0 = 1 �S(gSint) � 0 �O � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In situations where Sint ⊗ AI ∈ E′ Floc[[ℏ]][I] respects the composition (S, ∗, T) 36 WILLIAM NORLEDGE we can interpret free particles/wave packets labeled by T coming in from the far past, interacting in a compact region according to the adiabatically switched interaction Sint, and then emerging into the far future, labeled by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For AI ∈ EFloc[[ℏ]][I], let GI(AI) := ��T(AI) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If we fix the field polynomial of local observables to be P(Φ) = Φ, then AI �→ GI(AI) is the time-ordered n-point correlation function, or Green’s function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' They are usually presented in generalized function notation as follows, GI(bi1 ⊗ · · · ⊗ bin) = � X I � T �Φ(xi1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Φ(xin) �� 0bi1(xi1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' bin(xin)dxi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' dxin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Note that to obtain the realistic Green’s functions, we still have to take the adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' If the Hadamard vacuum state ⟨−⟩0 is stable with respect to Sint ∈ Floc[[ℏ]], and if Sint ⊗ AI ∈ E′ Floc[[ℏ]][I] respects the composition (S, ∗, T), then GI(AI) = 1 � S(gSint) � 0 � TS(AS) ⋆H S(gSint) ⋆H TT (AT ) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' We have GI(AI) = ��T(AI) � 0 = � ∞ � r=0 gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Rr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='I(S r int;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' AI) � 0 = � ∞ � r=0 � r1+r2=r gr r1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' r2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='T[r1]⊔∅(S r1 int) ⋆H T[r2]⊔I(S r2 intAI) � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' To obtain the final line, we expanded the retarded products according to (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Then, by causal factorization (21), we have T[r2]⊔I(S r2 intAI) = TS(AS) ⋆H T[r2]⊔∅(S r2 int) ⋆H TT (AT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Therefore GI(AI) = � ∞ � r=0 � r1+r2=r gr r1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' r2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='T[r1]⊔∅(S r1 int) ⋆H TS(AS) ⋆H T[r2]⊔∅(S r2 int) ⋆H TT (AT ) � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' = � ∞ � r=0 gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T[r]⊔∅(S r int) ⋆H TS(AS) ⋆H ∞ � r=0 gr r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' T[r]⊔∅(S r int) ⋆H TT (AT ) � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' = � S−1(gSint) ⋆H TS(AS) ⋆H S(gSint) ⋆H TT (AT ) � 0 = 1 � S(gSint) � 0 � TS(AS) ⋆H S(gSint) ⋆H TT (AT ) � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' For the final step, we used vacuum stability (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' □ 24 the element S(gSint) ∈ F((ℏ))[[g]] is called the perturbative S-matrix HOPF MONOIDS IN PERTURBATIVE ALGEBRAIC QUANTUM FIELD THEORY 37 References [AB08] Peter Abramenko and Kenneth S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Buildings, volume 248 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Springer, New York, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Theory and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 34 [Abd04] Abdelmalek Abdesselam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Feynman diagrams in algebraic combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Sém.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' With forewords by Kenneth Brown, Stephen Chase and André Joyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2, 3, 4, 8, 10, 11, 12, 14, 19, 26, 32 [AM13] Marcelo Aguiar and Swapneel Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Hopf monoids in the category of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In Hopf algebras and tensor categories, volume 585 of Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=', pages 17–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=', Providence, RI, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2, 3, 11, 15, 26, 29, 34 [AM17] Marcelo Aguiar and Swapneel Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Topics in hyperplane arrangements, volume 226 of Mathematical Surveys and Monographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' American Mathematical Society, Providence, RI, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 3, 5, 14, 15, 24 [AM20] Marcelo Aguiar and Swapneel Mahajan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Bimonoids for Hyperplane Arrangements, volume 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Cambridge University Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 24 [Ara61] Huzihiro Araki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Generalized retarded functions and analytic function in momentum space in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Journal of Mathematical Physics, 2(2):163–177, 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 6 [Bar78] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Barratt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Twisted Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' In Geometric applications of homotopy theory (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=', Evanston, Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=', 1977), II, volume 658 of Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=', pages 9–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Springer, Berlin, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' 2 [Bär15] Christian Bär.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Green-hyperbolic operators on globally hyperbolic spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content=' Phys.' metadata={'source': 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+page_content=' 3, 34 Pennsylvania State University Email address: wxn39@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf'} diff --git a/q9AzT4oBgHgl3EQfA_op/content/tmp_files/2301.00934v1.pdf.txt b/q9AzT4oBgHgl3EQfA_op/content/tmp_files/2301.00934v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..caef9416102d17b36f138af37952913c2012082e --- /dev/null +++ b/q9AzT4oBgHgl3EQfA_op/content/tmp_files/2301.00934v1.pdf.txt @@ -0,0 +1,1155 @@ +Finding the Most Transferable Tasks for Brain +Image Segmentation +Yicong Li, Yang Tan, Jingyun Yang, Yang Li, Xiao-Ping Zhang +Tsinghua-Berkeley Shenzhen Institute +Tsinghua University +Shenzhen, China +yangli@sz.tsinghua.edu.cn +Abstract—Although many studies have successfully applied +transfer learning to medical image segmentation, very few of +them have investigated the selection strategy when multiple +source tasks are available for transfer. In this paper, we propose +a prior knowledge guided and transferability based framework +to select the best source tasks among a collection of brain image +segmentation tasks, to improve the transfer learning performance +on the given target task. The framework consists of modality +analysis, RoI (region of interest) analysis, and transferability es- +timation, such that the source task selection can be refined step by +step. Specifically, we adapt the state-of-the-art analytical transfer- +ability estimation metrics to medical image segmentation tasks +and further show that their performance can be significantly +boosted by filtering candidate source tasks based on modality +and RoI characteristics. Our experiments on brain matter, brain +tumor, and white matter hyperintensities segmentation datasets +reveal that transferring from different tasks under the same +modality is often more successful than transferring from the same +task under different modalities. Furthermore, within the same +modality, transferring from the source task that has stronger RoI +shape similarity with the target task can significantly improve the +final transfer performance. And such similarity can be captured +using the Structural Similarity index in the label space. +Index Terms—transfer learning, medical image analysis, source +selection +I. INTRODUCTION +Supervised deep learning algorithm requires an abundant +amount of annotated data to obtain a well-trained model that +can make accurate predictions. Such a requirement severely +limits the application of deep learning in the medical do- +main since acquiring and labeling medical data can be very +expensive and time-consuming. A commonly used solution +is transfer learning [1]: pre-training a model on a source +task where sufficient annotated data exists and fine-tuning the +model on the desired target task where only a small amount +of annotated data is available. A key question to the success +of transfer learning between source and target tasks is what +source task shall we transfer from given a target task (i.e., +what to transfer). That is, if we transfer knowledge from a +less related source task, it may inversely hurt the performance +on the target task, a phenomenon known as negative transfer +[2]. Although many studies have already successfully applied +transfer learning to medical image analysis problems [3]–[5], +very few of them have investigated how to select the best +This research is funded by Natural Science Foundation of China 62001266. +Supplementary materials are available in this arXiv version. +source tasks. Hence in this work, we aim to systematically +tackle the source selection problem, focusing particularly on +the most dominant image segmentation tasks. +An effective source selection strategy requires a good un- +derstanding of what factors affect the transfer performance +for medical segmentation tasks. Several experimental studies +have investigated the transfer learning performance in medical +image analysis. In [6], advantages and disadvantages of several +transfer learning strategies for medical image segmentation +tasks are explored, such as which component of a CNN model +is better to transfer. Weatheritt et al. [7] analyze the impacts +of different choices of data pre-processing methods, tasks, and +amount of data used on the final transfer performance. Unfor- +tunately, these studies haven’t fully investigated how detailed +characteristics of source tasks can influence transfer learning +performance. Particularly, for medical image segmentation, +image characteristics of source tasks such as the modality and +the geometric shape of the segmentation region of interest +(RoI) vary greatly due to the fundamental differences in +scanner protocols, imaging procedures, lesion positions, etc. +Such variations will definitely affect the final transfer learning +performance on the target task. +To tackle the source selection problem systematically, we +need a quantitative way to rank the transfer performance of +source tasks. Recent studies have proposed different methods +to estimate the knowledge transferability between source and +target tasks for natural images. Transferability reveals how +easy it is to transfer knowledge learned from a source task +to a target task. Bao et al. [8] takes an information theoretic +approach and develops a computable metric called H-score +to estimate the knowledge transferability between datasets for +image classification problems. Under the assumption that the +inputs of source and target tasks share the same domain, +Tran et al. [9] uses NCE, negative conditional entropy, to +evaluate transferability. In [10], a new metric called LEEP is +proposed. It makes predictions based on the expected empirical +conditional distribution between source and target labels. More +recently, Tan et al. [11] proposes an optimal transport based +conditional entropy (OTCE) metric to analytically predict +transfer performance for supervised classification problems, +which has been further extended to evaluate segmentation +problems as well [12]. However, these works focus on finding +a general-purpose transferability metric purely based on fea- +978-1-6654-6819-0/22/$31.00 ©2022 IEEE +arXiv:2301.00934v1 [eess.IV] 3 Jan 2023 + +ture effectiveness without consideration of prior knowledge +(e.g., image characteristics) about tasks, such as modality +difference and RoI shape similarity between source and target +tasks in the medical image domain. +As such, in this work, we propose a novel source selection +framework that leverages the prior knowledge of medical +image segmentation tasks for reducing the computation time +and improving the selection accuracy of existing transferability +metrics, so as to create a systematic way to select the best +source tasks for a given target task. We choose brain image +segmentation datasets for our experiments since it is one of the +most challenging and time-consuming clinical procedures for +diagnosing brain disorders, whose demand has been increasing +in recent years. In summary, our main contributions are: +• A prior knowledge guided and transferability based +source selection framework. The framework incorpo- +rates prior knowledge of medical image segmentation +tasks with transferability estimation metrics to select the +best source tasks for transfer learning given a target task. +• An analysis of the relationship between image char- +acteristics and transfer learning performance. We +extensively conduct transfer learning experiments under +the cross-modality or cross-task setting and conclude that +transferring from a different task under the same modality +is often more beneficial than transferring from the same +task under a different modality. We also quantitatively +show that within the same modality, transferring from the +source task that has stronger RoI shape similarity with +the target task outperforms transferring from those less +similar ones. +II. METHODS +A. Problem Description +In a typical source selection problem, we are given K pre- +trained source models {θs}K +k=1 corresponding to K source +task {Ts}K +k=1 and a target task Tt. The problem goal is to find +out which source task Ts we should transfer from in order to +achieve the best transfer performance on the target task Tt. +The most common source selection approach is to fine- +tune each source model on the target task to obtain a transfer +accuracy on the target test set. This transfer accuracy is called +the ground truth transferability, which can be represented +by a certain segmentation accuracy evaluation metric, such +as Dice score. Then the task that corresponds to the source +model that achieves the best transfer accuracy will be selected +as the most appropriate source. However, this naive method +is very computationally expensive and may become very +inefficient when K is large. Therefore, analytical methods +are developed to estimate the transferability without fine- +tuning each source model on the target task to save time +and resources. The ranking of transferability scores between +source and target tasks produced by such analytical methods +should well correlate with the ranking given by the ground +truth transferability scores so that top-performing sources can +be selected accurately but in a less time-consuming way. +Transferability +Estimation +1 +2 +Potentially +Appropriate +Source(s) +Source +Datasets with +Pre-trained +Models +Modality +Analysis +Filtered +Subset 1 +RoI +Analysis +Filtered +Subset 2 +Select source tasks from the +same modality +Select source tasks with +stronger RoI shape similarity +Target: +Source: +Target: +Source: +T1 +T1 +T1 +FLAIR +ED +ET +WMH +CSF +Fig. 1. +Prior knowledge guided and transferability based source selec- +tion framework. Path 1: do modality analysis and RoI analysis, then use +transferability estimation metrics. Path 2: apply transferability estimation +metrics directly on the source tasks. T1: t1-weighted modality, FLAIR: fluid- +attenuated inversion recovery modality, ED: edematous tissue, ET: enhancing +tumor, WMH: white matter hyperintensities, CSF: cerebrospinal fluid. +B. Prior Knowledge Guided and Transferability Based Source +Selection Framework +Unfortunately, analytical transferability estimation metrics +like H-score and OTCE are designed for natural images, +thus their performance in the medical image domain is not +guaranteed. Inspired by the differences in modalities and RoI +shapes of medical image segmentation tasks, we propose a +source selection framework that incorporates the analysis of +image characteristics with current state-of-the-art transferabil- +ity estimation metrics, as shown in Fig. 1. +Given a pool of source tasks and a target task, we propose +to analyze the image characteristics of tasks before computing +transferability estimation metrics (Path 1 in Fig. 1). Specifi- +cally, our framework consists of three steps: +1) Modality Analysis: Select source tasks that are under +the same modality as the target task to generate Subset 1 (we +define a subset as a smaller group of tasks selected from a +bigger group of tasks). +2) RoI Analysis: Within Subset 1 (in which source tasks +are under the same modality as the target task), select source +tasks whose RoI shapes are more similar to that of the target +task by calculating the shape similarity to generate Subset +2. Specifically, we propose to use structural similarity index +measure (SSIM) [13] to quantify the RoI shape similarity. In +our transfer learning context, given a source task Ts with its +label set Ys and a target task Tt with its label set Yt, we +compute their RoI shape similarity as: +RoI-Sim(Ts, Tt) = SSIM(Ys, Yt). +(1) +More details on SSIM can be found in the supplementary +materials. + +3) Transferabiltiy Estimation: Within Subset 2, we apply +a certain analytical transferability estimation metric to select +potentially appropriate source tasks. In this work, we choose +H-score [8] or OTCE [12] as the metric. +In the task transfer learning setting, let (Xs, Ys) and +(Xt, Yt) represent the input and output random variables of +the source and target tasks, respectively. Given a source model +θs pre-trained on a source task, we denote the feature of +source and target data Xs, Xt extracted by the source model +θs as θs(Xs) and θs(Xt). Assuming the same input domain +P(θs(Xs)) = P(θs(Xt)), H-score measures the transferability +of θs with respect to the target task as +H(θs, Xt, Yt) = tr(cov(θs(Xt))−1cov(EPXt|Yt [θs(Xt)|Yt])), +(2) +where the covariance (cov) and the expectation are estimated +empirically from data [8]. H-score is originally designed to +handle classification tasks, therefore we adapt it to segmen- +tation tasks by considering each pixel in the image as an +individual classification task and calculating an H-score for +each of them. Then the final H-score is the arithmetic mean +of all pixel-wise scores: +H-score(Ts, Tt) = 1 +N +N +� +j=1 +H(θj +s, Xt, Y j +t ), +(3) +where N is the total number of pixels in the segmentation +label image, θj +s represents the source model feature mapping +corresponding to the jth pixel, and Y j +t represents the segmen- +tation label of the jth pixel. We choose H-score because it +is highly efficient and effective when the segmentation tasks +have similar input distributions but different RoIs. +The OTCE score is a versatile transferability metric for +both the cross-domain and the cross-task transfer scenarios. +To compute OTCE, we first use the pre-trained source model +θs to produce the pixel-wise feature sets Ds = {(vi +s, yi +s)}Ns +i=1 +and Dt += {(vi +t, yi +t)}Nt +i=1 for the source and target tasks +respectively, where vi +d, yi +d, and Nd denotes the pixel-wise +feature vector, pixel-wise label, and the total number of pixels +in all images of task d ∈ {s, t}, respectively. Next, we find the +optimal coupling matrix of size Ns × Nt between source and +target features by solving the following regularized Optimal +Transport (OT) problem: +OT(Ds, Dt) ≜ +min +π∈Π(Ds,Dt) +Ns,Nt +� +i,j=1 +||vi +s − vj +t ||2 +2πij + ϵH(π), (4) +where H(π) += +− �Ns +i=1 +�Nt +j=1 πij log πij is the entropic +regularizer with ϵ = 0.1. Since the optimal coupling matrix +π∗ represents the empirical joint probability distribution of +source and target pixel-wise features, under mild assumptions, +the empirical joint probability distribution of source and target +labels can be written as +ˆP(ys, yt) = +� +i,j:yis=ys,yj +t =yt +π∗ +ij. +(5) +Finally, the OTCE score can be computed as the negative +conditional entropy between the source and target labels: +OTCE(Ts, Tt) = −H(Yt|Ys) += +� +yt∈Yt +� +ys∈Ys +ˆP(ys, yt) log +ˆP(ys, yt) +� +yt∈Yt ˆP(ys, yt) +, +(6) +where Ys and Yt denote the source and target pixel-wise label +space. More details of OTCE for semantic segmentation can +be found in [12]. +As an ablation study of our source selection framework, a +baseline approach (Path 2 in Fig. 1) is directly computing the +transferability estimation metric on all source tasks without +considering the image characteristics of tasks. +III. EXPERIMENTAL SETTINGS AND RESULTS +A. Datasets +We perform experiments on three publicly available brain +MRI segmentation datasets: FeTS 2021 [14]–[16] for brain +tumor segmentation, iSeg-2019 [17] for brain matter seg- +mentation, and WMH [18] for white matter hyperintensities +segmentation. +For each sample in FeTS 2021 dataset, volumes of 4 +modalities are available, including T1-weighted (T1), T2- +weighted (T2), Fluid-Attenuated Inversion Recovery (FLAIR), +and T1-Weighted Contrast-Enhanced (T1CE). The volume size +is 240 × 240 × 155. Corresponding labels of edematous tissue +(ED), enhancing tumor (ET), and necrotic tumor core (NCR) +are manually segmented by clinical experts. For each sample +in iSeg-2019 dataset, volumes of 2 modalities are available, +including T1 and T2. The volume size is 144 × 192 × 256. +Corresponding labels of white matter (WM), gray matter +(GM), and cerebrospinal fluid (CSF) are manually segmented +by clinicians. The WMH dataset consists of 60 brain MRI +volumes of FLAIR modality with manual annotations of white +matter hyperintensities from three different institutes, namely, +VU Amsterdam (A), NUHS Singapore (S), and UMC Utrecht +(U). Volume sizes are 132 × 256 × 83, 256 × 232 × 48, +and 240 × 240 × 48 for the three institutes, respectively. +Corresponding labels of white matter hyperintensities (WMH) +are manually segmented by clinical experts. +To provide enough tasks for experimental analysis, we +reorganize these three datasets into a collection of binary +segmentation tasks on every available modality. Examples of +images from these three datasets with their corresponding +labels are visualized in Fig. 2. +B. Model Architecture and Transfer Learning Strategy +Since the goal of this work is to investigate the source +selection problem rather than trying to achieve state-of-the- +art performance on medical image segmentation tasks, we +use the same model architecture for all experiments presented +in this paper, a classic 2D U-Net [19]. As for the transfer +learning strategy, we follow the most common way which is +pre-training the model on a source task and fine-tuning it on a +target task. During the fine-tuning stage, the encoder is frozen + +Fig. 2. +Examples of images from FeTS 2021, iSeg-2019, and WMH +datasets with their corresponding labels. In FeTS 2021 dataset, e.g., “ED- +T1” denotes edematous tissue segmentation on T1 modality. In iSeg-2019 +dataset, e.g., “WM-T1” denotes white matter segmentation on T1 modality. In +WMH dataset, e.g., “WMH-A-FLAIR” denotes white matter hyperintensities +segmentation on FLAIR modality from institute A. +TABLE I +COMPARISONS OF CROSS-TASK TRANSFER AND CROSS-MODALITY +TRANSFER ON FETS 2021 (TOP) AND ISEG-2019 (BOTTOM) DATASETS. +Target +Sourcea +Average Diceb +ET-T1CE +ET-T1, ET-T2, ET-FLAIR +0.755 +ED-T1CE, NCR-T1CE +0.821 +ED-T1CE +ED-T1, ED-T2, ED-FLAIR +0.731 +ET-T1CE, NCR-T1CE +0.786 +NCR-T1CE +NCR-T1, NCR-T2, NCR-FLAIR +0.726 +ET-T1CE, ED-T1CE +0.782 +WM-T1 +WM-T2 +0.864 +GM-T1, CSF-T1 +0.877 +GM-T1 +GM-T2 +0.881 +WM-T1, CSF-T1 +0.892 +CSF-T1 +CSF-T2 +0.935 +WM-T1, GM-T1 +0.934 +aGreen/Blue: cross-modality/task transfer. +bRed: better transfer performance. +and only the parameters of the decoder are updated. See the +supplementary materials for details on the model architecture +and the training process. +C. Verification of Modality Analysis +In Modality Analysis, we select source tasks having the +same modality as the target task as the candidate sources +(Subset 1). To verify the correctness of this design, we conduct +experiments using source and target tasks from the multi- +modal datasets FeTS 2021 and iSeg-2019. +As shown in Table I, transferring from different tasks under +the same modality (sources in blue) outperforms transferring +from the same task under different modality (sources in green) +in almost all trials (5 out of 6, the scores of the only exception +are very close) with a significant margin (average Dice scores +in red). Such a finding suggests that for the common transfer +learning strategy of pre-training and fine-tuning, matching +different source and target data modalities is harder than re- +learning a new task within the same modality. +D. Verification of RoI Analysis +In RoI Analysis, within Subset 1, we select source tasks +whose RoI shapes are more similar to that of the target task by +calculating the shape similarity using SSIM, and then generate +Subset 2. To verify this choice, we conduct experiments using +source and target tasks from all three datasets. Specifically, +our experiment includes both the setting when the source and +target tasks are from the same dataset (same-dataset), and +when they are from different datasets (cross-dataset) to show +the effectiveness and generalizability of our framework. +As shown in Table II and III, we first compute the RoI shape +similarity score (RoI-Sim) between the source and the target +task of the same modality. Then, we also perform transfer +learning experiments to obtain the ground truth transfer accu- +racy between them. According to the results, we can conclude +that in both same-dataset (Table II) and cross-dataset (Table +III) settings, when the RoI shape similarity between the source +and the target task is stronger (the RoI-Sim score in red), the +transfer performance is often (16 out of 18 trials) better (the +Dice score in red). Such a finding suggests that within the +same modality, SSIM can serve as an indicator to rank the +performances of transfer learning from different source tasks +to the target task. +E. Results of Source Selection +For the source selection experiments, we use FeTS 2021 +dataset. This dataset is further split into 22 partitions by the +provider, according to different institutions and information +extracted from images. Thus, each partition can be seen as +an individual domain. Here, we additionally denote a task +by “Task-Partition-Modality”, e.g., “ET-14-T1” represents the +task of enhancing tumor segmentation on T1 modality using +data from partition 14. In total, 16 source tasks (ED/NCR- +13/14/17/18-T1/T2) and 2 target tasks (ET-22-T2 and ET- +20-T1) are used to conduct two groups of source selection +experiments. The ground truth transfer learning results and +the calculated transferability results on two target tasks are +shown in Table IV and Table V. +The result of source selection is a ranking of source tasks +according to their transfer performance on a given target task. +The ground truth ranking is obtained by sorting the Dice +scores after fine-tuning each source task on a given target task. +A higher Dice score indicates better transferability. The base- +line ranking prediction is obtained by directly computing and +sorting H-scores or OTCE scores on all source tasks (Path 2 +in Fig. 1). A higher H-score or OTCE score indicates better +transferability. The ranking prediction proposed by our +framework is obtained through combining prior knowledge +with transferability estimation metrics, as indicated by Path 1 +in Fig. 1. We take Table IV as an example to illustrate how to +obtain the ranking prediction with our proposed framework. +Given a target task of ET-22-T2, we notice that this task is + +ED-T1 +ET-T1 +NCR-T1 +WM-T1 +GM-T1 +CSF-T1 +WMH-A-FLAIR +WMH-S-FLAIR +WMH-U-FLAIRTABLE II +ANALYSIS OF RELATIONSHIP BETWEEN ROI SHAPE SIMILARITY AND TRANSFER PERFORMANCE ON FETS 2021 (TOP) AND ISEG-2019 (BOTTOM) +DATASETS. +Target +Source +Dicea +RoI-Simb +Target +Source +Dicea +RoI-Simb +Target +Source +Dicea +RoI-Simb +ED-T1 +ET-T1 +0.757 +0.987 +ED-T2 +ET-T2 +0.811 +0.987 +ED-FLAIR +ET-FLAIR +0.773 +0.987 +NCR-T1 +0.738 +0.984 +NCR-T2 +0.802 +0.984 +NCR-FLAIR +0.760 +0.984 +ET-T1 +ED-T1 +0.693 +0.987 +ET-T2 +ED-T2 +0.703 +0.987 +ET-FLAIR +ED-FLAIR +0.681 +0.987 +NCR-T1 +0.672 +0.985 +NCR-T2 +0.660 +0.985 +NCR-FLAIR +0.613 +0.985 +NCR-T1 +ED-T1 +0.585 +0.984 +NCR-T2 +ED-T2 +0.598 +0.984 +NCR-FLAIR +ED-FLAIR +0.558 +0.984 +ET-T1 +0.602 +0.985 +ET-T2 +0.578 +0.985 +ET-FLAIR +0.550 +0.985 +WM-T1 +GM-T1 +0.885 +0.844 +WM-T2 +GM-T2 +0.852 +0.844 +- +- +- +- +CSF-T1 +0.868 +0.824 +CSF-T2 +0.815 +0.824 +- +- +- +GM-T1 +WM-T1 +0.893 +0.844 +GM-T2 +WM-T2 +0.879 +0.844 +- +- +- +- +CSF-T1 +0.891 +0.838 +CSF-T2 +0.863 +0.838 +- +- +- +CSF-T1 +WM-T1 +0.925 +0.824 +CSF-T2 +WM-T2 +0.909 +0.824 +- +- +- +- +GM-T1 +0.942 +0.838 +GM-T2 +0.916 +0.838 +- +- +- +aRed: better transfer performance. bRed: stronger RoI shape similarity. +TABLE III +ANALYSIS OF RELATIONSHIP BETWEEN ROI SHAPE SIMILARITY AND +TRANSFER PERFORMANCE ACROSS FETS 2021 AND WMH DATASETS. +Target +Source +Dicea +RoI-Simb +NCR-FLAIR +0.650 +0.979 +ET-FLAIR +0.644 +0.978 +WMH-U-FLAIR +ED-FLAIR +0.578 +0.967 +NCR-FLAIR +0.542 +0.982 +ET-FLAIR +0.542 +0.981 +WMH-S-FLAIR +ED-FLAIR +0.531 +0.970 +NCR-FLAIR +0.572 +0.985 +ET-FLAIR +0.538 +0.984 +WMH-A-FLAIR +ED-FLAIR +0.547 +0.973 +aRed: better transfer performance. +bRed: stronger RoI shape similarity. +TABLE IV +TRANSFER LEARNING AND TRANSFERABILITY ESTIMATION RESULTS ON +THE TARGET TASK OF ET-22-T2. +Target +Sourcea +Dice +H-score +OTCE +ED-14-T1 +0.664 +-0.0380 +-0.0395 +ED-14-T2 +0.703 +0.1887 +-0.0226 +NCR-14-T1 +0.646 +0.8990 +-0.0395 +NCR-14-T2 +0.660 +0.5140 +-0.0383 +ED-13-T1 +0.657 +0.4142 +-0.0407 +ED-13-T2 +0.695 +1.4031 +-0.0356 +NCR-13-T1 +0.628 +5.0050 +-0.0407 +NCR-13-T2 +0.683 +10.5247 +-0.0401 +ED-17-T1 +0.697 +0.3525 +-0.0435 +ED-17-T2 +0.708 +1.3327 +-0.0389 +NCR-17-T1 +0.612 +1.5211 +-0.0436 +NCR-17-T2 +0.681 +6.6535 +-0.0433 +ED-18-T1 +0.675 +0.1070 +-0.0435 +ED-18-T2 +0.707 +0.2776 +-0.0273 +NCR-18-T1 +0.664 +0.9038 +-0.0436 +ET-22-T2 +NCR-18-T2 +0.666 +2.2038 +-0.0394 +aUnderlined sources: Subset 1. Sources in red: Subset 2. +under the T2 modality. According to our modality analysis in +Section II-B1, we should select those 8 source tasks under +the same modality as the target task (underlined in Table +IV). This procedure forms Subset 1. Next, according to our +RoI analysis in Section II-B2, within the T2 modality, the +TABLE V +TRANSFER LEARNING AND TRANSFERABILITY ESTIMATION RESULTS ON +THE TARGET TASK OF ET-20-T1. +Target +Sourcea +Dice +H-score +OTCE +ED-14-T1 +0.636 +1.5433 +-0.0320 +ED-14-T2 +0.609 +0.2268 +-0.0330 +NCR-14-T1 +0.560 +3.2383 +-0.0325 +NCR-14-T2 +0.627 +2.5139 +-0.0330 +ED-13-T1 +0.593 +1.7564 +-0.0333 +ED-13-T2 +0.636 +1.3293 +-0.0347 +NCR-13-T1 +0.498 +2.4574 +-0.0342 +NCR-13-T2 +0.610 +6.5037 +-0.0346 +ED-17-T1 +0.680 +2.5901 +-0.0351 +ED-17-T2 +0.571 +-3.2459 +-0.0363 +NCR-17-T1 +0.581 +25.7285 +-0.0361 +NCR-17-T2 +0.532 +40.6843 +-0.0363 +ED-18-T1 +0.613 +0.0901 +-0.0357 +ED-18-T2 +0.616 +0.3164 +-0.0355 +NCR-18-T1 +0.632 +0.2743 +-0.0361 +ET-20-T1 +NCR-18-T2 +0.637 +1.6508 +-0.0362 +aUnderlined sources: Subset 1. Sources in red: Subset 2. +RoI shape similarity between ED and ET estimated by SSIM +is higher than that between ED and NCR, thus we should +select those 4 source tasks (colored in red in Table IV) of ED +segmentation rather than NCR segmentation. This procedure +forms Subset 2. Finally, we apply the analytical transferability +estimation metric on source tasks in Subset 2 and obtain +their predicted ranking. As for Table V, the source selection +procedure using our proposed framework is similar. +The performance of source selection methods is often +evaluated by comparing the difference between the ground +truth transfer performance ranking and the predicted transfer +performance ranking. Here, we use Spearman’s footrule [20] +to quantify the difference between the two rankings. More +details on Spearman’s footrule can be found in the supple- +mentary materials. The performance evaluation on selecting +the top 1-4 source tasks is shown in Table VI. For both +target tasks and under all top 1-4 source selection settings, +when following our proposed prior knowledge guided and +transferability based framework, the difference between the + +TABLE VI +EVALUATION OF SOURCE SELECTION PERFORMANCE WITH/WITHOUT +PRIOR KNOWLEDGE. +Target +Methoda +Top 1b +Top 2b +Top 3b +Top 4b +H-score w/o PK +5 +10 +22 +27 +H-score w/ PK +4 +5 +6 +7 +OTCE w/o PK +2 +2 +4 +12 +ET-22-T2 +OTCE w/ PK +2 +2 +4 +7 +H-score w/o PK +14 +24 +30 +40 +H-score w/ PK +0 +9 +9 +13 +OTCE w/o PK +2 +14 +17 +23 +ET-20-T1 +OTCE w/ PK +2 +11 +13 +17 +aPK: prior knowledge. +bTop 1-4: required number of selected sources. Red: better performance. +predicted ranking and the ground truth ranking is reduced. +This suggests that prior knowledge about the medical image +segmentation tasks, including modality and RoI characteristics, +can indeed improve the current state-of-the-art transferability +metrics’ ability to successfully select source tasks with better +transfer performance. Besides, these results also reveal that +current transferability estimation metrics are not sufficient to +handle the large gaps between source and target tasks and thus +require further refinement, particularly in the medical domain. +IV. CONCLUSION +We propose a prior knowledge guided and transferability +based framework to tackle the source selection problem in +transfer learning for brain image segmentation tasks. We are +the first to apply state-of-the-art transferability estimation met- +rics to the medical image segmentation domain. Different from +the common procedure that directly applies these metrics, our +framework further considers the prior knowledge of the given +source and target tasks when selecting sources. Specifically, +we perform modality analysis and RoI analysis to select a +subset of source tasks and then only compute the metric +within this subset. Modality analysis shows that transferring +from different tasks under the same modality is better than +transferring from the same task under different modalities. RoI +Analysis shows that stronger RoI shape similarity between the +source and the target task often leads to better transfer perfor- +mance. 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SUPPLEMENTARY MATERIALS +A. Structural Similarity Index Measure (SSIM) +SSIM is often used to evaluate the visual similarity between +two images. The idea is that natural images often contain +highly structural information, i.e., neighboring pixels in nat- +ural images have a strong correlation. And such correlation +encompasses the structural information of the object in a given +environment. The human visual system is very used to extract +such structural information from natural images. Therefore, +the measurement of similarity given by SSIM is more in line +with the perception of human eyes [21], [22] compared to +other metrics like peak signal-to-noise ratio (PSNR). Given +two images x and y, SSIM can be calculated as: +SSIM(x, y) = +(2µxµy + C1) (2σxy + C2) +� +µ2x + µ2y + C1 +� � +σ2x + σ2y + C2 +�, +(7) +where µx is the average of x, µy is the average of y, σ2 +x is the +variance of x, σ2 +y is the variance of y, and σxy is the covariance +of x and y. C1 and C2 are constants for maintaining stability. +Higher SSIM indicates stronger similarity between x and y. +It ranges from 0 to 1 and when the two images are identical, +the value equals 1. +B. Model and Training Details +1) Model Architecture: We use the same model architecture +for all experiments presented in this paper, a classic 2D U- +Net [19], as shown in Fig. 3. The model includes an encoder +and a decoder. The encoder consists of 5 blocks, each of +2) Training Configurations: During the training stage, we +use Adam [23] as the optimizer with a learning rate of 1e- +4, a weight decay of 5e-5, and a batch size of 20. We set +10,000 iterations for the pre-training stage and 1,000 iterations +for the fine-tuning stage. Cross-entropy is used as the loss +function and Dice score is chosen as the metric to evaluate the +segmentation performance. The model is implemented using +Output +Segmentation +Input +Image +Conv Block +Skip Connection +Encoder +Block-1 +Max Pool 2x2 +Transpose Conv 2x2 +Conv 1x1 +Encoder +Block-2 +Encoder +Block-3 +Encoder +Block-4 +Encoder +Block-5 +Decoder +Block-4 +Decoder +Block-3 +Decoder +Block-2 +Decoder +Block-1 +Fig. 3. +U-Net architecture. Parameters of convolutional blocks in blue are +frozen during the fine-tuning stage. Parameters of convolutional blocks in +green are updated during the fine-tuning stage. Red box indicates the feature +we use to compute the transferability scores. This figure is partially reproduced +from [19]. +which contains 2 sub-blocks of 3 × 3 convolution, batch +normalization, and ReLU activation, followed by a 2 × 2 max +pooling layer. In the decoder, similar blocks are used, each +of which is followed by a 2 × 2 transpose convolution layer. +The final 1 × 1 convolution layer outputs a segmentation map +(logits) with 2 channels. During transfer learning, we freeze +the encoder and fine-tune the decoder. The final feature map +(output segmentation) produced by the decoder is used to +compute the transferability score. +Python 3.6.8 and the deep learning framework PyTorch 1.8.0 +[24]. All experiments are conducted on a CentOS 7.6.1810 +system with one GeForce RTX 3090 GPU. +C. Spearman’s Footrule +Spearman’s footrule [20] measures the absolute distance +between two rankings by calculating how many steps we need +to move the elements in the predicted ranking, in order to make +it the same as the ground truth ranking. Formally, given two +rankings A and B with the same number (denoted as N) of +elements, Spearman’s footrule is calculated as: +Spearman(A, B) = +N +� +n=1 +|A[n] − B[n]|, +(8) +For example, if A += +[1, 2, 3] and B += +[2, 1, 3], then +Spearman(A, B) = |1 − 2| + |2 − 1| + |3 − 3| = 2. + diff --git a/q9AzT4oBgHgl3EQfA_op/content/tmp_files/load_file.txt b/q9AzT4oBgHgl3EQfA_op/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fef94df01415cbc390c38c9be1c8c77d4c26e5d5 --- /dev/null +++ b/q9AzT4oBgHgl3EQfA_op/content/tmp_files/load_file.txt @@ -0,0 +1,513 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf,len=512 +page_content='Finding the Most Transferable Tasks for Brain Image Segmentation Yicong Li, Yang Tan, Jingyun Yang, Yang Li, Xiao-Ping Zhang Tsinghua-Berkeley Shenzhen Institute Tsinghua University Shenzhen, China yangli@sz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='cn Abstract—Although many studies have successfully applied transfer learning to medical image segmentation, very few of them have investigated the selection strategy when multiple source tasks are available for transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In this paper, we propose a prior knowledge guided and transferability based framework to select the best source tasks among a collection of brain image segmentation tasks, to improve the transfer learning performance on the given target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The framework consists of modality analysis, RoI (region of interest) analysis, and transferability es- timation, such that the source task selection can be refined step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Specifically, we adapt the state-of-the-art analytical transfer- ability estimation metrics to medical image segmentation tasks and further show that their performance can be significantly boosted by filtering candidate source tasks based on modality and RoI characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Our experiments on brain matter, brain tumor, and white matter hyperintensities segmentation datasets reveal that transferring from different tasks under the same modality is often more successful than transferring from the same task under different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Furthermore, within the same modality, transferring from the source task that has stronger RoI shape similarity with the target task can significantly improve the final transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' And such similarity can be captured using the Structural Similarity index in the label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Index Terms—transfer learning, medical image analysis, source selection I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' INTRODUCTION Supervised deep learning algorithm requires an abundant amount of annotated data to obtain a well-trained model that can make accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Such a requirement severely limits the application of deep learning in the medical do- main since acquiring and labeling medical data can be very expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' A commonly used solution is transfer learning [1]: pre-training a model on a source task where sufficient annotated data exists and fine-tuning the model on the desired target task where only a small amount of annotated data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' A key question to the success of transfer learning between source and target tasks is what source task shall we transfer from given a target task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', what to transfer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' That is, if we transfer knowledge from a less related source task, it may inversely hurt the performance on the target task, a phenomenon known as negative transfer [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Although many studies have already successfully applied transfer learning to medical image analysis problems [3]–[5], very few of them have investigated how to select the best This research is funded by Natural Science Foundation of China 62001266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Supplementary materials are available in this arXiv version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Hence in this work, we aim to systematically tackle the source selection problem, focusing particularly on the most dominant image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' An effective source selection strategy requires a good un- derstanding of what factors affect the transfer performance for medical segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Several experimental studies have investigated the transfer learning performance in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In [6], advantages and disadvantages of several transfer learning strategies for medical image segmentation tasks are explored, such as which component of a CNN model is better to transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Weatheritt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' [7] analyze the impacts of different choices of data pre-processing methods, tasks, and amount of data used on the final transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Unfor- tunately, these studies haven’t fully investigated how detailed characteristics of source tasks can influence transfer learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Particularly, for medical image segmentation, image characteristics of source tasks such as the modality and the geometric shape of the segmentation region of interest (RoI) vary greatly due to the fundamental differences in scanner protocols, imaging procedures, lesion positions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Such variations will definitely affect the final transfer learning performance on the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' To tackle the source selection problem systematically, we need a quantitative way to rank the transfer performance of source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Recent studies have proposed different methods to estimate the knowledge transferability between source and target tasks for natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Transferability reveals how easy it is to transfer knowledge learned from a source task to a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' [8] takes an information theoretic approach and develops a computable metric called H-score to estimate the knowledge transferability between datasets for image classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Under the assumption that the inputs of source and target tasks share the same domain, Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' [9] uses NCE, negative conditional entropy, to evaluate transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In [10], a new metric called LEEP is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' It makes predictions based on the expected empirical conditional distribution between source and target labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' More recently, Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' [11] proposes an optimal transport based conditional entropy (OTCE) metric to analytically predict transfer performance for supervised classification problems, which has been further extended to evaluate segmentation problems as well [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' However, these works focus on finding a general-purpose transferability metric purely based on fea- 978-1-6654-6819-0/22/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='00 ©2022 IEEE arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='00934v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='IV] 3 Jan 2023 ture effectiveness without consideration of prior knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', image characteristics) about tasks, such as modality difference and RoI shape similarity between source and target tasks in the medical image domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As such, in this work, we propose a novel source selection framework that leverages the prior knowledge of medical image segmentation tasks for reducing the computation time and improving the selection accuracy of existing transferability metrics, so as to create a systematic way to select the best source tasks for a given target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We choose brain image segmentation datasets for our experiments since it is one of the most challenging and time-consuming clinical procedures for diagnosing brain disorders, whose demand has been increasing in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In summary, our main contributions are: A prior knowledge guided and transferability based source selection framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The framework incorpo- rates prior knowledge of medical image segmentation tasks with transferability estimation metrics to select the best source tasks for transfer learning given a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' An analysis of the relationship between image char- acteristics and transfer learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We extensively conduct transfer learning experiments under the cross-modality or cross-task setting and conclude that transferring from a different task under the same modality is often more beneficial than transferring from the same task under a different modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We also quantitatively show that within the same modality, transferring from the source task that has stronger RoI shape similarity with the target task outperforms transferring from those less similar ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Problem Description In a typical source selection problem, we are given K pre- trained source models {θs}K k=1 corresponding to K source task {Ts}K k=1 and a target task Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The problem goal is to find out which source task Ts we should transfer from in order to achieve the best transfer performance on the target task Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The most common source selection approach is to fine- tune each source model on the target task to obtain a transfer accuracy on the target test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This transfer accuracy is called the ground truth transferability, which can be represented by a certain segmentation accuracy evaluation metric, such as Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Then the task that corresponds to the source model that achieves the best transfer accuracy will be selected as the most appropriate source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' However, this naive method is very computationally expensive and may become very inefficient when K is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Therefore, analytical methods are developed to estimate the transferability without fine- tuning each source model on the target task to save time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The ranking of transferability scores between source and target tasks produced by such analytical methods should well correlate with the ranking given by the ground truth transferability scores so that top-performing sources can be selected accurately but in a less time-consuming way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Transferability Estimation 1 2 Potentially Appropriate Source(s) Source Datasets with Pre-trained Models Modality Analysis Filtered Subset 1 RoI Analysis Filtered Subset 2 Select source tasks from the same modality Select source tasks with stronger RoI shape similarity Target: Source: Target: Source: T1 T1 T1 FLAIR ED ET WMH CSF Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Prior knowledge guided and transferability based source selec- tion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Path 1: do modality analysis and RoI analysis, then use transferability estimation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Path 2: apply transferability estimation metrics directly on the source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' T1: t1-weighted modality, FLAIR: fluid- attenuated inversion recovery modality, ED: edematous tissue, ET: enhancing tumor, WMH: white matter hyperintensities, CSF: cerebrospinal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Prior Knowledge Guided and Transferability Based Source Selection Framework Unfortunately, analytical transferability estimation metrics like H-score and OTCE are designed for natural images, thus their performance in the medical image domain is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Inspired by the differences in modalities and RoI shapes of medical image segmentation tasks, we propose a source selection framework that incorporates the analysis of image characteristics with current state-of-the-art transferabil- ity estimation metrics, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Given a pool of source tasks and a target task, we propose to analyze the image characteristics of tasks before computing transferability estimation metrics (Path 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Specifi- cally, our framework consists of three steps: 1) Modality Analysis: Select source tasks that are under the same modality as the target task to generate Subset 1 (we define a subset as a smaller group of tasks selected from a bigger group of tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 2) RoI Analysis: Within Subset 1 (in which source tasks are under the same modality as the target task), select source tasks whose RoI shapes are more similar to that of the target task by calculating the shape similarity to generate Subset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Specifically, we propose to use structural similarity index measure (SSIM) [13] to quantify the RoI shape similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In our transfer learning context, given a source task Ts with its label set Ys and a target task Tt with its label set Yt, we compute their RoI shape similarity as: RoI-Sim(Ts, Tt) = SSIM(Ys, Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' (1) More details on SSIM can be found in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 3) Transferabiltiy Estimation: Within Subset 2, we apply a certain analytical transferability estimation metric to select potentially appropriate source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In this work, we choose H-score [8] or OTCE [12] as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In the task transfer learning setting, let (Xs, Ys) and (Xt, Yt) represent the input and output random variables of the source and target tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Given a source model θs pre-trained on a source task, we denote the feature of source and target data Xs, Xt extracted by the source model θs as θs(Xs) and θs(Xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Assuming the same input domain P(θs(Xs)) = P(θs(Xt)), H-score measures the transferability of θs with respect to the target task as H(θs, Xt, Yt) = tr(cov(θs(Xt))−1cov(EPXt|Yt [θs(Xt)|Yt])), (2) where the covariance (cov) and the expectation are estimated empirically from data [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' H-score is originally designed to handle classification tasks, therefore we adapt it to segmen- tation tasks by considering each pixel in the image as an individual classification task and calculating an H-score for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Then the final H-score is the arithmetic mean of all pixel-wise scores: H-score(Ts, Tt) = 1 N N � j=1 H(θj s, Xt, Y j t ), (3) where N is the total number of pixels in the segmentation label image, θj s represents the source model feature mapping corresponding to the jth pixel, and Y j t represents the segmen- tation label of the jth pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We choose H-score because it is highly efficient and effective when the segmentation tasks have similar input distributions but different RoIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The OTCE score is a versatile transferability metric for both the cross-domain and the cross-task transfer scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' To compute OTCE, we first use the pre-trained source model θs to produce the pixel-wise feature sets Ds = {(vi s, yi s)}Ns i=1 and Dt = {(vi t, yi t)}Nt i=1 for the source and target tasks respectively, where vi d, yi d, and Nd denotes the pixel-wise feature vector, pixel-wise label, and the total number of pixels in all images of task d ∈ {s, t}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Next, we find the optimal coupling matrix of size Ns × Nt between source and target features by solving the following regularized Optimal Transport (OT) problem: OT(Ds, Dt) ≜ min π∈Π(Ds,Dt) Ns,Nt � i,j=1 ||vi s − vj t ||2 2πij + ϵH(π), (4) where H(π) = − �Ns i=1 �Nt j=1 πij log πij is the entropic regularizer with ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Since the optimal coupling matrix π∗ represents the empirical joint probability distribution of source and target pixel-wise features, under mild assumptions, the empirical joint probability distribution of source and target labels can be written as ˆP(ys, yt) = � i,j:yis=ys,yj t =yt π∗ ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' (5) Finally, the OTCE score can be computed as the negative conditional entropy between the source and target labels: OTCE(Ts, Tt) = −H(Yt|Ys) = � yt∈Yt � ys∈Ys ˆP(ys, yt) log ˆP(ys, yt) � yt∈Yt ˆP(ys, yt) , (6) where Ys and Yt denote the source and target pixel-wise label space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' More details of OTCE for semantic segmentation can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As an ablation study of our source selection framework, a baseline approach (Path 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1) is directly computing the transferability estimation metric on all source tasks without considering the image characteristics of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' EXPERIMENTAL SETTINGS AND RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Datasets We perform experiments on three publicly available brain MRI segmentation datasets: FeTS 2021 [14]–[16] for brain tumor segmentation, iSeg-2019 [17] for brain matter seg- mentation, and WMH [18] for white matter hyperintensities segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' For each sample in FeTS 2021 dataset, volumes of 4 modalities are available, including T1-weighted (T1), T2- weighted (T2), Fluid-Attenuated Inversion Recovery (FLAIR), and T1-Weighted Contrast-Enhanced (T1CE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The volume size is 240 × 240 × 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Corresponding labels of edematous tissue (ED), enhancing tumor (ET), and necrotic tumor core (NCR) are manually segmented by clinical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' For each sample in iSeg-2019 dataset, volumes of 2 modalities are available, including T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The volume size is 144 × 192 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Corresponding labels of white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF) are manually segmented by clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The WMH dataset consists of 60 brain MRI volumes of FLAIR modality with manual annotations of white matter hyperintensities from three different institutes, namely, VU Amsterdam (A), NUHS Singapore (S), and UMC Utrecht (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Volume sizes are 132 × 256 × 83, 256 × 232 × 48, and 240 × 240 × 48 for the three institutes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Corresponding labels of white matter hyperintensities (WMH) are manually segmented by clinical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' To provide enough tasks for experimental analysis, we reorganize these three datasets into a collection of binary segmentation tasks on every available modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Examples of images from these three datasets with their corresponding labels are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Model Architecture and Transfer Learning Strategy Since the goal of this work is to investigate the source selection problem rather than trying to achieve state-of-the- art performance on medical image segmentation tasks, we use the same model architecture for all experiments presented in this paper, a classic 2D U-Net [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As for the transfer learning strategy, we follow the most common way which is pre-training the model on a source task and fine-tuning it on a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' During the fine-tuning stage, the encoder is frozen Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Examples of images from FeTS 2021, iSeg-2019, and WMH datasets with their corresponding labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In FeTS 2021 dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', “ED- T1” denotes edematous tissue segmentation on T1 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In iSeg-2019 dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', “WM-T1” denotes white matter segmentation on T1 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In WMH dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', “WMH-A-FLAIR” denotes white matter hyperintensities segmentation on FLAIR modality from institute A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' TABLE I COMPARISONS OF CROSS-TASK TRANSFER AND CROSS-MODALITY TRANSFER ON FETS 2021 (TOP) AND ISEG-2019 (BOTTOM) DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Sourcea Average Diceb ET-T1CE ET-T1, ET-T2, ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='755 ED-T1CE, NCR-T1CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='821 ED-T1CE ED-T1, ED-T2, ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='731 ET-T1CE, NCR-T1CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='786 NCR-T1CE NCR-T1, NCR-T2, NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='726 ET-T1CE, ED-T1CE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='782 WM-T1 WM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='864 GM-T1, CSF-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='877 GM-T1 GM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='881 WM-T1, CSF-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='892 CSF-T1 CSF-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='935 WM-T1, GM-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='934 aGreen/Blue: cross-modality/task transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' bRed: better transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' and only the parameters of the decoder are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' See the supplementary materials for details on the model architecture and the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Verification of Modality Analysis In Modality Analysis, we select source tasks having the same modality as the target task as the candidate sources (Subset 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' To verify the correctness of this design, we conduct experiments using source and target tasks from the multi- modal datasets FeTS 2021 and iSeg-2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As shown in Table I, transferring from different tasks under the same modality (sources in blue) outperforms transferring from the same task under different modality (sources in green) in almost all trials (5 out of 6, the scores of the only exception are very close) with a significant margin (average Dice scores in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Such a finding suggests that for the common transfer learning strategy of pre-training and fine-tuning, matching different source and target data modalities is harder than re- learning a new task within the same modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Verification of RoI Analysis In RoI Analysis, within Subset 1, we select source tasks whose RoI shapes are more similar to that of the target task by calculating the shape similarity using SSIM, and then generate Subset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' To verify this choice, we conduct experiments using source and target tasks from all three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Specifically, our experiment includes both the setting when the source and target tasks are from the same dataset (same-dataset), and when they are from different datasets (cross-dataset) to show the effectiveness and generalizability of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As shown in Table II and III, we first compute the RoI shape similarity score (RoI-Sim) between the source and the target task of the same modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Then, we also perform transfer learning experiments to obtain the ground truth transfer accu- racy between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' According to the results, we can conclude that in both same-dataset (Table II) and cross-dataset (Table III) settings, when the RoI shape similarity between the source and the target task is stronger (the RoI-Sim score in red), the transfer performance is often (16 out of 18 trials) better (the Dice score in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Such a finding suggests that within the same modality, SSIM can serve as an indicator to rank the performances of transfer learning from different source tasks to the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Results of Source Selection For the source selection experiments, we use FeTS 2021 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This dataset is further split into 22 partitions by the provider, according to different institutions and information extracted from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Thus, each partition can be seen as an individual domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Here, we additionally denote a task by “Task-Partition-Modality”, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', “ET-14-T1” represents the task of enhancing tumor segmentation on T1 modality using data from partition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In total, 16 source tasks (ED/NCR- 13/14/17/18-T1/T2) and 2 target tasks (ET-22-T2 and ET- 20-T1) are used to conduct two groups of source selection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The ground truth transfer learning results and the calculated transferability results on two target tasks are shown in Table IV and Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The result of source selection is a ranking of source tasks according to their transfer performance on a given target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The ground truth ranking is obtained by sorting the Dice scores after fine-tuning each source task on a given target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' A higher Dice score indicates better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The base- line ranking prediction is obtained by directly computing and sorting H-scores or OTCE scores on all source tasks (Path 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' A higher H-score or OTCE score indicates better transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The ranking prediction proposed by our framework is obtained through combining prior knowledge with transferability estimation metrics, as indicated by Path 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We take Table IV as an example to illustrate how to obtain the ranking prediction with our proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Given a target task of ET-22-T2, we notice that this task is ED-T1 ET-T1 NCR-T1 WM-T1 GM-T1 CSF-T1 WMH-A-FLAIR WMH-S-FLAIR WMH-U-FLAIRTABLE II ANALYSIS OF RELATIONSHIP BETWEEN ROI SHAPE SIMILARITY AND TRANSFER PERFORMANCE ON FETS 2021 (TOP) AND ISEG-2019 (BOTTOM) DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Source Dicea RoI-Simb Target Source Dicea RoI-Simb Target Source Dicea RoI-Simb ED-T1 ET-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 ED-T2 ET-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 ED-FLAIR ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 NCR-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 NCR-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 ET-T1 ED-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 ET-T2 ED-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 ET-FLAIR ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='987 NCR-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 NCR-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 NCR-T1 ED-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 NCR-T2 ED-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 NCR-FLAIR ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 ET-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 ET-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 WM-T1 GM-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='844 WM-T2 GM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='844 CSF-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='824 CSF-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='824 GM-T1 WM-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='844 GM-T2 WM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='844 CSF-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='838 CSF-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='838 CSF-T1 WM-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='824 CSF-T2 WM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='824 GM-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='838 GM-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='838 aRed: better transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' bRed: stronger RoI shape similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' TABLE III ANALYSIS OF RELATIONSHIP BETWEEN ROI SHAPE SIMILARITY AND TRANSFER PERFORMANCE ACROSS FETS 2021 AND WMH DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Source Dicea RoI-Simb NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='979 ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='978 WMH-U-FLAIR ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='967 NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='982 ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='981 WMH-S-FLAIR ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='970 NCR-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='985 ET-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='984 WMH-A-FLAIR ED-FLAIR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='973 aRed: better transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' bRed: stronger RoI shape similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' TABLE IV TRANSFER LEARNING AND TRANSFERABILITY ESTIMATION RESULTS ON THE TARGET TASK OF ET-22-T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Sourcea Dice H-score OTCE ED-14-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0395 ED-14-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='1887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0226 NCR-14-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='8990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0395 NCR-14-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='660 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='5140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0383 ED-13-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='4142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0407 ED-13-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='695 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='4031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0356 NCR-13-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='628 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='6535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0433 ED-18-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='1070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0435 ED-18-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='2776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0273 NCR-18-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='9038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0436 ET-22-T2 NCR-18-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='666 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='2038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0394 aUnderlined sources: Subset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Sources in red: Subset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' under the T2 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' According to our modality analysis in Section II-B1, we should select those 8 source tasks under the same modality as the target task (underlined in Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This procedure forms Subset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Next, according to our RoI analysis in Section II-B2, within the T2 modality, the TABLE V TRANSFER LEARNING AND TRANSFERABILITY ESTIMATION RESULTS ON THE TARGET TASK OF ET-20-T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Sourcea Dice H-score OTCE ED-14-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='636 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='5433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0320 ED-14-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='609 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0330 ED-13-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='593 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='7564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0333 ED-13-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='636 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='3293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0347 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+page_content='680 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='5901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0351 ED-17-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='571 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='2459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0363 NCR-17-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='581 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='7285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0361 NCR-17-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='532 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='6843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0363 ED-18-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0357 ED-18-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='3164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0355 NCR-18-T1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='2743 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0361 ET-20-T1 NCR-18-T2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='637 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='6508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0362 aUnderlined sources: Subset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Sources in red: Subset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' RoI shape similarity between ED and ET estimated by SSIM is higher than that between ED and NCR, thus we should select those 4 source tasks (colored in red in Table IV) of ED segmentation rather than NCR segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This procedure forms Subset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Finally, we apply the analytical transferability estimation metric on source tasks in Subset 2 and obtain their predicted ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' As for Table V, the source selection procedure using our proposed framework is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The performance of source selection methods is often evaluated by comparing the difference between the ground truth transfer performance ranking and the predicted transfer performance ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Here, we use Spearman’s footrule [20] to quantify the difference between the two rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' More details on Spearman’s footrule can be found in the supple- mentary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The performance evaluation on selecting the top 1-4 source tasks is shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' For both target tasks and under all top 1-4 source selection settings, when following our proposed prior knowledge guided and transferability based framework, the difference between the TABLE VI EVALUATION OF SOURCE SELECTION PERFORMANCE WITH/WITHOUT PRIOR KNOWLEDGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Target Methoda Top 1b Top 2b Top 3b Top 4b H-score w/o PK 5 10 22 27 H-score w/ PK 4 5 6 7 OTCE w/o PK 2 2 4 12 ET-22-T2 OTCE w/ PK 2 2 4 7 H-score w/o PK 14 24 30 40 H-score w/ PK 0 9 9 13 OTCE w/o PK 2 14 17 23 ET-20-T1 OTCE w/ PK 2 11 13 17 aPK: prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' bTop 1-4: required number of selected sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Red: better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' predicted ranking and the ground truth ranking is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This suggests that prior knowledge about the medical image segmentation tasks, including modality and RoI characteristics, can indeed improve the current state-of-the-art transferability metrics’ ability to successfully select source tasks with better transfer performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Besides, these results also reveal that current transferability estimation metrics are not sufficient to handle the large gaps between source and target tasks and thus require further refinement, particularly in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' CONCLUSION We propose a prior knowledge guided and transferability based framework to tackle the source selection problem in transfer learning for brain image segmentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We are the first to apply state-of-the-art transferability estimation met- rics to the medical image segmentation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Different from the common procedure that directly applies these metrics, our framework further considers the prior knowledge of the given source and target tasks when selecting sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Specifically, we perform modality analysis and RoI analysis to select a subset of source tasks and then only compute the metric within this subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Modality analysis shows that transferring from different tasks under the same modality is better than transferring from the same task under different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' RoI Analysis shows that stronger RoI shape similarity between the source and the target task often leads to better transfer perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Consequently, by incorporating image characteristics of modality difference and RoI similarity into the framework, source selection 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“Pytorch: An imperative style, high-performance deep learning library,” Advances in neural information processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' SUPPLEMENTARY MATERIALS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Structural Similarity Index Measure (SSIM) SSIM is often used to evaluate the visual similarity between two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The idea is that natural images often contain highly structural information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=', neighboring pixels in nat- ural images have a strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' And such correlation encompasses the structural information of the object in a given environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The human visual system is very used to extract such structural information from natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Therefore, the measurement of similarity given by SSIM is more in line with the perception of human eyes [21], [22] compared to other metrics like peak signal-to-noise ratio (PSNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Given two images x and y, SSIM can be calculated as: SSIM(x, y) = (2µxµy + C1) (2σxy + C2) � µ2x + µ2y + C1 � � σ2x + σ2y + C2 �, (7) where µx is the average of x, µy is the average of y, σ2 x is the variance of x, σ2 y is the variance of y, and σxy is the covariance of x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' C1 and C2 are constants for maintaining stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Higher SSIM indicates stronger similarity between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' It ranges from 0 to 1 and when the two images are identical, the value equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Model and Training Details 1) Model Architecture: We use the same model architecture for all experiments presented in this paper, a classic 2D U- Net [19], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The model includes an encoder and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The encoder consists of 5 blocks, each of 2) Training Configurations: During the training stage, we use Adam [23] as the optimizer with a learning rate of 1e- 4, a weight decay of 5e-5, and a batch size of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' We set 10,000 iterations for the pre-training stage and 1,000 iterations for the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Cross-entropy is used as the loss function and Dice score is chosen as the metric to evaluate the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The model is implemented using Output Segmentation Input Image Conv Block Skip Connection Encoder Block-1 Max Pool 2x2 Transpose Conv 2x2 Conv 1x1 Encoder Block-2 Encoder Block-3 Encoder Block-4 Encoder Block-5 Decoder Block-4 Decoder Block-3 Decoder Block-2 Decoder Block-1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Parameters of convolutional blocks in blue are frozen during the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Parameters of convolutional blocks in green are updated during the fine-tuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Red box indicates the feature we use to compute the transferability scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' This figure is partially reproduced from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' which contains 2 sub-blocks of 3 × 3 convolution, batch normalization, and ReLU activation, followed by a 2 × 2 max pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' In the decoder, similar blocks are used, each of which is followed by a 2 × 2 transpose convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The final 1 × 1 convolution layer outputs a segmentation map (logits) with 2 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' During transfer learning, we freeze the encoder and fine-tune the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' The final feature map (output segmentation) produced by the decoder is used to compute the transferability score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='8 and the deep learning framework PyTorch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='0 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' All experiments are conducted on a CentOS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content='1810 system with one GeForce RTX 3090 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Spearman’s Footrule Spearman’s footrule [20] measures the absolute distance between two rankings by calculating how many steps we need to move the elements in the predicted ranking, in order to make it the same as the ground truth ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} +page_content=' Formally, given two rankings A and B with the same number (denoted as N) of elements, Spearman’s footrule is calculated as: Spearman(A, B) = N � n=1 |A[n] − B[n]|, (8) For example, if A = [1, 2, 3] and B = [2, 1, 3], then Spearman(A, B) = |1 − 2| + |2 − 1| + |3 − 3| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9AzT4oBgHgl3EQfA_op/content/2301.00934v1.pdf'} diff --git a/qdAzT4oBgHgl3EQfrP0p/vector_store/index.faiss b/qdAzT4oBgHgl3EQfrP0p/vector_store/index.faiss new file mode 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France +Associate Professor of Quantitative Analysis +Mississippi State University +PO Box 9582, Mississippi State, MS 39762 +Email: sfrance@business.msstate.edu + +Frank G. Adams +Associate Professor of Marketing +Mississippi State University +PO Box 9582, Mississippi State, MS 39762 +Email: fadams@business.msstate.edu + +V. Myles Landers +Assistant Professor of Marketing +Mississippi State University +PO Box 9582, Mississippi State, MS 39762 +Email: vlanders@business.msstate.edu + + + +Worst-Case Resistance Testing: +A Nonresponse Bias Solution for Today’s Behavioral Research Realities + +Abstract +This study proposes a method of nonresponse assessment based on meta-analytical file-drawer +techniques, also known as worst-case resistance testing (WCRT), and suitable for a wide range of +data collection scenarios. A general method is devised to estimate the number of significantly +different nonrespondents it would take to significantly alter the results of an analysis. Estimates +of nonrespondents can be plotted against effect sizes using “n-curves”, with similar interpretation +to p-curves or power curves. Variants of the general method are derived for tests of means and +correlations. A sample using a well-established survey instrument from previous behavioral +research is used to test the method. The results suggest that employing worst-case resistance testing +can be used on its own or in conjunction with wave analysis to precisely flag nonresponse risks. + +Keywords: Nonresponse Bias, Worst-Case Resistance, Hypothesis Testing, Validity Testing + + +1 + +Introduction +All quantitative empirical methods rely on the assumption that the sample participants +represent the population of interest sufficiently to justify extrapolation of findings beyond the +sample measured (Chesney & Obrecht, 2012). However, some portion of the participants +solicited in almost every study do not respond, and as the proportion of those non-respondents +grows larger, the study’s results suffer from potential bias (Boyd & Westfall, 1965). This +participation nonresponse bias is the focus of this paper. +Participant nonresponse bias has been attributed to the variation of characteristics +between respondents and nonrespondents (Deming, 1953), and this variance has the potential to +confound the variance observed between the constructs measured in any given empirical test +(Groves and Peytcheva, 2008) and introduce bias into statistical tests. This bias can be thought +of as a type of selection bias and unlike bias for nonresponse of individual items this bias cannot +be corrected without gathering data from nonrespondents (Berg, 2005). +The growing use of internet surveys for behavioral survey research has changed the +nature of survey response and nonresponse. While scholars have well accepted means of +assessing nonresponse bias – most notably, wave analysis – those methods were developed based +upon physical mail collection of surveys. By contrast, much survey and experimental research +today employs electronically curated samples that can be gathered in hours, or even minutes and +that does not have well defined participant response data. +Accordingly, this study develops a set of methods independent of survey delivery mode +allowing researchers to examine the robustness of statistical tests against participant nonresponse + + +bias1 by calculating the number of cases needed to reverse a statistical test over a range of +different effect sizes. The resulting “n-curves” provide similar insight to similar methods such as +power curves and p-curves and provide a measure of robustness for the results of statistical tests +in situations where participant nonresponse may affect the results and conclusions from such +tests. This study then tests the proposed methods on an empirical survey of customer satisfaction +and shows how these methods can be used on their own or combined with wave analysis to flag +statistical tests where nonresponse bias may be problematic. All data files and code used in the +creation of this paper are available at https://github.com/MDSOPT/WCRT. +Background +In recent years, dedicated efforts have examined practices such as “p-hacking” to recall +the academy to replicable research methods (Simmons et. al, 2011). Similarly, recent failures to +replicate psychological research have been a cause for concern (Stanley et al., 2018), leading to a +call for more transparency in research (Inman et al., 2018). This includes reporting data +collection techniques, power analysis, effect sizes and potential biases that might influence +results. +Arising from a combination of sampling error and coverage error, nonresponse error +results in a sufficient difference between the data sought by a researcher and the data actually +obtained to compromise a study’s validity (Collier & Bienstock, 2007). Conceptually, +nonresponse error holds that the potential responses of subjects who do not answer a solicitation +to participate in a given research study might be different enough from the responses recorded to +alter the findings of the study and higher nonresponse rates can negatively affect the + +1 For the sake of parsimony, as this paper focuses on participant nonresponse issues, subsequent mentions of +nonresponse refer to participant nonresponse rather than item nonresponse. + + +representativeness of a sample (Cook et al., 2000). When records of response data are available, +it is quite easy to assess participant nonresponse, but “there is no magical response rate below +which an observed mean, standard deviation, or correlation becomes automatically invalid” +(Newman 2009, pp. 7). Still, the larger the percentage of the solicited sample measured, the +lower the error resulting from nonresponse bias tends to be (Olson, 2006). +Scholars have developed several procedures to adjust survey results to account for survey +nonresponse bias, as detailed in Table 1 (following Halbesleben & Whitman, 2013), but they all +inherently rest on a key assumption: “…respondents and nonrespondents within a weighting +class have the same values on key variables…” (Groves 2006, pp. 653). Accordingly, attempts to +assess nonresponse bias rely on an assumption that differences causing some solicited subjects to +forego answering a survey are related to how a nonrespondent might react to a study’s constructs +of interest. Based on this assumption, for decades, the most widely used method of assessing +nonresponse bias has been Armstrong and Overton’s wave analysis technique (1977). +Table 1: Summary of Nonresponse Bias Assessments and Remedies +Technique +Description +References +Comparison of +Sample and +Population +Compare demographics of known population +characteristics to collected sample +characteristics. +Armstrong and Overton +(1977); Groves (2006); +Beebe, et al. (2011) +Wave Analysis +Compare early respondent answers to later +respondent answers. +Armstrong and Overton +(1977) +Follow-up +Analysis +Obtain responses from subjects who did not +respond to the original data collection in order +to test for differences. +Aiken (1981), Sosdian +and Sharp (1980) +Bayesian +Analysis +Utilize Bayes rule to estimate nonresponse +data, assuming independence of attributes and +Daniel and Schott (1982) + + +known characteristics across respondents and +nonrespondents +Passive and +Active +Nonresponse +Analysis +Attempt to assess why active nonrespondents +declined to participate through focus groups, +interviews, and surveys about the original data +collection. Resend the survey to address +passive nonrespondents. +Rogelberg and Stanton +(2007); Rogelberg, et al. +(2003); Roth (1994) +Interest-level +Analysis +Include questions about subject interest in the +survey topic among the measured items and +statistically control for interest when analyzing +responses. +Rogelberg and Stanton +(2007); Rogelberg, et al. +(2000) +Benchmarking +Compare sample demographics with those of +other studies of similar phenomena to see if +there are inconsistencies of means or standard +deviations. +Rogelberg and Staton +(2007) +Replication +Conduct multiple surveys using different +samples to assess whether findings remain +consistent. +Rogelberg and Staton +(2007) + +Wave Analysis +Simply put, wave analysis compares relationships between variables observed among +early respondents to a measurement instrument with those observed among later respondents +(Armstrong & Overton, 1977). “The basic assumption … is that subjects who respond less +readily are more like those who do not respond at all than those who do respond readily (i.e., +those who respond sooner and those who need less prodding to answer)” (Kanuk & Berenson, +1975, pg. 449). The method poses that a lack of significant difference between early and late + + +respondents to a research solicitation implies that potential subjects that did not respond do not +represent observations that might alter an analysis’s results. +For all its long-proven utility (over 19,000 citations at this writing), wave analysis was +explicitly built around mail surveys, which generally required considerable periods of time to +collect (Kanuk & Berenson, 1975) and where information on early and late response waves can +easily be found. Studies employing postal mail and citing wave analysis have including follow +up prompts of up to four weeks (Diamantopoulous & Winklhofer, 2001; Mohr & Spekman, +1994; Sirdeshmukh et al. 2002). Even surveys distributed over email have noted time spent +awaiting response from subjects (Pavlou 2003). As of 2007, internet surveys constituted the +majority of surveys, and as of 2020 the vast majority of surveys are completed via the internet +(Daikeler et al., 2020). The “internet age” of surveys has seen a growth of third-party survey +platforms, such as Qualtrics and Prolific, who recruit participants well in advance of any study, +and pay participants fees to complete studies. The resulting participants are more likely to reply +quickly because they have pre-agreed to participate in studies (Qualtrics, 2020). On some +research platforms, such as Prolific (Peer et al., 2017) and the Amazon Mechanical Turk +(Chandler et al., 2019), potential respondents when logging on, will pick from a list of potential +surveys or work to complete. In this situation, unless user click/screen viewing behavior is +analyzed, it is difficult to identify and quantify nonresponses (e.g., Boas et al. 2020; Paolacci et +al. 2010) and the early and late response waves required by wave analysis. +Resampling +A typical means of addressing potential nonresponse bias is to simply resolicit sample +nonrespondents (Aiken, 1981; Hartman, et al. 1986; MacDonald, et al. 2009), often employing +shorter surveys that assess only the items whose constructs are of critical importance to the + + +observed findings, to look for differences from the findings of the original (Lambert & +Harrington, 1990). However, the absence of a specific response rate below which nonresponse +bias is considered problematic (Newman, 2009) implies that supplemental sampling – whether +among the originally solicited group, or from a different group of potential respondents – may +not necessarily address nonresponse bias of a sample relative to the population of interest. A +different potential solution may lie in meta-analysis techniques to address a bias issue known as +the file drawer problem. +Meta-Analysis and the File Draw Problem +The file drawer problem is a term used in meta-analytic literature to describe a +conceptual, but quantifiable sampling bias. Because meta-analyses examine the standardized +results of extant literature, they are presumed to be biased by the tendency of statistically +significant findings to achieve academic publication, and the corollary tendency of non- +significant results of similar phenomena never entering the scholarly body of knowledge +(Rosenberg, 2005). The direst assumptions hold that 95% of contrary findings do not survive the +academic publication process, and that the body of knowledge is, therefore, a victim of Type 1 +error (Rosenthal, 1979). +Rosenthal proposed a solution to the file drawer problem, sometimes known as worst- +case resistance testing (1979), or fail-safe number calculation (Rosenberg, 2005). The technique +calculates the number of studies required to significantly alter an observed mean of effect sizes, +assuming the hypothetical unobserved studies have a collective mean significantly different than +that of the observed effect sizes. As this calculated number of studies increases, the likelihood of +a file drawer bias decreases. In other words, the larger the effect size observed in tests of a given +sample, and/or the less stringent the standard of testing significance, the more hypothetical + + +contradictory cases it would take to cast doubt on the observed findings. The technique has been +used in varying meta-analytic studies including (but by no means limited to) electronic word of +mouth (Babić et al. ,2016), interstitial space impacts on consumer appeal (Sevilla and Townsend +2016), and consumer responses to humanoid robots (Mende et al., 2020). +The nonresponse bias problem is very similar to the file drawer problem in that both seek +to assess a difficult-to-quantify bias of findings stemming from uncollected data presumed to +contradict results based on observed data. It stands to reason that the file drawer solution – +WCRT – should also be efficacious in assessing nonresponse bias. +Methodology +To illustrate how file-draw concepts can be applied to the nonresponse bias problem, the +problem is given in general in terms of the basic NHST (null hypothesis significance test) +paradigm. Though this paradigm has been much criticized (e.g., Gill, 1999; Hubbard & +Armstrong, 2006; Hunter, 1997; Schneider, 2015) it is still by far the predominantly used +framework for building theory in empirical management and social science research. +Furthermore, most alternative approaches proposed to replace NHST also have criticisms, +for example the use of confidence intervals for inference leads to the same “inverse inference” +that is criticized in NHST testing and Bayesian analysis requires specification of prior +distributions, which can be conceptually difficult (e.g., Masson, 2011; Trafimow, 2017). While +at least one journal has banned significance testing (Woolston, 2015), most journals and +scientific associations in the behavior sciences and business disciplines have focused on best +practice to improve the use of NHST results and to put these results into context. +Scholars have advanced several recommendations to improve the implementation of +NHST methods. These include putting p-values into context and avoiding erroneous overly + + +strong conclusions from p-values (Wasserstein & Lazar, 2016), focusing on the magnitude and +size of any statistical effect and incorporating information from prior beliefs (Harvey, 2017), +reporting of descriptive statistics and reporting guidelines for major statistical tests (JCR, 2021), +including detailed graphs and discussions of effects and utilizing robust error statistics (Schwab +et al., 2011), and calculating power values for each statistical test and ensuring that the Type II +error rate (β) is less than 0.05 when making conclusions on a lack of “effect” relative to a null +hypothesis (Baroudi & Orlikowski, 1989; Cashen & Geiger, 2004). +A theme in most of the rules and suggestions described above is the “triangulation” of +NHST results with other metrics to build evidence for hypothesis test conclusions. As such, the +methodology described in this paper fits in with this theme. The aim is to provide a set of +measures of robustness of statistical results to problems caused by nonresponse bias. However, +the methods described can be used beyond the realm of nonresponse bias to examine robustness +to other sources of error, such as the experimental design. +In this study, the WCRT methodology is described using a generic NHST hypothesis +testing procedure. Examples are included for problems with simple hypothesis testing of means +and of correlations, where equations are given for “finding the number of additional studies” +required to negate a conclusion and then models are developed to solve these equations. +The General Model +The general problem is outlined as follows: Consider a situation with a NHST performed +on data collected from a survey. The purpose of the test is to find sufficient evidence to reject a +null hypothesis (H0) in favor of an alternative hypothesis (HA). There is some critical value at +which enough evidence is gathered so that the researcher flips from failing to reject the null +hypothesis to rejecting the null hypothesis. If the researcher finds enough evidence to reject H0, + + +but H0 is in fact true, then the researcher is considered to have committed a Type I error, with a +probability denoted as α. The value of α is usually defined in terms of extreme results in the +distribution of expected sample values in the H0 distribution, which can be denoted as 𝛼 = +𝑃(𝑅|H0), where R is rejection of H0. Given the distribution of H0, H0 is rejected if there is +enough evidence, operationalized by the sample statistic being far enough away from a “null +effect” in a sampling distribution. +A researcher will often make the “opposite assertion”, that given insufficient evidence to +reject H0, one can conclude that H0 is in fact true. However, there is a danger with this assertion +in that researchers may assume a trivial effect without understanding the implications of the +power of the statistical test (Baroudi & Orlikowski, 1999; Cashen & Geiger, 2004; Sawyer & +Ball, 1981). If the researcher fails to reject H0 and in fact HA is true, then the researcher has made +a Type II error, i.e., 𝛽 = 𝑃(𝑅𝑐|H𝐴), where the power of the test is 1 − 𝛽. An issue here is that +HA can take multiple values and that the power varies with the “effect size” difference between +H0 and the HA used to calculate power. Solutions to this issue include calculating power using a +reasonable effect size based on prior studies, standard small, medium, and large effect sizes +(Cohen, 1992), and graphing power values across a range of effect sizes, a “so called” power +curve (Faul et al., 2007). +In the context of nonresponse bias and WCRT, the focus is to find the number of non- +respondents who can reverse a statistical conclusion and use this as a measure of robustness of +the solution. But how is the effect size for these studies chosen? Is it a “zero effect”, the opposite +effect, or a smaller effect in the same direction? The methodology outlined in this paper mirrors +the work described above in choosing effect sizes for power analyses. The number of non- +respondents needed to reverse a statistical test can be calculated for a range of feasible effect + + +sizes, which can be estimated from wave analysis or by examining effect sizes for similar +studies. These values can be plotted, creating an “n curve”, which is similar to curves used for +determining quality bounds for confidence intervals (e.g., Trafimow, 2018) or p-curves used to +map sample sizes for different p-values at different power levels (Simonsohn et al., 2014). +At the core of the analyses in this paper is the idea of a standardized effect size (Cohen +1998). An effect size can be thought of as a quantitative measure of the phenomenon being +studied (Kelley & Preacher, 2012). For example, for a single sample t test, the effect size d, is +given in (1). +𝑑 = 𝑥̅ − 𝜇0 +𝑠 += +(𝑥̅ − 𝜇0) √𝑛 +⁄ +𝑠 √𝑛 +⁄ += 𝑡 +√𝑛 +(1) +, where 𝑥̅ is the sample mean, 𝑠 is the sample standard deviation, 𝜇0 is the hypothesized population +mean, and n is the sample size. This invariance towards n is particularly useful for large sample +size experiments, as effect sizes can put into context results that are significant with only a small +effect size, but a very large sample size (e.g., Coe, 2002). Different statistical tests have different +effect size calculations. For example, effect sizes for the comparison of two group means, such as +Cohen’s d and Glass’s g, have a similar format to the effect size given in (1), while for Pearson’s +correlation, the sample regression coefficient r is often used as a measure of effect size (Hemphill, +2003). +In the context of a WCRT analysis of a NHST test, we define a general effect size , +which can be substituted by the appropriate metric for a specific test (e.g., d for sample mean +tests). Consider the following situations: +1. With a sample size of n1 there is enough evidence to reject H0. There is an effect size 1 +that is associated with the test. We wish to find n2, where this is the number of items or + + +non-respondents with effect size 2 required to negate the result, so that H0 is no longer +rejected. +2. As above, but with a sample size of n1 there is not enough evidence to reject H0 and a +sufficient n2 with effect size 2 required to negate the result, so that H0 is now longer +rejected. +A set of candidate effect sizes needs to be defined for 2. This is key to the methods in +this paper and the appropriate range can be informed by previous research, the results from a +wave analysis of the data, and the effect size 1 (for example, if there is a significant effect, an +effect size greater or equal to 1 and in the same direction is not going to negate the hypothesis +test). For each 2, the procedure will give the n2 value needed to reverse the result of the +statistical test. +Inference for Single Sample t-test +Consider a single sample t-test of a population mean being equal to hypothesized mean +𝜇0. The notation is as per (1) and the null hypothesis is 𝐻𝑜: 𝜇 = 𝜇0. The methodology outlined in +this section covers both two tailed tests where the alternative hypothesis is defined as 𝐻𝑎: 𝜇 ≠ 𝜇0 +and one-tailed tests where the alternative hypothesis can be defined as 𝐻𝑎: 𝜇 > 𝜇0 or 𝐻𝑎: 𝜇 < 𝜇0. +The test statistic derived from the sampling distribution is defined as (2) by rearranging (1). +𝑡 = +(𝑥̅1 − 𝜇0) +𝑠1 +√𝑛1 += 𝑑1√𝑛1 +(2) +Here, the subscript 1 indicates that the sample values are based on the responses, while +the subscript 2 will be used for the sample values for hypothesized nonresponses. The t +distribution has n-1 degrees of freedom and varies with n. Let 𝑡∗ be the critical boundary + + +between rejecting and failing to reject H0. Dependent on n and the strictness of the test (using the +Type I error α), H0 is rejected if |𝑡| > 𝑡∗ = 𝑡𝛼/2 for a two tailed test and 𝑡 > 𝑡∗ = 𝑡𝛼. +Here we consider four different scenarios2. +1. For a one-tailed upper test or a two-tailed test with 𝑡 > 0, H0 is rejected as 𝑡 > 𝑡∗. We wish +to find, for some nonrespondents with effect size 𝑑2, the n2 for required to reverse this +conclusion, so that 𝑡 ≤ 𝑡∗. +2. For a one-tailed upper test or a two-tailed test with 𝑡 > 0, H0 is not rejected as 𝑡 ≤ 𝑡∗. We +wish to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this +conclusion, so that 𝑡 > 𝑡∗. +3. For a one-tailed lower test or a two-tailed test with 𝑡 < 0, H0 is rejected as 𝑡 < 𝑡∗. We wish +to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this +conclusion, so that 𝑡 ≥ 𝑡∗. +4. For a one-tailed lower test or a two-tailed test with 𝑡 < 0, H0 is not rejected as 𝑡 ≥ 𝑡∗. We +wish to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this +conclusion, so that 𝑡 < 𝑡∗. +A simplifying assumption is to assume that +1 +2 +s +s += + , i.e., the nonresponse and response +data have the same standard deviations. However, if the nonresponse data have different +characteristics than the original data then this assumption will not hold. A solution is to set some +range for s2, so that (1 − 𝜃)𝑠1 ≤ 𝑠2 ≤ (1 + 𝜃)𝑠1, where 0 ≤ 𝜃 ≤ 1 and θ is set based on some +prior inferences regarding the data. Given s2, the effect size for the nonresponse data is given in +(3). + +2 The only scenario not covered by the above is the “so-called” type III error scenario (Leventhal, and Huynh 1996), +where the sample mean is in the opposite direction to the population mean. + + +𝑑2 = 𝑥̄2 − 𝜇0 +𝑠2 + +(3) +The sample mean for the nonresponse data is found by rearranging (3) to give (4). +𝑥̄2 = 𝑑2𝑠2 + 𝜇0 +(4) +This value can be used to find the sample mean for the combined response and nonresponse +samples. +𝑥̄𝑐 = 𝑛1𝑥̄1 + 𝑛2𝑥̄2 +𝑛1 + 𝑛2 + +(5) +The pooled standard deviation can be calculated using the meta-analysis formulation given in +Higgins et al. (2019). +𝑠𝑐 = √(𝑛1 − 1)𝑠1 +2 + (𝑛2 − 1)𝑠2 +2 + +𝑛1𝑛2 +𝑛1 + 𝑛2 (𝑥̄1 +2 + 𝑥̄2 +2 + 2𝑥̄1𝑥̄2) +𝑛1 + 𝑛2 − 2 + +(6) +Consider the overall t test with the combined data for scenario 1. We wish to find the lowest n2 for +which 𝑡 ≤ 𝑡∗, and define some small quantity , such that 𝑡 + 𝜀 = 𝑡∗, with 𝜀 ≥ 0, so that 𝑡 = 𝑡∗ − +𝜀. +𝑡∗ − 𝜀 = 𝑥̄𝑐 − 𝜇0 +𝑠𝑐 +√𝑛1 + 𝑛2 + +(7) +Rearrange to put in terms of the total sample size. +√𝑛1 + 𝑛2 = 𝑠𝑐(𝑡∗ − 𝜀) +𝑥̄𝑐 − 𝜇0 + +(8) +Rearrange to put in terms of n2. +𝑛2 = 𝑠𝑐 +2(𝑡∗ − 𝜀)2 +(𝑥̄𝑐 − 𝜇0)2 − 𝑛1 +(9) +The task is to find the smallest integer value of n2 for which 𝜀 ≥ 0. By making minor alterations, +(9) can be used for scenarios (2)-(4). The four scenarios are summarized in Table 2. + + + +Table 2: Scenarios for Single Sample t-test +Scenario Test +Direction +Test Result +Find min{𝒏𝟐} +to make + +Noneresponse +d Bounded +1 +Upper +Significant +Non- +significant +𝜀 ≥ 0 +Upper +2 +Upper +Non-significant +Significant +𝜀 < 0 +Lower +3 +Lower +Significant +Non- +significant +𝜀 ≤ 0 +Lower +4 +Lower +Non-significant +Significant +𝜀 > 0 +Upper + +Each scenario has a direction of test (upper or lower), the result of the test on the +response data, the opposite result, the range of 𝜀 for which the minimum integer n2 is being +found, and how the nonresponse effect size is bounded3. Given that the t-value in (9) is +dependent on 𝑛2, giving a cross-dependency, 𝑛2 cannot be calculated directly. A fixed-point +optimization procedure for finding n2 is given in the Appendix. +A similar process can be followed for two-sample independent sample tests. Sample sizes +can be calculated for Student’s t test (for equal variances) and Welch’s t-test (for unequal +variances) and a two-group measure, such as Glass’s g or Cohen’s d can be used to calculate +effect sizes (Rosenthal & Rubin, 1982). If sample sizes are uneven, some constraints need to be +placed on relative group sizes. For the sake of parsimony, full derivations are not included4. In +addition, similar inference can be used for z tests of means and proportions. + +3 The exact bounds are not given here as there is a nonlinear dependence between sample size and effect size. For +scenarios 1 and 3, there is an upper bound on effect size at which n2 goes to infinity. For scenarios 2 and 4, there is a +lower bound on effect size at which n2 goes to infinity. +4 These are available from the authors on request. + + +Inference for Correlation Test +Consider a situation, where a correlation is being tested for significance. The null +hypothesis is 𝐻𝑜: 𝜌 = 0, where 𝜌 is the population correlation. Standard alternate hypothesis are +𝐻𝑎: 𝜌 > 0 (or 𝜌 < 0) for a one-tailed test and 𝐻𝑎: 𝜌 ≠ 0 for a two tailed test. A population +hypothesis is tested with a Pearson sample correlation coefficient r. The correlation r is +essentially an effect size (Cohen, 1988), with small ( 0.1 ≤ 𝑟 < 0.3), medium (0.3 ≤ 𝑟 < 0.5), +and large (𝑟 ≥ 0.5) effect sizes defined. +There are several different tests for the significance of correlations. The one most +commonly used in meta-analysis involves transforming the correlation 𝑟 ∈ [−1,1] into a z score +using the inverse hyperbolic tangent transformation (Cox, 2008) and is given in (10). +𝑧𝑟1 = 𝑡𝑎𝑛ℎ−1(𝑟1) = 1 +2 𝑙𝑛 {(1 + 𝑟1) +(1 − 𝑟1)} +(10) +, where r1 is the correlation coefficient for the response data. Now, this value is still +essentially an effect size and does not depend on the sample size n. A standard error of +√1 (𝑛 − 3) +⁄ + is defined by Fisher (1921), which can be used to give the z statistic in (11). +𝑧1 = +𝑧𝑟1 +𝑆𝐸(𝑧𝑟1) = +𝑧𝑟1 +√1 (𝑛 − 3) +⁄ + +(11) +Given that the z test is a simple two-way directional test, the four scenarios for finding +the n2 values needed to change a hypothesis test result are similar to the scenarios outlined for +the one sample t-test. The only changes are that “z” replaces “t” for the test statistical and critical +values and that the effect size defined for the nonresponse data is a correlation coefficient r2, +which can be transformed into a z score 𝑧𝑟2 using the same transformation as given in (10). The +z-scores for the response and hypothesized nonresponse data can be combined (Field, 2001; +Hedges & Vevea, 1998; Higgins et al., 2019) using (12). + + +𝑧𝑟𝑐 = (𝑛1 − 3)𝑧𝑟1 + (𝑛2 − 3)𝑧𝑟2 +𝑛1 + 𝑛2 − 6 + +(12) +, which has the standard error given in (13). +𝑆𝐸(𝑧𝑟𝑐) = √ +1 +𝑛1 + 𝑛2 − 6 +(13) +A similar process can be carried out as for the single sample t test. For scenario 1, we wish to find +the lowest n for which 𝑧 ≤ 𝑧∗, where 𝑧∗ is the boundary value for significance and define some +small quantity , such that 𝑧 + 𝜀 = 𝑧∗, with 𝜀 ≥ 0, so that 𝑧 = 𝑧∗ − 𝜀. +𝑧∗ − 𝜀 = +𝑧𝑟𝑐 +𝑆𝐸(𝑧𝑟𝑐) = +𝑧𝑟𝑐 +√ +1 +𝑛1 + 𝑛2 − 6 + +(14) +This equation can be rearranged to give n2. +𝑛2 = (𝑧∗ − 𝜀 +𝑧𝑟𝑐 +) +2 +− 𝑛1 + 6 +(15) +Now n2 can be found in a similar manner to the single sample hypothesis test. The other three +scenarios can be taken from Table 2 (but with r replacing d in the final column). A grid search +optimization procedure for finding n2 is given in the Appendix. +Empirical Example +To assess the efficacy of the proposed WCRT, a simple survey was administered to a +sample curated through Qualtrics. The goal of the survey was not to investigate any substantive +empirical point, but to apply WCRT methods to assess robustness to participant nonresponse bias +for a series of correlation tests. As the retailing scales employed by Szymanski & Henard (2001) +have over 3,000 citations, and pose relatively simple questions, they were judged as liable to +provide stable results, and unlikely to represent confounding factors due to their complexity. + + +The Dataset +The survey includes five different multi-item measurement scales, each of which relates +to some measure of customer satisfaction for a recent retail transaction. Each of the individual +items is measured using a seven-point Likert scale. The full list of scales and items within these +scales is given in Table 3. Overall, there are five different scales, consisting of 19 subitems. The +first three scales deal with the actual shopping experience being evaluated, the fourth scale +examines how this experience impacts behavioral intent, and the fifth is a general scale +measuring retail/shopping enjoyment. Thus, the first three scales should be strongly correlated, +while scale five may have some positive correlation with the other scales (someone who has +positive views of retail shopping is more likely to select a positive shopping experience), but the +level of correlation should be lower. Two of the items (item two on INTENT and item one on +ENJOY) were negative direction items and were reversed. +Table 3: Survey Scale Information +Information +Description +Name +Shopping Experience (EXP) +Prompt +Thinking about this retail shopping experience, please rate your overall +feelings about the shopping experience. +Sub-items +unpleasant:pleasant +dislike very much:like very much +left me feeling bad:left me with a good feeling +Name +Satisfaction (SAT) +Prompt +My overall impression of this retail shopping experience is +Sub-items +Bad:Good +Unfavorable:Favorable +Unsatisfactory:Satisfactory +Negative:Positive +Dislike:Liked +Name +Positive Word of Mouth (PWOM) +Prompt +Thinking about your shopping experience, please rate your agreement with +the following statements. +Sub-items +(All strongly disagree:strongly agree) +I would say positive things about this retailer. +I would recommend this retailer to people I know. + + +I would encourage relatives and friends to do business with this retailer. +Name +Behavioral Intentions (INTENT) +Prompt +Thinking about your shopping experience, please rate your agreement with +the following statements. +Sub-items +(All strongly disagree:strongly agree) +I expect to be coming to this retailer for a long time. +I do not expect to visit this retailer in the future. +I expect my relationship with this retailer to be enduring. +It is likely that I will visit this retailer in the future. +Name +Shopping Enjoyment (ENJOY) +Prompt +Please rate your agreement with the following statements. +Sub-items +(All strongly disagree:strongly agree) +I consider shopping a big hassle. +When traveling, I enjoy visiting new and interesting shops. +I enjoy browsing for things even if I cannot buy them yet. +I often visit shopping malls or markets just for something to do. + +The data were collected via a Qualtrics panel. There were n = 415 fully completed +surveys out of a total of n = 463 surveys. In line with the focus on participant non-response bias, +participant responses with missing participants were removed rather than imputed using a +missing data technique. +Exploratory Data Analysis +As a preface to analyzing the correlation tests using WCRT, some analysis was +performed on the consistency of the rating scale and on the correlations. To examine the +consistency of the summated ratings scales, the Cronbach’s alpha (Cronbach, 1951), was +calculated for each of the summated rating scales. The values are EXP (0.96), SAT (0.99), PWM +(0.96), INTENT (0.78), ENJOY (0.78). From past literature (e.g., Bland & Altman, 1997; +Tavakol & Dennick, 2011), cut-offs for “good” to “excellent” values of alpha range from 0.7- +0.95, so these values are in the correct range. + + +Figure 1: Multi-Item Scale Correlations + +A summary matrix plot of the overall correlations between the values in the summated +rating scales is given in Figure 1. Here, the diagonal values give histogram distributions of the +summated values, the upper triangle of the matrix contains the correlations between the +summated values (*** represents p<0.001 for a statistical test of correlation), and the lower +triangle contains scatterplots, each overlaid with a linear regression best fit line and a confidence +circle for the multivariate mean of the distribution. +Wave Analysis +A simple wave analysis was performed as follows. For this experiment, respondents were +taken from a panel. As noted previously, there are n = 415 fully completed survey forms out of n += 463. For a panel, it is difficult to estimate the number of missing responses, but it is possible to +estimate the percentage of missing participants given reported percentages for previous similar + +5 +10 +15 +20 +25 +30 +35 +10 +15 +20 +25 +EXP.SUM +0.94*** +0.84*** +0.62*** +0.27*** +0 +8 +SAT.SUM +0.84*** +0.63*** +0.25*** +8- +PWOM.SUM +8 +0.26*** +5 +0.73*** +2 +INTENT.SUM +8 +0.24*** +ENJOY.SUM +2 +5 +0 +10 +15 +20 +10 +15 +20 +10 +15 +20 +25 +studies in the literature. For the purpose of this analysis, three scenarios were assumed, one with +50% response, one with 25% response, and one with 10% response. +The wave analysis approach described in Armstrong and Overton (1977), considers two +different waves of responses, an early wave and a late wave, and then a “virtual” wave of +nonresponses. While the responses to the survey were not split into waves, for the purpose of this +illustrative example, it was assumed that the first 50% belong to the early wave and the second +50% belong to the late wave. The three response scenarios give the number of participant +responses for the third wave as 415 (50% response), 1245 (25% response), and 3735 (10% +response). Armstrong and Overton (1977) lay out three methods of calculating values for a third +wave. +1. Assume that the nonresponses have the same mean as the second wave. +2. Assume that the nonresponses are at the same level as the responses at the end of the +second wave. +3. Assume a linear interpolation through the nonresponse third wave. +For the measure of interest, let be the mean values for waves 1 and 2 respectively be 𝐱̅1 +and 𝐱̅2. The number of item values in the three waves are denoted n1, n2, and n3. Wave analysis +aims to give a prediction for 𝐱̅3 in the nonresponse value. For case 1, 𝐱̅3 = 𝐱̅2. Cases 2 and 3 +assume a linear relationship for the variable of interest over time. From Armstrong and Overton +(1977)5, for case 2, 𝐱̅3 is calculated as in (16). +𝐱̅3 = 𝐱̅2 + (𝐱̅2 − 𝐱̅1) +𝑛2 +(𝑛1 + 𝑛2) +(16) + +5 To be consistent with the development of the WCRT method, the calculations are given using group means rather +than upper and lower boundaries, but the calculations are equivalent. + + +Here, a straight line is drawn between the midpoint of group 1 and the midpoint of group 2. The +line is extrapolated to the end of group 2. For the third scenario, the line is extrapolated to the +middle of group 3, giving (17). +𝐱̅3 = 𝐱̅2 + (𝐱̅2 − 𝐱̅1) (𝑛2 + 𝑛3) +(𝑛1 + 𝑛2) +(17) +A wave analysis was performed for each combination of the correlations given in Figure +1 and the three different nonresponse scenarios. The first two wave analysis methods are +independent of the number of nonresponses 𝑛3, but the third is not, so can be calculated for all +three nonresponse scenarios. +Table 4: Wave Analysis Results +Correlation +𝐱̅𝟏 +𝐱̅𝟐 (M1) +M2: End +wave 2 +M3: 415 +(50%) +M3: 1245 +(25%) +M3: 3735 +(10%) +EXP, SAT +0.928 +0.955 +0.969 +0.997 +1.000 +1.000 +EXP, PWOM +0.859 +0.817 +0.797 +0.755 +0.672 +0.422 +EXP, INTENT +0.709 +0.517 +0.421 +0.230 +-0.154 +-1.000 +EXP, ENJOY +0.313 +0.225 +0.181 +0.093 +-0.084 +-0.612 +SAT, PWOM +0.881 +0.797 +0.755 +0.671 +0.504 +0.001 +SAT, INTENT +0.736 +0.498 +0.380 +0.142 +-0.333 +-1.000 +SAT, ENJOY +0.287 +0.204 +0.163 +0.080 +-0.086 +-0.585 +PWOM, INTENT +0.815 +0.625 +0.531 +0.342 +-0.036 +-1.000 +PWOM, ENJOY +0.266 +0.249 +0.240 +0.223 +0.188 +0.085 +INTENT, ENJOY +0.264 +0.210 +0.183 +0.129 +0.021 +-0.304 + +The results of the wave analysis are given in Table 4. Results are given for each of the 10 +possible correlations between the summated rating scales. The first two columns contain the +values of the mean correlation values from waves one and two. The means for the second wave +are taken as the M1 (method 1) estimate of the third wave. The next column contains the M2 +estimate of the value at the end of the second wave and the subsequent columns contain the three +M3 estimates for the three levels of participant response (50%, 25%, and 10%). For the moderate +response scenarios, (50%, 25%), the correlations all stayed within bounds, but for the 10% + + +scenario, several values need to be truncated at either -1 or 1. This shows the difficulty of a linear +interpolation that extends well beyond the range of data. It is likely that as n3 increases, any +change in the dependent variable will lessen. However, the values for 10% response provide +useful “extreme bounds”, which can be utilized by the WCRT procedure. +WCRT Procedure +As previously shown in Figure 1, all multi-item scale correlations are strongly (p<0.001) +significant, with correlations between scales related to the actual shopping experience (EXP, +SAT, POW) being over 0.8, correlations between these scales and the future shopping intention +(INTENT) scale being in the 0.6-0.7 range and the correlations between the general shopping +enjoyment measure and the other scales being in the 0.2-0.3 range. +For each correlation, the WCRT procedure was calculated for effect sizes with +increments of 0.01 ranging from -0.99 to the maximum effect size with a finite n (-0.01) for α +values from 0.01 to 0.1. Selected results are examined in Figures 2 and 3 in what we call “n- +curves”, which are similar to n-curves used to determine sample sizes (e.g., Trafimow 2018) or +the probability of replication (Killeen, 2015) and the previously discussed p-curves for statistical +power (Simonsohn et al., 2014). +For contrast, curves are given for the highest correlation (EXT and SAT) where r1 = 0.94 +and for the lowest correlation (INTENT and JOY) where r1 = 0.24. For each of these +correlations, curves are given for α = 0.05, though any value of α can be chosen. The x-axis +contains the r2 required to negate the significance of the significance test6. In the case of +correlations, due to the asymptotic behavior of the significance test being a tradeoff between the + +6 A similar n-curve could be drawn where the aim is to find n to make a non-significant test significant. + + +overall effect size and n, only negative r2 values give a finite n and the graphs go off to infinity at +approximately r2 = 0. +As the relationship between the value of r and n is strongly exponential, it is difficult to +plot n versus r on a linear scale, so a logarithmic scale is used for n. This makes it more difficult +to read the values of n, but to make up for this, values of n are explicitly given for negatives of +the standard effect sizes defined by Cohen (1988), giving r =- 0.1 (small), r = -0.3 (medium) and +r = -0.5 (large) effect sizes, along with r = -0.7 and r = -0.9. +Looking at Figure 2, which is for an α = 0.05 test for the pair of scales with the strongest +correlation (r1 = 0.94), for a small negative effect (r2 = -0.1), n = 5670 would be required to +negate significance, while for a large negative effect (r2 = -0.5), n = 1175 would be required to +negate significance. This would be very unlikely, given the large negative effect. Even an almost +“complete reversal” of the correlation (r2 = -0.9) would require n = 454 in order to negate +significance. + + +Figure 2: n-Curve for EXP and SAT: α = 0.05. + +The graph in Figure 3 is for an α = 0.05 test for the lowest correlation of r = 0.24 between +INTENT and ENJOY with α = 0.05 and shows much lower values of n. For α = 0.05, for a small +negative effect (r2 = -0.1), n = 427 would be required to negate significance, while for a large +negative effect (r2 = -0.5), n = 103 would be required to negate significance. The extreme r2 = - +0.9 case would require n = 43 to negate significance. + +Effect size and n reguired for non-sig. at alpha=0.05: EXP and SAT (r=0.94) +30 000- +-0.9, n=453.879 +n=757.553 +r=-0.3, n=2021.42 +10000- += 3000- +1000- +300- +-1.00 +-0.75 +0.50 +-0.25 +0.00 +P +Figure 3: n-Curve for INTENT and ENJOY: α = 0.05. + +Combining WCRT with Wave Analysis +In any scenario where response times can be calculated, wave analysis can provide +estimates of sample statistics for nonresponding participants, which can be converted to effect +sizes. These effect sizes can be used to help choose a realistic range of effect sizes in the +outlined WCRT procedure. Accordingly, we propose a method combining wave analysis results +with WCRT to create a set of “warning” metrics for results that may be called in to question by +possible nonresponse bias. An outline of the method is given below. +Assume a situation where a statistical test has been performed, with some level of Type I +error α and there are two possible results; either H0 is rejected in favor of HA or there is not +enough evidence to reject H0. The test will have some measure of effect size (e.g., Cohen’s d for + +Effect size and n reguired for non-sig. at alpha=0.05: INTENT and ENJOY (r=0.24) +1000 +r--0.9, n=42.0255 +n=67.8879 +r--0.5, n=102.661 +F-0.3, n-170.649 +300 +100 +30 +-1.00 +-0.75 +0.50 +-0.25 +0.00 +P +a two-sample test or the sample correlation r for a correlation test). There is some number of +nonresponses n3 (for panel data, multiple scenarios can be given). +1. Calculate the three different wave analysis effect size values: M1: average of second +wave, M2: end of second wave, M3: extrapolation to mean of third (nonresponse) wave. +2. For WCRT, calculate the effect size r needed to reverse the statistical test given the +number of nonresponses n3. This is the inverse procedure of finding n given an effect +size, i.e., for a correlation r, if the calculation of n from r is defined as the function +𝑓(𝑟) = 𝑛 then 𝑓−1(𝑛) = 𝑟. +3. Record if each of the three effect sizes found by wave analysis that will reverse the result +of the statistical test. For example, for positive correlation r that is statistically significant, +if the correlation found by the WCRT is greater than the nonresponse correlation value +predicted by wave analysis then the wave analysis correlation value will reverse the test. +The three wave analysis predictions give different levels of future extrapolations. For M1, +where the predicted nonresponse mean is the mean of the second wave, unless a statistical result +is close to a boundary, it is unlikely that a nonresponse mean effect size value equal to the value +for the second wave will change the result of a statistical test. However, a linear extrapolation for +M3 to the middle of the nonresponse wave for large nonresponse n is liable to change a test +result and the extrapolation is likely to be over-exaggerated, as it is unlikely that the trend from +the 1st wave to the 2nd wave would continue linearly for a large nonresponse wave. Some +damping is likely. However, the M3 scenario can provide a good “worst-case” scenario. +The combined method was applied to the previously discussed correlation example for all +10 correlations, two significance levels (𝛼 = 0.05, 0.01), and the previously discussed +participant nonresponse scenarios (nonresponse n = 415, 1245, 3735). The results are given in + + +Tables 5 to 7, with each table containing one of the three nonresponse scenarios. Each table +contains a row for each of the ten tested correlations. There are columns for the sample +correlation value, the three wave analysis values, and the two WCRT values for the tested values +of the Type I error α. As all correlations are significant and positive, the wave analysis results are +flagged/counted as reversing the result of the statistical test if the correlations are less than the +WCRT value. +Table 5: Combining Wave Analysis and Worst-case Resistance Testing for 50% Response +(Nonresponse n3 = 415) +Correlation +r +r3 (M1) r3 (M2) +r3 (M3) +Wr3 α=0.05 +Wr3 α=0.01 +EXP, SAT +0.94 +0.955 +0.969 +0.997 +-0.924 +-0.917 +EXP, PWOM +0.84 +0.817 +0.797 +0.755 +-0.792 +-0.775 +EXP, INTENT +0.62 +0.517 +0.421 +0.230 +-0.526 +-0.495 +EXP, ENJOY +0.27 +0.225 +0.181 +0.093 +-0.139 +-0.097 +SAT, PWOM +0.84 +0.797 +0.755 +0.671 +-0.795 +-0.779 +SAT, INTENT +0.63 +0.498 +0.380 +0.142 +-0.536 +-0.505 +SAT, ENJOY +0.25 +0.204 +0.163 +0.080 +-0.114 +-0.072 +PWOM, INTENT 0.73 +0.625 +0.531 +0.342 +-0.662 +-0.637 +PWOM, ENJOY +0.26 +0.249 +0.240 +0.223 +-0.128 +-0.085 +INTENT, ENJOY 0.24 +0.210 +0.183 +0.129 +-0.104 +-0.061 + + In Table 5, no values are flagged and none of the wave analysis scenarios will reverse +the result of the statistical test. In part, this is because all the test correlations are quite “strong”. +Even the correlations that include the ENJOY measure (0.24 ≤ r ≤ 0.27), while less than the +other correlations, are strongly significant with a sample size of n = 415. As the nonresponse n3 +increases from 415 to 3735, the magnitude of the correlations found by the inverse WCRT +procedure decrease. This is intuitive, as given that statistical significance is a function of both +effect size and sample size, for a larger sample size, a smaller negative effect is needed to reverse +the results of a statistical test. + + + +Table 6: Combining Wave Analysis and Worst-Case Resistance Testing for 25% Response +(Nonresponse n3 = 1245) +Correlation +r +r3 (M1) r3 (M2) r3 (M3) +Wr3 α=0.05 +Wr3 α=0.01 +EXP, SAT +0.94 +0.955 +0.969 +1.000 +-0.475 +-0.428 +EXP, PWOM +0.84 +0.817 +0.797 +0.672 +-0.326 +-0.308 +EXP, INTENT +0.62 +0.517 +0.421 +-0.154 +-0.173 +-0.154 +EXP, ENJOY +0.27 +0.225 +0.181 +-0.084 +-0.028 (1) +-0.007 (1) +SAT, PWOM +0.84 +0.797 +0.755 +0.504 +-0.329 +-0.310 +SAT, INTENT +0.63 +0.498 +0.380 +-0.333 +-0.178 (1) +-0.158 (1) +SAT, ENJOY +0.25 +0.204 +0.163 +-0.086 +-0.019 (1) +0.001 (1) +PWOM, INTENT 0.73 +0.625 +0.531 +-0.036 +-0.241 +-0.221 +PWOM, ENJOY +0.26 +0.249 +0.240 +0.188 +-0.024 +-0.004 +INTENT, ENJOY 0.24 +0.210 +0.183 +0.021 +-0.016 +0.005 + +In Table 6, the extrapolated r3 for M3 goes outside of the testing “flip” boundaries +defined by WCRT for three correlations, which increases to six correlations for the n3 = 3735 +results given in Table 7. This includes all the “enjoy” correlations except for the “PWOM, +ENJOY” correlation, for which there is only a very slight linear trend. Despite negative linear +trends, the “EXP, PWOM” and “SAT, PWOM” correlations are not flagged, as the correlations +are high relative to the negative linear trends. +Table 7: Combining Wave Analysis and Worst-Case Resistance Testing for 10% Response +(Nonresponse n3 = 3735) +Correlation +r +r3 (M1) r3 (M2) +r3 (M3) Wr3 α=0.05 +Wr3 α=0.01 +EXP, SAT +0.94 +0.955 +0.969 +1.000 +-0.158 +-0.148 +EXP, PWOM +0.84 +0.817 +0.797 +0.422 +-0.100 +-0.089 +EXP, INTENT +0.62 +0.517 +0.421 +-1.000 +-0.046 (1) +-0.035 (1) +EXP, ENJOY +0.27 +0.225 +0.181 +-0.612 +0.003 (1) +0.014 (1) +SAT, PWOM +0.84 +0.797 +0.755 +0.001 +-0.101 +-0.090 +SAT, INTENT +0.63 +0.498 +0.380 +-1.000 +-0.047 (1) +-0.037 (1) +SAT, ENJOY +0.25 +0.204 +0.163 +-0.585 +0.006 (1) +0.017 (1) +PWOM, INTENT 0.73 +0.625 +0.531 +-1.000 +-0.069 (1) +-0.058 (1) +PWOM, ENJOY +0.26 +0.249 +0.240 +0.085 +0.005 +0.015 +INTENT, ENJOY 0.24 +0.210 +0.183 +-0.304 +0.007 (1) +0.018 (1) +Discussion + + +This study has presented a methodology and set of statistical tools for analyzing +nonresponse bias situations. A methodology based on the file-drawer problem and worst-case +resistance testing (WCRT) is given to help researchers quantify and understand the “robustness” +of results with respect to nonresponse bias. Researchers can examine the number of non- +responders to reverse the results of a statistical test for a range of feasible effect sizes for the +nonresponse data. This relationship can be plotted using an “n-curve”. The range of feasible +effect sizes can be decided using evidence from past research, guidance on standard effect sizes, +and the results of a wave analysis. Conversely, researchers can find the effect size needed to +reverse the results of a statistical test for a given number of experimental nonresponses and then +evaluate if these effect sizes are feasible using the guidance described above. +The basic WCRT methodology was developed in this paper as a method for analyzing +robustness towards nonresponse bias. However, the methodology is more generally applicable to +other scenarios. For any situation where there is a statistical test and some idea of possible +“negative effect sizes”, the WCRT methodology can be used to measure robustness. As noted in +the introduction, there is a strong push to improve experimental rigor in the behavioral sciences +and in marketing. An added urgency was added to this process by reports finding a low level of +replicability in behavioral science studies (e.g., Open Science Collaboration 2015; Stanley et al. +2018) and by high-profile behavioral research scandals and retractions (e.g., Inman, et al. 2018; +Stricker and Günther 2019). In addition to the focus on improving statistical rigor described +earlier in the paper (e.g., JCR 2021; Harvey 2017; Schwab et al. 2011; Wasserstein and Lazar +2016), there has been a move towards requiring preregistration of experiments (Simmons et al. +2021), i.e., the process of researchers stating the experimental procedure and expected results +and storing this information externally in a third-party repository, and to improved sharing and + + +availability of research data (Towse et al., 2021). Including the preregistration information along +with a paper submission ensures that the experiment is not altered in an ad-hoc manner to +account for unexpected results. +The methods outlined in this paper can easily be incorporated into the behavioral science +environment outlined above. Possible nonresponse bias should be noted, and procedures should +be outlined for measuring bias. Even in a pure experimental setting, some type of nonresponse +bias may be present; for example, for a student experiment, a certain number of students in a +subject pool could be notified of a study, with only a few participating. When nonresponse bias +is not an issue, WCRT can still be used to help examine the robustness of the results. Gelman +and Loken (2013) noted that even with preregistration and no p-hacking, researchers can still +bend the rules, for example, choosing the regression technique that gives the best results or +choosing whether to use a main effect or interaction effect to justify a hypothesis. Given +continued publication bias towards significant results (e.g., Franco et al. 2014; Harrison et al. +2017), there will always be an incentive to choose the research path to give the most significant +results, in what statisticians sometimes call “the garden of forking paths”. Rules to increase +experimental rigor, such as preregistration, may prune some of these paths, but without being +overly restrictive, cannot prevent researchers finding new paths. This is somewhat analogous to +the situation of accountants finding new workarounds as rules on tax avoidance are strengthened. +In the context outlined above, WCRT could be utilized as a measure of robustness of +results with respect to all possible experimental errors and biases. A range of possible effect sizes +for the nonresponse bias could be derived and combined. Feasible nonresponse effect sizes could +be derived for nonresponses using wave analysis or a similar method and generally by collating +effect sizes the past literature in the area or through a meta-analytic p-curve analysis (Simonsohn + + +et al. 2014). In time, a set of “n” thresholds could be developed to flag results with insufficient +robustness to the factors outlined above. +Limitations and Future Research +This paper develops WCRT methods for correlations and single-sample hypothesis tests7. +To be widely utilized, WCRT methods would need to be developed for a wider range of +statistical tests, such as regression, ANOVA and SEM, as these methods are the most widely +used methods in behavioral research. This is a similar scenario to effect size and power +calculations, where over time, methods have been developed for a wide range of statistical tests. +For the methods to be widely used, it would be important to package them together into a single +cohesive software package, in a similar manner to G*Power (Faul 2007), which has become the +de-facto standard software package for power analysis. +In the modern internet-mediated environment, more surveys are being conducted using +online panels designed to represent certain population characteristics and through co- +working/online hiring platforms, such as the Amazon Mechanical Turk (Kees et al. 2017). +Determining nonresponse in online environments is difficult, as the survey platform recruitment +procedure may be opaque. What exactly constitutes nonresponse in a panel or online working +platform? If a set of respondents are notified about an opportunity, then the number of +nonresponses can be calculated only if the number notified is reported by the platform. In a co- +working platform where respondents search through lists of opportunities, calculating +nonresponse may be more difficult. If views of an opportunity are recorded (e.g., through a +scroll-down list), then some measure of nonresponse of “aware” respondents can be calculated, + +7 Two sample hypothesis tests have also been developed and material is available from the authors on request. + + +but determining how to set a threshold for awareness would be difficult. There has been some +initial work on analyzing nonresponse for the Mechanical Turk for longitudinal studies (Daly +and Nataraajan 2015) and several studies have tried to quantify possible nonresponse bias for +online platforms (Boas et al. 2020; Paolacci et al. 2010). However, there is strong scope for a +systematic analysis of nonresponse for online surveys. Such analysis could include work from +both information systems and experimental standpoints and include aspects such as data +reporting, human-computer interaction, and nonresponse behavior. +The wave analysis method utilized in this paper is a simple linear extrapolation method. +Linear extrapolation may not be reliable outside of the range of the data. It is likely that +significant linear trends would probably “damp” outside of the range of the data, particularly in +situations where there are many non-respondents. This is a reason why damped trend forecasting +methods that give conservative forecasts are often successful (e.g., Armstrong et al. 2015; +Gardner 2015). In the use of wave analysis in the experimental section, this lack of conservatism +is an advantage, as linear extrapolation is used to create worst case bounds for correlations. +However, given the advances in forecasting over the 40 plus years since the introduction of wave +analysis (e.g., Makridakis et al. 2020), there is scope to bring new methodology to bear on wave +analysis and develop methods to improve forecasts of nonresponse bias. + +APPENDIX A: Optimization Algorithms +Optimization Algorithm for Single Sample t-Test Inference +Equation (9) in the main paper cannot be solved outright as both 𝑠𝑐 and 𝑡∗ are dependent +on n2, creating cross-dependencies. However, the equation can be solved using a simple fixed- +point algorithm. + + +1) Utilize an initial starting value of n2 = n1 and call this nOpt. +2) Calculate +2x , +cx , and sc using nOpt. +3) Calculate n2 from equation (9) and store this in variable nCalc. +4) Recalculate nOpt as (nOpt+nCalc)/2. +5) Repeat steps 2-4 until |nOpt-nCalc|<, where  is some pre-set convergence criterion. +In practice, the values of nOpt and nCalc always converge so that |nOpt-nCalc|<. +Optimization Algorithm for Correlation Inference +The fixed-point method used for the previous tests did not converge for the correlation +test, due to Equation (16) in the main paper having both a negative and positive root. Thus, a +divide and conquer optimization method was employed. It takes advantage of the fact that given +a candidate value of zrc, the value of n2 can be calculated by rearranging Equation (12) in the +main paper as follows: +𝑧𝑟𝑐(𝑛1 + 𝑛2 − 6) = (𝑛1 − 3)𝑧𝑟1 + (𝑛2 − 3)𝑧𝑟2 +(A-18) +Collecting n2 terms gives (A-2). +𝑛2(𝑧𝑟𝑐 − 𝑧𝑟2) = (𝑛1 − 3)𝑧𝑟1 − 3𝑧𝑟2 − 𝑧𝑟𝑐(𝑛1 − 6) +(A-19) +Rearranging in terms of n2 gives (A-3). +𝑛2 = (𝑛1 − 3)𝑧𝑟1 − 3𝑧𝑟2 − 𝑧𝑟𝑐(𝑛1 − 6) +(𝑧𝑟𝑐 − 𝑧𝑟2) + +(A-20) +1. The algorithm works by exploring the possible values of zrc, calculating z and then +constraining z towards 𝑧∗ ± 𝜀. Calculate zr1 from r1. Calculate zr2 from r2. For a +nonresponse effect size r2, the steps are as follows: + + +2. From (12) in the main paper, zrc is a linear combination of zr1 and zr2, so lies between +these two values. For cases 1 and 4, set LB = zr1 and UB = zr2; for cases 2 and 3 set LB = +zr2 and UB = zr1. +3. Set +2 +c +LB +UB +zr ++ += + and then use this value of +c +zr to calculate n2, using (A-3). +4. 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Nature News, 519(7541), 9-9. + diff --git a/stE_T4oBgHgl3EQf8xx6/content/tmp_files/load_file.txt b/stE_T4oBgHgl3EQf8xx6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bd99f85f335f9d1179df17101838fd42cebbe9f9 --- /dev/null +++ b/stE_T4oBgHgl3EQf8xx6/content/tmp_files/load_file.txt @@ -0,0 +1,1419 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf,len=1418 +page_content='1 Worst Case Resistance Testing: A Nonresponse Bias Solution for Today’s Behavioral Research Realities Stephen L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' France Associate Professor of Quantitative Analysis Mississippi State University PO Box 9582, Mississippi State, MS 39762 Email: sfrance@business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='edu Frank G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Adams Associate Professor of Marketing Mississippi State University PO Box 9582, Mississippi State, MS 39762 Email: fadams@business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='edu V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Myles Landers Assistant Professor of Marketing Mississippi State University PO Box 9582, Mississippi State, MS 39762 Email: vlanders@business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='edu Worst-Case Resistance Testing: A Nonresponse Bias Solution for Today’s Behavioral Research Realities Abstract This study proposes a method of nonresponse assessment based on meta-analytical file-drawer techniques, also known as worst-case resistance testing (WCRT), and suitable for a wide range of data collection scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A general method is devised to estimate the number of significantly different nonrespondents it would take to significantly alter the results of an analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Estimates of nonrespondents can be plotted against effect sizes using “n-curves”, with similar interpretation to p-curves or power curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Variants of the general method are derived for tests of means and correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A sample using a well-established survey instrument from previous behavioral research is used to test the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The results suggest that employing worst-case resistance testing can be used on its own or in conjunction with wave analysis to precisely flag nonresponse risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Keywords: Nonresponse Bias, Worst-Case Resistance, Hypothesis Testing, Validity Testing 1 Introduction All quantitative empirical methods rely on the assumption that the sample participants represent the population of interest sufficiently to justify extrapolation of findings beyond the sample measured (Chesney & Obrecht, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, some portion of the participants solicited in almost every study do not respond, and as the proportion of those non-respondents grows larger, the study’s results suffer from potential bias (Boyd & Westfall, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This participation nonresponse bias is the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Participant nonresponse bias has been attributed to the variation of characteristics between respondents and nonrespondents (Deming, 1953), and this variance has the potential to confound the variance observed between the constructs measured in any given empirical test (Groves and Peytcheva, 2008) and introduce bias into statistical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This bias can be thought of as a type of selection bias and unlike bias for nonresponse of individual items this bias cannot be corrected without gathering data from nonrespondents (Berg, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The growing use of internet surveys for behavioral survey research has changed the nature of survey response and nonresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' While scholars have well accepted means of assessing nonresponse bias – most notably, wave analysis – those methods were developed based upon physical mail collection of surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' By contrast, much survey and experimental research today employs electronically curated samples that can be gathered in hours, or even minutes and that does not have well defined participant response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Accordingly, this study develops a set of methods independent of survey delivery mode allowing researchers to examine the robustness of statistical tests against participant nonresponse bias1 by calculating the number of cases needed to reverse a statistical test over a range of different effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The resulting “n-curves” provide similar insight to similar methods such as power curves and p-curves and provide a measure of robustness for the results of statistical tests in situations where participant nonresponse may affect the results and conclusions from such tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This study then tests the proposed methods on an empirical survey of customer satisfaction and shows how these methods can be used on their own or combined with wave analysis to flag statistical tests where nonresponse bias may be problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' All data files and code used in the creation of this paper are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='com/MDSOPT/WCRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Background In recent years, dedicated efforts have examined practices such as “p-hacking” to recall the academy to replicable research methods (Simmons et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' al, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Similarly, recent failures to replicate psychological research have been a cause for concern (Stanley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2018), leading to a call for more transparency in research (Inman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This includes reporting data collection techniques, power analysis, effect sizes and potential biases that might influence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Arising from a combination of sampling error and coverage error, nonresponse error results in a sufficient difference between the data sought by a researcher and the data actually obtained to compromise a study’s validity (Collier & Bienstock, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Conceptually, nonresponse error holds that the potential responses of subjects who do not answer a solicitation to participate in a given research study might be different enough from the responses recorded to alter the findings of the study and higher nonresponse rates can negatively affect the 1 For the sake of parsimony, as this paper focuses on participant nonresponse issues, subsequent mentions of nonresponse refer to participant nonresponse rather than item nonresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' representativeness of a sample (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' When records of response data are available, it is quite easy to assess participant nonresponse, but “there is no magical response rate below which an observed mean, standard deviation, or correlation becomes automatically invalid” (Newman 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Still, the larger the percentage of the solicited sample measured, the lower the error resulting from nonresponse bias tends to be (Olson, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Scholars have developed several procedures to adjust survey results to account for survey nonresponse bias, as detailed in Table 1 (following Halbesleben & Whitman, 2013), but they all inherently rest on a key assumption: “…respondents and nonrespondents within a weighting class have the same values on key variables…” (Groves 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 653).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Accordingly, attempts to assess nonresponse bias rely on an assumption that differences causing some solicited subjects to forego answering a survey are related to how a nonrespondent might react to a study’s constructs of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Based on this assumption, for decades, the most widely used method of assessing nonresponse bias has been Armstrong and Overton’s wave analysis technique (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 1: Summary of Nonresponse Bias Assessments and Remedies Technique Description References Comparison of Sample and Population Compare demographics of known population characteristics to collected sample characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Armstrong and Overton (1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Groves (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Beebe, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2011) Wave Analysis Compare early respondent answers to later respondent answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Armstrong and Overton (1977) Follow-up Analysis Obtain responses from subjects who did not respond to the original data collection in order to test for differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Aiken (1981), Sosdian and Sharp (1980) Bayesian Analysis Utilize Bayes rule to estimate nonresponse data, assuming independence of attributes and Daniel and Schott (1982) known characteristics across respondents and nonrespondents Passive and Active Nonresponse Analysis Attempt to assess why active nonrespondents declined to participate through focus groups, interviews, and surveys about the original data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Resend the survey to address passive nonrespondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg and Stanton (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Roth (1994) Interest-level Analysis Include questions about subject interest in the survey topic among the measured items and statistically control for interest when analyzing responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg and Stanton (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2000) Benchmarking Compare sample demographics with those of other studies of similar phenomena to see if there are inconsistencies of means or standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg and Staton (2007) Replication Conduct multiple surveys using different samples to assess whether findings remain consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rogelberg and Staton (2007) Wave Analysis Simply put, wave analysis compares relationships between variables observed among early respondents to a measurement instrument with those observed among later respondents (Armstrong & Overton, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' “The basic assumption … is that subjects who respond less readily are more like those who do not respond at all than those who do respond readily (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', those who respond sooner and those who need less prodding to answer)” (Kanuk & Berenson, 1975, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 449).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The method poses that a lack of significant difference between early and late respondents to a research solicitation implies that potential subjects that did not respond do not represent observations that might alter an analysis’s results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For all its long-proven utility (over 19,000 citations at this writing), wave analysis was explicitly built around mail surveys, which generally required considerable periods of time to collect (Kanuk & Berenson, 1975) and where information on early and late response waves can easily be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Studies employing postal mail and citing wave analysis have including follow up prompts of up to four weeks (Diamantopoulous & Winklhofer, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Mohr & Spekman, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sirdeshmukh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Even surveys distributed over email have noted time spent awaiting response from subjects (Pavlou 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As of 2007, internet surveys constituted the majority of surveys, and as of 2020 the vast majority of surveys are completed via the internet (Daikeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The “internet age” of surveys has seen a growth of third-party survey platforms, such as Qualtrics and Prolific, who recruit participants well in advance of any study, and pay participants fees to complete studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The resulting participants are more likely to reply quickly because they have pre-agreed to participate in studies (Qualtrics, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' On some research platforms, such as Prolific (Peer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2017) and the Amazon Mechanical Turk (Chandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2019), potential respondents when logging on, will pick from a list of potential surveys or work to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In this situation, unless user click/screen viewing behavior is analyzed, it is difficult to identify and quantify nonresponses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Boas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Paolacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2010) and the early and late response waves required by wave analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Resampling A typical means of addressing potential nonresponse bias is to simply resolicit sample nonrespondents (Aiken, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Hartman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' MacDonald, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2009), often employing shorter surveys that assess only the items whose constructs are of critical importance to the observed findings, to look for differences from the findings of the original (Lambert & Harrington, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the absence of a specific response rate below which nonresponse bias is considered problematic (Newman, 2009) implies that supplemental sampling – whether among the originally solicited group, or from a different group of potential respondents – may not necessarily address nonresponse bias of a sample relative to the population of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A different potential solution may lie in meta-analysis techniques to address a bias issue known as the file drawer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Meta-Analysis and the File Draw Problem The file drawer problem is a term used in meta-analytic literature to describe a conceptual, but quantifiable sampling bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Because meta-analyses examine the standardized results of extant literature, they are presumed to be biased by the tendency of statistically significant findings to achieve academic publication, and the corollary tendency of non- significant results of similar phenomena never entering the scholarly body of knowledge (Rosenberg, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The direst assumptions hold that 95% of contrary findings do not survive the academic publication process, and that the body of knowledge is, therefore, a victim of Type 1 error (Rosenthal, 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rosenthal proposed a solution to the file drawer problem, sometimes known as worst- case resistance testing (1979), or fail-safe number calculation (Rosenberg, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The technique calculates the number of studies required to significantly alter an observed mean of effect sizes, assuming the hypothetical unobserved studies have a collective mean significantly different than that of the observed effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As this calculated number of studies increases, the likelihood of a file drawer bias decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In other words, the larger the effect size observed in tests of a given sample, and/or the less stringent the standard of testing significance, the more hypothetical contradictory cases it would take to cast doubt on the observed findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The technique has been used in varying meta-analytic studies including (but by no means limited to) electronic word of mouth (Babić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' ,2016), interstitial space impacts on consumer appeal (Sevilla and Townsend 2016), and consumer responses to humanoid robots (Mende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The nonresponse bias problem is very similar to the file drawer problem in that both seek to assess a difficult-to-quantify bias of findings stemming from uncollected data presumed to contradict results based on observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' It stands to reason that the file drawer solution – WCRT – should also be efficacious in assessing nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Methodology To illustrate how file-draw concepts can be applied to the nonresponse bias problem, the problem is given in general in terms of the basic NHST (null hypothesis significance test) paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Though this paradigm has been much criticized (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Gill, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Hubbard & Armstrong, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Hunter, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Schneider, 2015) it is still by far the predominantly used framework for building theory in empirical management and social science research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Furthermore, most alternative approaches proposed to replace NHST also have criticisms, for example the use of confidence intervals for inference leads to the same “inverse inference” that is criticized in NHST testing and Bayesian analysis requires specification of prior distributions, which can be conceptually difficult (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Masson, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Trafimow, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' While at least one journal has banned significance testing (Woolston, 2015), most journals and scientific associations in the behavior sciences and business disciplines have focused on best practice to improve the use of NHST results and to put these results into context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Scholars have advanced several recommendations to improve the implementation of NHST methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' These include putting p-values into context and avoiding erroneous overly strong conclusions from p-values (Wasserstein & Lazar, 2016), focusing on the magnitude and size of any statistical effect and incorporating information from prior beliefs (Harvey, 2017), reporting of descriptive statistics and reporting guidelines for major statistical tests (JCR, 2021), including detailed graphs and discussions of effects and utilizing robust error statistics (Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2011), and calculating power values for each statistical test and ensuring that the Type II error rate (β) is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 when making conclusions on a lack of “effect” relative to a null hypothesis (Baroudi & Orlikowski, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Cashen & Geiger, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A theme in most of the rules and suggestions described above is the “triangulation” of NHST results with other metrics to build evidence for hypothesis test conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As such, the methodology described in this paper fits in with this theme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The aim is to provide a set of measures of robustness of statistical results to problems caused by nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the methods described can be used beyond the realm of nonresponse bias to examine robustness to other sources of error, such as the experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In this study, the WCRT methodology is described using a generic NHST hypothesis testing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Examples are included for problems with simple hypothesis testing of means and of correlations, where equations are given for “finding the number of additional studies” required to negate a conclusion and then models are developed to solve these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The General Model The general problem is outlined as follows: Consider a situation with a NHST performed on data collected from a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The purpose of the test is to find sufficient evidence to reject a null hypothesis (H0) in favor of an alternative hypothesis (HA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There is some critical value at which enough evidence is gathered so that the researcher flips from failing to reject the null hypothesis to rejecting the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If the researcher finds enough evidence to reject H0, but H0 is in fact true, then the researcher is considered to have committed a Type I error, with a probability denoted as α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The value of α is usually defined in terms of extreme results in the distribution of expected sample values in the H0 distribution, which can be denoted as 𝛼 = 𝑃(𝑅|H0), where R is rejection of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Given the distribution of H0, H0 is rejected if there is enough evidence, operationalized by the sample statistic being far enough away from a “null effect” in a sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A researcher will often make the “opposite assertion”, that given insufficient evidence to reject H0, one can conclude that H0 is in fact true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, there is a danger with this assertion in that researchers may assume a trivial effect without understanding the implications of the power of the statistical test (Baroudi & Orlikowski, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Cashen & Geiger, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sawyer & Ball, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If the researcher fails to reject H0 and in fact HA is true, then the researcher has made a Type II error, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 𝛽 = 𝑃(𝑅𝑐|H𝐴), where the power of the test is 1 − 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' An issue here is that HA can take multiple values and that the power varies with the “effect size” difference between H0 and the HA used to calculate power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Solutions to this issue include calculating power using a reasonable effect size based on prior studies, standard small, medium, and large effect sizes (Cohen, 1992), and graphing power values across a range of effect sizes, a “so called” power curve (Faul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the context of nonresponse bias and WCRT, the focus is to find the number of non- respondents who can reverse a statistical conclusion and use this as a measure of robustness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' But how is the effect size for these studies chosen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Is it a “zero effect”, the opposite effect, or a smaller effect in the same direction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The methodology outlined in this paper mirrors the work described above in choosing effect sizes for power analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The number of non- respondents needed to reverse a statistical test can be calculated for a range of feasible effect sizes, which can be estimated from wave analysis or by examining effect sizes for similar studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' These values can be plotted, creating an “n curve”, which is similar to curves used for determining quality bounds for confidence intervals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Trafimow, 2018) or p-curves used to map sample sizes for different p-values at different power levels (Simonsohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' At the core of the analyses in this paper is the idea of a standardized effect size (Cohen 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' An effect size can be thought of as a quantitative measure of the phenomenon being studied (Kelley & Preacher, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For example, for a single sample t test, the effect size d, is given in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑑 = 𝑥̅ − 𝜇0 𝑠 = (𝑥̅ − 𝜇0) √𝑛 ⁄ 𝑠 √𝑛 ⁄ = 𝑡 √𝑛 (1) , where 𝑥̅ is the sample mean, 𝑠 is the sample standard deviation, 𝜇0 is the hypothesized population mean, and n is the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This invariance towards n is particularly useful for large sample size experiments, as effect sizes can put into context results that are significant with only a small effect size, but a very large sample size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Coe, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Different statistical tests have different effect size calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For example, effect sizes for the comparison of two group means, such as Cohen’s d and Glass’s g, have a similar format to the effect size given in (1), while for Pearson’s correlation, the sample regression coefficient r is often used as a measure of effect size (Hemphill, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the context of a WCRT analysis of a NHST test, we define a general effect size \uf066, which can be substituted by the appropriate metric for a specific test (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', d for sample mean tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Consider the following situations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' With a sample size of n1 there is enough evidence to reject H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There is an effect size \uf0661 that is associated with the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find n2, where this is the number of items or non-respondents with effect size \uf0662 required to negate the result, so that H0 is no longer rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As above, but with a sample size of n1 there is not enough evidence to reject H0 and a sufficient n2 with effect size \uf0662 required to negate the result, so that H0 is now longer rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A set of candidate effect sizes needs to be defined for \uf0662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is key to the methods in this paper and the appropriate range can be informed by previous research, the results from a wave analysis of the data, and the effect size \uf0661 (for example, if there is a significant effect, an effect size greater or equal to \uf0661 and in the same direction is not going to negate the hypothesis test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For each \uf0662, the procedure will give the n2 value needed to reverse the result of the statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Inference for Single Sample t-test Consider a single sample t-test of a population mean being equal to hypothesized mean 𝜇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The notation is as per (1) and the null hypothesis is 𝐻𝑜: 𝜇 = 𝜇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The methodology outlined in this section covers both two tailed tests where the alternative hypothesis is defined as 𝐻𝑎: 𝜇 ≠ 𝜇0 and one-tailed tests where the alternative hypothesis can be defined as 𝐻𝑎: 𝜇 > 𝜇0 or 𝐻𝑎: 𝜇 < 𝜇0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The test statistic derived from the sampling distribution is defined as (2) by rearranging (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑡 = (𝑥̅1 − 𝜇0) 𝑠1 √𝑛1 = 𝑑1√𝑛1 (2) Here, the subscript 1 indicates that the sample values are based on the responses, while the subscript 2 will be used for the sample values for hypothesized nonresponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The t distribution has n-1 degrees of freedom and varies with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Let 𝑡∗ be the critical boundary between rejecting and failing to reject H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Dependent on n and the strictness of the test (using the Type I error α), H0 is rejected if |𝑡| > 𝑡∗ = 𝑡𝛼/2 for a two tailed test and 𝑡 > 𝑡∗ = 𝑡𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Here we consider four different scenarios2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a one-tailed upper test or a two-tailed test with 𝑡 > 0, H0 is rejected as 𝑡 > 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find, for some nonrespondents with effect size 𝑑2, the n2 for required to reverse this conclusion, so that 𝑡 ≤ 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a one-tailed upper test or a two-tailed test with 𝑡 > 0, H0 is not rejected as 𝑡 ≤ 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this conclusion, so that 𝑡 > 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a one-tailed lower test or a two-tailed test with 𝑡 < 0, H0 is rejected as 𝑡 < 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this conclusion, so that 𝑡 ≥ 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a one-tailed lower test or a two-tailed test with 𝑡 < 0, H0 is not rejected as 𝑡 ≥ 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find, for some nonrespondents with effect size 𝑑2, the n2 required to reverse this conclusion, so that 𝑡 < 𝑡∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A simplifying assumption is to assume that 1 2 s s = , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', the nonresponse and response data have the same standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, if the nonresponse data have different characteristics than the original data then this assumption will not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A solution is to set some range for s2, so that (1 − 𝜃)𝑠1 ≤ 𝑠2 ≤ (1 + 𝜃)𝑠1, where 0 ≤ 𝜃 ≤ 1 and θ is set based on some prior inferences regarding the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Given s2, the effect size for the nonresponse data is given in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2 The only scenario not covered by the above is the “so-called” type III error scenario (Leventhal, and Huynh 1996), where the sample mean is in the opposite direction to the population mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑑2 = 𝑥̄2 − 𝜇0 𝑠2 (3) The sample mean for the nonresponse data is found by rearranging (3) to give (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑥̄2 = 𝑑2𝑠2 + 𝜇0 (4) This value can be used to find the sample mean for the combined response and nonresponse samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑥̄𝑐 = 𝑛1𝑥̄1 + 𝑛2𝑥̄2 𝑛1 + 𝑛2 (5) The pooled standard deviation can be calculated using the meta-analysis formulation given in Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑠𝑐 = √(𝑛1 − 1)𝑠1 2 + (𝑛2 − 1)𝑠2 2 + 𝑛1𝑛2 𝑛1 + 𝑛2 (𝑥̄1 2 + 𝑥̄2 2 + 2𝑥̄1𝑥̄2) 𝑛1 + 𝑛2 − 2 (6) Consider the overall t test with the combined data for scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' We wish to find the lowest n2 for which 𝑡 ≤ 𝑡∗, and define some small quantity \uf065, such that 𝑡 + 𝜀 = 𝑡∗, with 𝜀 ≥ 0, so that 𝑡 = 𝑡∗ − 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑡∗ − 𝜀 = 𝑥̄𝑐 − 𝜇0 𝑠𝑐 √𝑛1 + 𝑛2 (7) Rearrange to put in terms of the total sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' √𝑛1 + 𝑛2 = 𝑠𝑐(𝑡∗ − 𝜀) 𝑥̄𝑐 − 𝜇0 (8) Rearrange to put in terms of n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑛2 = 𝑠𝑐 2(𝑡∗ − 𝜀)2 (𝑥̄𝑐 − 𝜇0)2 − 𝑛1 (9) The task is to find the smallest integer value of n2 for which 𝜀 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' By making minor alterations, (9) can be used for scenarios (2)-(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The four scenarios are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 2: Scenarios for Single Sample t-test Scenario Test Direction Test Result Find min{𝒏𝟐} to make \uf065 Noneresponse d Bounded 1 Upper Significant Non- significant 𝜀 ≥ 0 Upper 2 Upper Non-significant Significant 𝜀 < 0 Lower 3 Lower Significant Non- significant 𝜀 ≤ 0 Lower 4 Lower Non-significant Significant 𝜀 > 0 Upper Each scenario has a direction of test (upper or lower),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' the result of the test on the response data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' the opposite result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' the range of 𝜀 for which the minimum integer n2 is being found,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' and how the nonresponse effect size is bounded3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Given that the t-value in (9) is dependent on 𝑛2, giving a cross-dependency, 𝑛2 cannot be calculated directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A fixed-point optimization procedure for finding n2 is given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A similar process can be followed for two-sample independent sample tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sample sizes can be calculated for Student’s t test (for equal variances) and Welch’s t-test (for unequal variances) and a two-group measure, such as Glass’s g or Cohen’s d can be used to calculate effect sizes (Rosenthal & Rubin, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If sample sizes are uneven, some constraints need to be placed on relative group sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the sake of parsimony, full derivations are not included4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In addition, similar inference can be used for z tests of means and proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3 The exact bounds are not given here as there is a nonlinear dependence between sample size and effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For scenarios 1 and 3, there is an upper bound on effect size at which n2 goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For scenarios 2 and 4, there is a lower bound on effect size at which n2 goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 4 These are available from the authors on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Inference for Correlation Test Consider a situation, where a correlation is being tested for significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The null hypothesis is 𝐻𝑜: 𝜌 = 0, where 𝜌 is the population correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Standard alternate hypothesis are 𝐻𝑎: 𝜌 > 0 (or 𝜌 < 0) for a one-tailed test and 𝐻𝑎: 𝜌 ≠ 0 for a two tailed test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A population hypothesis is tested with a Pearson sample correlation coefficient r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The correlation r is essentially an effect size (Cohen, 1988), with small ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='1 ≤ 𝑟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3), medium (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3 ≤ 𝑟 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5), and large (𝑟 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5) effect sizes defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There are several different tests for the significance of correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The one most commonly used in meta-analysis involves transforming the correlation 𝑟 ∈ [−1,1] into a z score using the inverse hyperbolic tangent transformation (Cox, 2008) and is given in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑧𝑟1 = 𝑡𝑎𝑛ℎ−1(𝑟1) = 1 2 𝑙𝑛 {(1 + 𝑟1) (1 − 𝑟1)} (10) , where r1 is the correlation coefficient for the response data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Now, this value is still essentially an effect size and does not depend on the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A standard error of √1 (𝑛 − 3) ⁄ is defined by Fisher (1921), which can be used to give the z statistic in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑧1 = 𝑧𝑟1 𝑆𝐸(𝑧𝑟1) = 𝑧𝑟1 √1 (𝑛 − 3) ⁄ (11) Given that the z test is a simple two-way directional test, the four scenarios for finding the n2 values needed to change a hypothesis test result are similar to the scenarios outlined for the one sample t-test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The only changes are that “z” replaces “t” for the test statistical and critical values and that the effect size defined for the nonresponse data is a correlation coefficient r2, which can be transformed into a z score 𝑧𝑟2 using the same transformation as given in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The z-scores for the response and hypothesized nonresponse data can be combined (Field, 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Hedges & Vevea, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Higgins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2019) using (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑧𝑟𝑐 = (𝑛1 − 3)𝑧𝑟1 + (𝑛2 − 3)𝑧𝑟2 𝑛1 + 𝑛2 − 6 (12) , which has the standard error given in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑆𝐸(𝑧𝑟𝑐) = √ 1 𝑛1 + 𝑛2 − 6 (13) A similar process can be carried out as for the single sample t test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For scenario 1, we wish to find the lowest n for which 𝑧 ≤ 𝑧∗, where 𝑧∗ is the boundary value for significance and define some small quantity \uf065, such that 𝑧 + 𝜀 = 𝑧∗, with 𝜀 ≥ 0, so that 𝑧 = 𝑧∗ − 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑧∗ − 𝜀 = 𝑧𝑟𝑐 𝑆𝐸(𝑧𝑟𝑐) = 𝑧𝑟𝑐 √ 1 𝑛1 + 𝑛2 − 6 (14) This equation can be rearranged to give n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑛2 = (𝑧∗ − 𝜀 𝑧𝑟𝑐 ) 2 − 𝑛1 + 6 (15) Now n2 can be found in a similar manner to the single sample hypothesis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The other three scenarios can be taken from Table 2 (but with r replacing d in the final column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A grid search optimization procedure for finding n2 is given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Empirical Example To assess the efficacy of the proposed WCRT, a simple survey was administered to a sample curated through Qualtrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The goal of the survey was not to investigate any substantive empirical point, but to apply WCRT methods to assess robustness to participant nonresponse bias for a series of correlation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As the retailing scales employed by Szymanski & Henard (2001) have over 3,000 citations, and pose relatively simple questions, they were judged as liable to provide stable results, and unlikely to represent confounding factors due to their complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The Dataset The survey includes five different multi-item measurement scales, each of which relates to some measure of customer satisfaction for a recent retail transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Each of the individual items is measured using a seven-point Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The full list of scales and items within these scales is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Overall, there are five different scales, consisting of 19 subitems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The first three scales deal with the actual shopping experience being evaluated, the fourth scale examines how this experience impacts behavioral intent, and the fifth is a general scale measuring retail/shopping enjoyment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Thus, the first three scales should be strongly correlated, while scale five may have some positive correlation with the other scales (someone who has positive views of retail shopping is more likely to select a positive shopping experience), but the level of correlation should be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Two of the items (item two on INTENT and item one on ENJOY) were negative direction items and were reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 3: Survey Scale Information Information Description Name Shopping Experience (EXP) Prompt Thinking about this retail shopping experience, please rate your overall feelings about the shopping experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sub-items unpleasant:pleasant dislike very much:like very much left me feeling bad:left me with a good feeling Name Satisfaction (SAT) Prompt My overall impression of this retail shopping experience is Sub-items Bad:Good Unfavorable:Favorable Unsatisfactory:Satisfactory Negative:Positive Dislike:Liked Name Positive Word of Mouth (PWOM) Prompt Thinking about your shopping experience, please rate your agreement with the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sub-items (All strongly disagree:strongly agree) I would say positive things about this retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I would recommend this retailer to people I know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I would encourage relatives and friends to do business with this retailer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Name Behavioral Intentions (INTENT) Prompt Thinking about your shopping experience, please rate your agreement with the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sub-items (All strongly disagree:strongly agree) I expect to be coming to this retailer for a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I do not expect to visit this retailer in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I expect my relationship with this retailer to be enduring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' It is likely that I will visit this retailer in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Name Shopping Enjoyment (ENJOY) Prompt Please rate your agreement with the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Sub-items (All strongly disagree:strongly agree) I consider shopping a big hassle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' When traveling, I enjoy visiting new and interesting shops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I enjoy browsing for things even if I cannot buy them yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' I often visit shopping malls or markets just for something to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The data were collected via a Qualtrics panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There were n = 415 fully completed surveys out of a total of n = 463 surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In line with the focus on participant non-response bias, participant responses with missing participants were removed rather than imputed using a missing data technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Exploratory Data Analysis As a preface to analyzing the correlation tests using WCRT, some analysis was performed on the consistency of the rating scale and on the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' To examine the consistency of the summated ratings scales, the Cronbach’s alpha (Cronbach, 1951), was calculated for each of the summated rating scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The values are EXP (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='96), SAT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='99), PWM (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='96), INTENT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='78), ENJOY (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' From past literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Bland & Altman, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Tavakol & Dennick, 2011), cut-offs for “good” to “excellent” values of alpha range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='7- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='95, so these values are in the correct range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Figure 1: Multi-Item Scale Correlations A summary matrix plot of the overall correlations between the values in the summated rating scales is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Here, the diagonal values give histogram distributions of the summated values, the upper triangle of the matrix contains the correlations between the summated values (*** represents p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='001 for a statistical test of correlation), and the lower triangle contains scatterplots, each overlaid with a linear regression best fit line and a confidence circle for the multivariate mean of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Wave Analysis A simple wave analysis was performed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For this experiment, respondents were taken from a panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As noted previously, there are n = 415 fully completed survey forms out of n = 463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a panel, it is difficult to estimate the number of missing responses, but it is possible to estimate the percentage of missing participants given reported percentages for previous similar 5 10 15 20 25 30 35 10 15 20 25 EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='SUM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='84*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='62*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='27*** 0 8 SAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='SUM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='84*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='63*** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='25*** 8- PWOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='SUM 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='26*** 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='73*** 2 INTENT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='SUM 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24*** ENJOY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='SUM 2 5 0 10 15 20 10 15 20 10 15 20 25 studies in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the purpose of this analysis, three scenarios were assumed, one with 50% response, one with 25% response, and one with 10% response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The wave analysis approach described in Armstrong and Overton (1977), considers two different waves of responses, an early wave and a late wave, and then a “virtual” wave of nonresponses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' While the responses to the survey were not split into waves, for the purpose of this illustrative example, it was assumed that the first 50% belong to the early wave and the second 50% belong to the late wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The three response scenarios give the number of participant responses for the third wave as 415 (50% response), 1245 (25% response), and 3735 (10% response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Armstrong and Overton (1977) lay out three methods of calculating values for a third wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Assume that the nonresponses have the same mean as the second wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Assume that the nonresponses are at the same level as the responses at the end of the second wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Assume a linear interpolation through the nonresponse third wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the measure of interest, let be the mean values for waves 1 and 2 respectively be 𝐱̅1 and 𝐱̅2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The number of item values in the three waves are denoted n1, n2, and n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Wave analysis aims to give a prediction for 𝐱̅3 in the nonresponse value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For case 1, 𝐱̅3 = 𝐱̅2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Cases 2 and 3 assume a linear relationship for the variable of interest over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' From Armstrong and Overton (1977)5, for case 2, 𝐱̅3 is calculated as in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝐱̅3 = 𝐱̅2 + (𝐱̅2 − 𝐱̅1) 𝑛2 (𝑛1 + 𝑛2) (16) 5 To be consistent with the development of the WCRT method, the calculations are given using group means rather than upper and lower boundaries, but the calculations are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Here, a straight line is drawn between the midpoint of group 1 and the midpoint of group 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The line is extrapolated to the end of group 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the third scenario, the line is extrapolated to the middle of group 3, giving (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝐱̅3 = 𝐱̅2 + (𝐱̅2 − 𝐱̅1) (𝑛2 + 𝑛3) (𝑛1 + 𝑛2) (17) A wave analysis was performed for each combination of the correlations given in Figure 1 and the three different nonresponse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The first two wave analysis methods are independent of the number of nonresponses 𝑛3, but the third is not, so can be calculated for all three nonresponse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 4: Wave Analysis Results Correlation 𝐱̅𝟏 𝐱̅𝟐 (M1) M2: End wave 2 M3: 415 (50%) M3: 1245 (25%) M3: 3735 (10%) EXP, SAT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='997 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 EXP, PWOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='422 EXP, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='154 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 EXP, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='612 SAT, PWOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='001 SAT, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 SAT, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='585 PWOM, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='036 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 PWOM, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='085 INTENT, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='304 The results of the wave analysis are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Results are given for each of the 10 possible correlations between the summated rating scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The first two columns contain the values of the mean correlation values from waves one and two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The means for the second wave are taken as the M1 (method 1) estimate of the third wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The next column contains the M2 estimate of the value at the end of the second wave and the subsequent columns contain the three M3 estimates for the three levels of participant response (50%, 25%, and 10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the moderate response scenarios, (50%, 25%), the correlations all stayed within bounds, but for the 10% scenario, several values need to be truncated at either -1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This shows the difficulty of a linear interpolation that extends well beyond the range of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' It is likely that as n3 increases, any change in the dependent variable will lessen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the values for 10% response provide useful “extreme bounds”, which can be utilized by the WCRT procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' WCRT Procedure As previously shown in Figure 1, all multi-item scale correlations are strongly (p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='001) significant, with correlations between scales related to the actual shopping experience (EXP, SAT, POW) being over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='8, correlations between these scales and the future shopping intention (INTENT) scale being in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='6-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='7 range and the correlations between the general shopping enjoyment measure and the other scales being in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For each correlation, the WCRT procedure was calculated for effect sizes with increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01 ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='99 to the maximum effect size with a finite n (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01) for α values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Selected results are examined in Figures 2 and 3 in what we call “n- curves”, which are similar to n-curves used to determine sample sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Trafimow 2018) or the probability of replication (Killeen, 2015) and the previously discussed p-curves for statistical power (Simonsohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For contrast, curves are given for the highest correlation (EXT and SAT) where r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94 and for the lowest correlation (INTENT and JOY) where r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For each of these correlations, curves are given for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05, though any value of α can be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The x-axis contains the r2 required to negate the significance of the significance test6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the case of correlations, due to the asymptotic behavior of the significance test being a tradeoff between the 6 A similar n-curve could be drawn where the aim is to find n to make a non-significant test significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' overall effect size and n, only negative r2 values give a finite n and the graphs go off to infinity at approximately r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As the relationship between the value of r and n is strongly exponential, it is difficult to plot n versus r on a linear scale, so a logarithmic scale is used for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This makes it more difficult to read the values of n, but to make up for this, values of n are explicitly given for negatives of the standard effect sizes defined by Cohen (1988), giving r =- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='1 (small), r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3 (medium) and r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5 (large) effect sizes, along with r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='7 and r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Looking at Figure 2, which is for an α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 test for the pair of scales with the strongest correlation (r1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94), for a small negative effect (r2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='1), n = 5670 would be required to negate significance, while for a large negative effect (r2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5), n = 1175 would be required to negate significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This would be very unlikely, given the large negative effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Even an almost “complete reversal” of the correlation (r2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='9) would require n = 454 in order to negate significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Figure 2: n-Curve for EXP and SAT: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The graph in Figure 3 is for an α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 test for the lowest correlation of r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24 between INTENT and ENJOY with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 and shows much lower values of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05, for a small negative effect (r2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='1), n = 427 would be required to negate significance, while for a large negative effect (r2 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5), n = 103 would be required to negate significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The extreme r2 = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='9 case would require n = 43 to negate significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Effect size and n reguired for non-sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' at alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05: EXP and SAT (r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94) 30 000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='9, n=453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='879 n=757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='553 r=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3, n=2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='42 10000- = 3000- 1000- 300- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='00 P Figure 3: n-Curve for INTENT and ENJOY: α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Combining WCRT with Wave Analysis In any scenario where response times can be calculated, wave analysis can provide estimates of sample statistics for nonresponding participants, which can be converted to effect sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' These effect sizes can be used to help choose a realistic range of effect sizes in the outlined WCRT procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Accordingly, we propose a method combining wave analysis results with WCRT to create a set of “warning” metrics for results that may be called in to question by possible nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' An outline of the method is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Assume a situation where a statistical test has been performed, with some level of Type I error α and there are two possible results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' either H0 is rejected in favor of HA or there is not enough evidence to reject H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The test will have some measure of effect size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Cohen’s d for Effect size and n reguired for non-sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' at alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05: INTENT and ENJOY (r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24) 1000 r--0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='9, n=42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='0255 n=67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='8879 r--0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='5, n=102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='661 F-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='3, n-170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='649 300 100 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='00 P a two-sample test or the sample correlation r for a correlation test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There is some number of nonresponses n3 (for panel data, multiple scenarios can be given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Calculate the three different wave analysis effect size values: M1: average of second wave, M2: end of second wave, M3: extrapolation to mean of third (nonresponse) wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For WCRT, calculate the effect size r needed to reverse the statistical test given the number of nonresponses n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is the inverse procedure of finding n given an effect size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', for a correlation r, if the calculation of n from r is defined as the function 𝑓(𝑟) = 𝑛 then 𝑓−1(𝑛) = 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Record if each of the three effect sizes found by wave analysis that will reverse the result of the statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For example, for positive correlation r that is statistically significant, if the correlation found by the WCRT is greater than the nonresponse correlation value predicted by wave analysis then the wave analysis correlation value will reverse the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The three wave analysis predictions give different levels of future extrapolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For M1, where the predicted nonresponse mean is the mean of the second wave, unless a statistical result is close to a boundary, it is unlikely that a nonresponse mean effect size value equal to the value for the second wave will change the result of a statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, a linear extrapolation for M3 to the middle of the nonresponse wave for large nonresponse n is liable to change a test result and the extrapolation is likely to be over-exaggerated, as it is unlikely that the trend from the 1st wave to the 2nd wave would continue linearly for a large nonresponse wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Some damping is likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the M3 scenario can provide a good “worst-case” scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The combined method was applied to the previously discussed correlation example for all 10 correlations, two significance levels (𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01), and the previously discussed participant nonresponse scenarios (nonresponse n = 415, 1245, 3735).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The results are given in Tables 5 to 7, with each table containing one of the three nonresponse scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Each table contains a row for each of the ten tested correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There are columns for the sample correlation value, the three wave analysis values, and the two WCRT values for the tested values of the Type I error α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As all correlations are significant and positive, the wave analysis results are flagged/counted as reversing the result of the statistical test if the correlations are less than the WCRT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 5: Combining Wave Analysis and Worst-case Resistance Testing for 50% Response (Nonresponse n3 = 415) Correlation r r3 (M1) r3 (M2) r3 (M3) Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01 EXP, SAT 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='061 In Table 5, no values are flagged and none of the wave analysis scenarios will reverse the result of the statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In part, this is because all the test correlations are quite “strong”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Even the correlations that include the ENJOY measure (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24 ≤ r ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='27), while less than the other correlations, are strongly significant with a sample size of n = 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As the nonresponse n3 increases from 415 to 3735, the magnitude of the correlations found by the inverse WCRT procedure decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is intuitive, as given that statistical significance is a function of both effect size and sample size, for a larger sample size, a smaller negative effect is needed to reverse the results of a statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 6: Combining Wave Analysis and Worst-Case Resistance Testing for 25% Response (Nonresponse n3 = 1245) Correlation r r3 (M1) r3 (M2) r3 (M3) Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01 EXP, SAT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='221 PWOM, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='004 INTENT, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='210 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='005 In Table 6, the extrapolated r3 for M3 goes outside of the testing “flip” boundaries defined by WCRT for three correlations, which increases to six correlations for the n3 = 3735 results given in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This includes all the “enjoy” correlations except for the “PWOM, ENJOY” correlation, for which there is only a very slight linear trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Despite negative linear trends, the “EXP, PWOM” and “SAT, PWOM” correlations are not flagged, as the correlations are high relative to the negative linear trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Table 7: Combining Wave Analysis and Worst-Case Resistance Testing for 10% Response (Nonresponse n3 = 3735) Correlation r r3 (M1) r3 (M2) r3 (M3) Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='05 Wr3 α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='01 EXP, SAT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='969 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='148 EXP, PWOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='089 EXP, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='421 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='046 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='035 (1) EXP, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='003 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='014 (1) SAT, PWOM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='090 SAT, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='380 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='047 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='037 (1) SAT, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='006 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='017 (1) PWOM, INTENT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='531 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='069 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='058 (1) PWOM, ENJOY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} 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+page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='007 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='018 (1) Discussion This study has presented a methodology and set of statistical tools for analyzing nonresponse bias situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A methodology based on the file-drawer problem and worst-case resistance testing (WCRT) is given to help researchers quantify and understand the “robustness” of results with respect to nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Researchers can examine the number of non- responders to reverse the results of a statistical test for a range of feasible effect sizes for the nonresponse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This relationship can be plotted using an “n-curve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The range of feasible effect sizes can be decided using evidence from past research, guidance on standard effect sizes, and the results of a wave analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Conversely, researchers can find the effect size needed to reverse the results of a statistical test for a given number of experimental nonresponses and then evaluate if these effect sizes are feasible using the guidance described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The basic WCRT methodology was developed in this paper as a method for analyzing robustness towards nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the methodology is more generally applicable to other scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For any situation where there is a statistical test and some idea of possible “negative effect sizes”, the WCRT methodology can be used to measure robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' As noted in the introduction, there is a strong push to improve experimental rigor in the behavioral sciences and in marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' An added urgency was added to this process by reports finding a low level of replicability in behavioral science studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Open Science Collaboration 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Stanley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2018) and by high-profile behavioral research scandals and retractions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Inman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Stricker and Günther 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In addition to the focus on improving statistical rigor described earlier in the paper (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', JCR 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Harvey 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Schwab et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Wasserstein and Lazar 2016), there has been a move towards requiring preregistration of experiments (Simmons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', the process of researchers stating the experimental procedure and expected results and storing this information externally in a third-party repository, and to improved sharing and availability of research data (Towse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Including the preregistration information along with a paper submission ensures that the experiment is not altered in an ad-hoc manner to account for unexpected results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The methods outlined in this paper can easily be incorporated into the behavioral science environment outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Possible nonresponse bias should be noted, and procedures should be outlined for measuring bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Even in a pure experimental setting, some type of nonresponse bias may be present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' for example, for a student experiment, a certain number of students in a subject pool could be notified of a study, with only a few participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' When nonresponse bias is not an issue, WCRT can still be used to help examine the robustness of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Gelman and Loken (2013) noted that even with preregistration and no p-hacking, researchers can still bend the rules, for example, choosing the regression technique that gives the best results or choosing whether to use a main effect or interaction effect to justify a hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Given continued publication bias towards significant results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2017), there will always be an incentive to choose the research path to give the most significant results, in what statisticians sometimes call “the garden of forking paths”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Rules to increase experimental rigor, such as preregistration, may prune some of these paths, but without being overly restrictive, cannot prevent researchers finding new paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is somewhat analogous to the situation of accountants finding new workarounds as rules on tax avoidance are strengthened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the context outlined above, WCRT could be utilized as a measure of robustness of results with respect to all possible experimental errors and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A range of possible effect sizes for the nonresponse bias could be derived and combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Feasible nonresponse effect sizes could be derived for nonresponses using wave analysis or a similar method and generally by collating effect sizes the past literature in the area or through a meta-analytic p-curve analysis (Simonsohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In time, a set of “n” thresholds could be developed to flag results with insufficient robustness to the factors outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Limitations and Future Research This paper develops WCRT methods for correlations and single-sample hypothesis tests7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' To be widely utilized, WCRT methods would need to be developed for a wider range of statistical tests, such as regression, ANOVA and SEM, as these methods are the most widely used methods in behavioral research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is a similar scenario to effect size and power calculations, where over time, methods have been developed for a wide range of statistical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For the methods to be widely used, it would be important to package them together into a single cohesive software package, in a similar manner to G*Power (Faul 2007), which has become the de-facto standard software package for power analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the modern internet-mediated environment, more surveys are being conducted using online panels designed to represent certain population characteristics and through co- working/online hiring platforms, such as the Amazon Mechanical Turk (Kees et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Determining nonresponse in online environments is difficult, as the survey platform recruitment procedure may be opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' What exactly constitutes nonresponse in a panel or online working platform?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If a set of respondents are notified about an opportunity, then the number of nonresponses can be calculated only if the number notified is reported by the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In a co- working platform where respondents search through lists of opportunities, calculating nonresponse may be more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If views of an opportunity are recorded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', through a scroll-down list), then some measure of nonresponse of “aware” respondents can be calculated, 7 Two sample hypothesis tests have also been developed and material is available from the authors on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' but determining how to set a threshold for awareness would be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' There has been some initial work on analyzing nonresponse for the Mechanical Turk for longitudinal studies (Daly and Nataraajan 2015) and several studies have tried to quantify possible nonresponse bias for online platforms (Boas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Paolacci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, there is strong scope for a systematic analysis of nonresponse for online surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Such analysis could include work from both information systems and experimental standpoints and include aspects such as data reporting, human-computer interaction, and nonresponse behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The wave analysis method utilized in this paper is a simple linear extrapolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Linear extrapolation may not be reliable outside of the range of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' It is likely that significant linear trends would probably “damp” outside of the range of the data, particularly in situations where there are many non-respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' This is a reason why damped trend forecasting methods that give conservative forecasts are often successful (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Gardner 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In the use of wave analysis in the experimental section, this lack of conservatism is an advantage, as linear extrapolation is used to create worst case bounds for correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, given the advances in forecasting over the 40 plus years since the introduction of wave analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Makridakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2020), there is scope to bring new methodology to bear on wave analysis and develop methods to improve forecasts of nonresponse bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' APPENDIX A: Optimization Algorithms Optimization Algorithm for Single Sample t-Test Inference Equation (9) in the main paper cannot be solved outright as both 𝑠𝑐 and 𝑡∗ are dependent on n2, creating cross-dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' However, the equation can be solved using a simple fixed- point algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 1) Utilize an initial starting value of n2 = n1 and call this nOpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 2) Calculate 2x , cx , and sc using nOpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3) Calculate n2 from equation (9) and store this in variable nCalc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 4) Recalculate nOpt as (nOpt+nCalc)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 5) Repeat steps 2-4 until |nOpt-nCalc|<\uf064, where \uf064 is some pre-set convergence criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' In practice, the values of nOpt and nCalc always converge so that |nOpt-nCalc|<\uf064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Optimization Algorithm for Correlation Inference The fixed-point method used for the previous tests did not converge for the correlation test, due to Equation (16) in the main paper having both a negative and positive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Thus, a divide and conquer optimization method was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' It takes advantage of the fact that given a candidate value of zrc, the value of n2 can be calculated by rearranging Equation (12) in the main paper as follows: 𝑧𝑟𝑐(𝑛1 + 𝑛2 − 6) = (𝑛1 − 3)𝑧𝑟1 + (𝑛2 − 3)𝑧𝑟2 (A-18) Collecting n2 terms gives (A-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑛2(𝑧𝑟𝑐 − 𝑧𝑟2) = (𝑛1 − 3)𝑧𝑟1 − 3𝑧𝑟2 − 𝑧𝑟𝑐(𝑛1 − 6) (A-19) Rearranging in terms of n2 gives (A-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 𝑛2 = (𝑛1 − 3)𝑧𝑟1 − 3𝑧𝑟2 − 𝑧𝑟𝑐(𝑛1 − 6) (𝑧𝑟𝑐 − 𝑧𝑟2) (A-20) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The algorithm works by exploring the possible values of zrc, calculating z and then constraining z towards 𝑧∗ ± 𝜀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Calculate zr1 from r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Calculate zr2 from r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For a nonresponse effect size r2, the steps are as follows: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' From (12) in the main paper, zrc is a linear combination of zr1 and zr2, so lies between these two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' For cases 1 and 4, set LB = zr1 and UB = zr2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' for cases 2 and 3 set LB = zr2 and UB = zr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Set 2 c LB UB zr + = and then use this value of c zr to calculate n2, using (A-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Calculate the value of 1 2 6 c z zr n n = + − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Now, if 𝑧 < 𝑧∗, set LB = c zr else set UB = c zr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If UB LB \uf064 − \uf03c , where \uf064 is a convergence criterion then exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Otherwise go to step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' If the value of 𝑧∗ is not in the range of the initial [LB,UB] this indicates that there is no possible n2 for the selected r2 that can give a z that reaches the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Typically, for case 1 and 4, when increasing effect size, this occurs 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Journal of the Academy of Marketing Science, 29(1), 16-35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Towse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', Ellis, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', & Towse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Opening Pandora’s Box: Peeking inside Psychology’s data sharing practices, and seven recommendations for change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Behavior Research Methods, 53(4), 1455-1468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Tavakol, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', & Dennick, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=" Making sense of Cronbach's alpha." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' International Journal of Medical Education, 2, 53-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Trafimow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Why it is problematic to calculate probabilities of findings given range null hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Open Journal of Statistics, 7(3), 483-499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Trafimow, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Confidence intervals, precision and confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' New Ideas in Psychology 5048-5053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Wasserstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=', & Lazar, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The ASA Statement on p-Values: Context, Process, and Purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' The American Statistician, 70(2), 129-133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Woolston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Psychology journal bans P values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} +page_content=' Nature News, 519(7541), 9-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf'} diff --git a/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/2301.03039v1.pdf.txt b/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/2301.03039v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aabcfa30c4a0fd132bb45984dadc1d64d3824eb8 --- /dev/null +++ b/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/2301.03039v1.pdf.txt @@ -0,0 +1,991 @@ +1 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +Equivalence of Two Expressions of Principal Line + +Cheng-Yen Hsu, Hsin-Yi Chen and Jen-Hui Chuang + +Abstract—Geometry-based camera calibration using principal +line is more precise and robust than calibration using optimization +approaches; therefore, several researches try to re-derive the +principal line from different views of 2D projective geometry to +increase alternatives of the calibration process. In this report, +algebraical equivalence of two expressions of principal line, one +derived w.r.t homography and the other using for two sets of +orthogonal vanishing points, is proved. Moreover, the extension of +the second expression to incorporate infinite vanishing point is +carried out with simple mathematics. + +Index Terms—Camera calibration, homography, principal line, +vanishing points + +I. INTRODUCTION +eometry-based camera calibration is more precise and +robust than traditional calibration approaches based on +algebraic optimization [1-3] due to the consideration of +camera parameters individually. Chen [4] exploited geometric +relationship between a vanishing line on image plane and a +plane which contains optical center and principal point (PP), +and is perpendicular to object plane, to sequentially derive three +rotation angles, three translation distances and a focal length. +However, the use of non-planar calibrated target, limit its usage. +Chuang et al. [5-7] consider similar geometry but formally +define principal line (PL) as the axis of symmetry of the 3D +plane containing a checkerboard pattern (CB plane), i.e., a +symmetric image pattern corresponds to a symmetric pattern on +the CB plane. +Fig. 1 illustrates the geometry of a PL, wherein both x-axis +and X-axis are parallel to the intersection of the two plane and +y-axis correspond to the PL. As each PL thus defined for each +camera-pattern pose will pass through the camera principal +point (PP), the PP can then be estimated by finding the +intersection of linearly independent PLs, possibly from a least +square solution. Moreover, outliers in checkerboard patterns +can also be identified easily in the early stage of calibration and +removed from the subsequent estimation of other camera +parameters. + +Figure 1. The geometry of a PL. +Wang et al. [8] rederive the same closed-from solution of PL +for light field camera calibration. The same geometric +relationship among image plane, a rectangle plane, and PL is +considered with a different projection model, i.e., a new +homography is derived according to projection model of the +standard light field camera. Although the derivation of PL starts +from line equations of four edges of a rectangular pattern, the +final results are exactly the same as [7]. Higher accuracy is also +demonstrated with the final results obtained from their PL- +based calibration algorithm. +In [9], Yang and Zhao derive a different closed-from +expression of PL for orthogonal vanishing points. Without +knowing the homography between image plane and CB plane, +as in [7], two sets of orthogonal vanishing points projected from +the 3D space are used for PL derivation, with experimental +results showing comparable accuracy in the calibration results. +However, no proof of the equivalence of two closed-form +expressions of PL are elaborated in [9]. Moreover, the +expression in [9] is derived for finite orthogonal vanishing +points (nonparallel image lines), while there is no such +constraint in [7]. +In this report, to address the above issues, the two PL +expressions are first reviewed in Sec. II, followed by the proof +of their equivalence in Sec. III. Then, simple extensions of the +PL expression in [9] for infinite orthogonal vanishing points are +provided in Sec. IV, before some concluding remarks are given +in Sec. V. +II. RELATED WORK +The principal line introduced in [7] plays an important role +in the camera calibration, as a PL can be obtained for each +checkerboard pattern and will pass through the principal point +of the camera. Therefore, the PP can be determined as the +intersection of a set of linearly independent PLs. Based on +mathematical derivations provided in [7], if the homographic +relationship between a CB plane and the image plane is +represented by the homography matrix, +𝐻 = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] +(1) + +the principal line can be expressed as + +(−ℎ1ℎ8 + ℎ2ℎ7)𝑢 + (−ℎ4ℎ8 + ℎ5ℎ7)𝑣 + 𝑐 = 0, +(2) + +where 𝑐 = +− (ℎ2 +2 + ℎ5 +2 − ℎ1 +2 − ℎ4 +2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 +2 − ℎ8 +2) +ℎ7 +2 + ℎ8 +2 + + +G + +Aa=45°2 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +On the other hand, a different expression of PL is also derived +in [9] based on orthogonal vanishing points (OVPs) of a planar +scene. In particular, given the first set of two OVPs (Pv1 = (m1, +n1, 1) and Pv2 = (m2, n2, 1)), and the second set (Pv3 = (m3, n3, 1), +and Pv4 = (m4, n4, 1)), the PL can be expressed as + +(𝑚2 − 𝑚1)𝑢 + (𝑛2 − 𝑛1)𝑣 + 𝑐 = 0, +(3) + +where 𝑐 = +−[(𝑚2 − 𝑚1) +𝑚1𝑚2 − 𝑚3𝑚4 +𝑚1 + 𝑚2 − 𝑚3 − 𝑚4 ++ (𝑛2 − 𝑛1) +𝑛1𝑛2 − 𝑛3𝑛4 +𝑛1 + 𝑛2 − 𝑛3 − 𝑛4 +] +In this report, the equivalence of the two expressions of PL, +i.e., Equations (2) and (3), will first be derived in the next +section, for two sets of (finite) orthogonal vanishing points of +two squares on the CB plane, as shown in Fig. 2, following +procedure outlined in [9] for a homographic transformation +between image plane and CB plane. Similar derivations for +more general situations will be considered in Sec. IV. +A +B +C +D +E +F +(0,0) +(1,0) +(0,1) +(-1,1) +(-1,0) +(1,1) +X +Y +I +II + +Figure 2. Two unit squares used in this report to prove the +equivalence of (2) and (3). +III. SHOWING THE EQUIVALENCE OF (2) AND (3) +A. Obtaining 𝑃𝑣1 and 𝑃𝑣2 +Consider the homogeneous coordinates of the four vertices of +Square I, i.e., A = (0,0,1), B = (1,0,1), C = (1,1,1), and D = +(0,1,1). Based on the homography matrix given in (1), the +homogeneous coordinates of images of these vertices can be +expressed as + +𝐴′ = +[ + + + + ℎ3 +ℎ9 +ℎ6 +ℎ9 +1 ] + + + + + , 𝐵′ = +[ + + + + ℎ1 + ℎ3 +ℎ7 + ℎ9 +ℎ4 + ℎ6 +ℎ7 + ℎ9 +1 +] + + + + + , +𝐶′ = +[ + + + + ℎ1 + ℎ2 + ℎ3 +ℎ7 + ℎ8 + ℎ9 +ℎ4 + ℎ5 + ℎ6 +ℎ7 + ℎ8 + ℎ9 +1 +] + + + + + , 𝐷′ = +[ + + + + ℎ2 + ℎ3 +ℎ8 + ℎ9 +ℎ5 + ℎ6 +ℎ8 + ℎ9 +1 +] + + + + +. +(4) + + +In this subsection, it is assumed that P𝑣1 is the intersection of +the 𝐴′𝐵′ +⃡ and 𝐶′𝐷′ +⃡ and P𝑣2 is the intersection of the 𝐴′𝐷′ +⃡ and +𝐵′𝐶′ +⃡ . For P𝑣1, we first calculate the directional vector + +𝐴′𝐵′ + = +[ + + + ℎ1ℎ9 − ℎ3ℎ7 +ℎ9(ℎ7 + ℎ9) +ℎ4ℎ9 − ℎ6ℎ7 +ℎ9(ℎ7 + ℎ9)] + + + +. +(5) + +Assume 𝐴′𝐵′ + exists, both ℎ9 and (ℎ7+ℎ9) will be nonzero, and +we have + +𝐴′𝐵′ + = [ℎ1ℎ9 − ℎ3ℎ7 +ℎ4ℎ9 − ℎ6ℎ7] , +(6) + +and the line equation of 𝐴′𝐵′ +⃡ can be expressed as + +(ℎ4ℎ9 − ℎ6ℎ7)𝑢 + (−ℎ1ℎ9 + ℎ3ℎ7)𝑣 + 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡1 = 0 +(7) +By substituting A′ in equation (4) into (7), we will have + +Constant1 = −ℎ3ℎ4 + ℎ1ℎ6. +(8) + +Similarly, the line equation of 𝐶’𝐷’ +⃡ can be expressed as + +(ℎ5ℎ7 + ℎ6ℎ7 − ℎ4ℎ8 − ℎ4ℎ9)𝑢 ++(−ℎ2ℎ7 − ℎ3ℎ7 + ℎ1ℎ8 + ℎ1ℎ9)𝑣 + 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡2 = 0. (9) + +By substituting D′ in equation (4) into (9), we will have + +Constant2 = ℎ2ℎ4 + ℎ3ℎ4 − ℎ1ℎ5 − ℎ1ℎ6 +(10) + +Therefore, P𝑣1 can be obtained as the intersection of the two +lines represented by (7) and (9), e.g., by evaluating the outer +product + +𝑃𝑣1 = 𝐴′𝐵′ +⃡ × 𝐶′𝐷′ +⃡ = | +𝑖 +𝑗 +𝑘 +𝑎1 +𝑏1 +𝑐1 +𝑎2 +𝑏2 +𝑐2 +| , +(11) + +where ai, bi, and ci are the coefficients in (7) and (9). Then, the +three components in (11) can be evaluated as + +𝑀1 = 𝑏1𝑐2 − 𝑏2𝑐1 += ℎ1(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 ++ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += ℎ1 ∙ det(H), +(12) + +𝑁1 = 𝑐1𝑎2 − 𝑐2𝑎1 += ℎ4(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 ++ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += ℎ4 ∙ det(H), +(13) + +and + 𝑘1 = 𝑎1𝑏2 − 𝑎2𝑏1 += ℎ7(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 ++ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += ℎ7 ∙ det(H), +(14) + +or, if 𝑘1 ≠ 0, + +3 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +𝑃𝑣1 = (𝑚1, 𝑛1, 1) = (ℎ1 +ℎ7 +, ℎ4 +ℎ7 +, 1) . +(15) + +Note that if no three vertices of Square I are colinear, we will +have det(H)≠0 as the homographic transformation is bijection. +Besides, if 𝑃𝑣1 does not locate at infinity we will have ℎ7≠0. +As for obtaining the location of Pv2, the exact procedure used +to derive (7) and (9) can be employed, and line equations of +𝐴′𝐷′ +⃡ and 𝐵′𝐶′ +⃡ can be found as + +(ℎ4ℎ9 − ℎ6ℎ7)𝑢 + (−ℎ1ℎ9 + ℎ3ℎ7)𝑣 − ℎ3ℎ5 + ℎ2ℎ6 = 0 + +(16) +and +(ℎ5ℎ7 + ℎ5ℎ9 − ℎ4ℎ8 − ℎ6ℎ8)𝑢 ++(−ℎ2ℎ7 − ℎ2ℎ9 + ℎ1ℎ8 + ℎ3ℎ8)𝑣 +−ℎ1ℎ5 − ℎ3ℎ5 + ℎ2ℎ4 + ℎ2ℎ6 = 0, +(17) + +respectively. + Thus, the three components of the intersection of 𝐴′𝐷′ +⃡ and +𝐵′𝐶′ +⃡ can be obtained, similar to that formulated in (11), as + +𝑀2 = ℎ2(ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 + ℎ3ℎ4ℎ8 +−ℎ3ℎ5ℎ7 + ℎ2ℎ6ℎ7 + ℎ1ℎ6ℎ8) += ℎ2 ∙ det(H), +(18) + +𝑁2 = ℎ5(−ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8 + ℎ2ℎ6ℎ7 ++ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 − ℎ1ℎ6ℎ8) += ℎ5 ∙ det(H), +(19) + +and +𝑘2 = ℎ8(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +−ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += ℎ8 ∙ det(H), +(20) + +Or, if 𝑘2 ≠ 0, +𝑃𝑣2 = (𝑚2, 𝑛2, 1) = (ℎ2 +ℎ8 +, ℎ5 +ℎ8 +, 1) . +(21) +B. Obtaining 𝑃𝑣3 and 𝑃𝑣4 +Consider Square II in Fig. 2, and homogeneous coordinates +of its two vertices E = (−1,1,1) and F = (−1,0,1). Based on the +homography matrix in (1), the homogeneous coordinates of +images of these two vertices can be expressed as + +𝐸′ = +[ + + + + −ℎ1 + ℎ2 + ℎ3 +−ℎ7 + ℎ8 + ℎ9 +−ℎ4 + ℎ5 + ℎ6 +−ℎ7 + ℎ8 + ℎ9 +1 +] + + + + +, 𝐹′ = +[ + + + + −ℎ1 + ℎ3 +ℎ7 + ℎ9 +−ℎ4 + ℎ6 +ℎ7 + ℎ9 +1 +] + + + + +. +(22) + +Therefore, another pair of orthogonal vanishing points can be +identified as the intersection of 𝐴′𝐶′ +⃡ and 𝐷′𝐹′ +⃡ (P𝑣3) and the +intersection of 𝐴′𝐸′ +⃡ and 𝐵′𝐷′ +⃡ (P𝑣4). +By procedure adopted earlier to find other line equations, line +equations of 𝐷′𝐹′ +⃡ and 𝐴′𝐶′ +⃡ can be found as +(−ℎ5ℎ7 + ℎ5ℎ9 − ℎ6ℎ7+ℎ4ℎ8+ℎ4ℎ9 − ℎ6ℎ8)𝑢 ++(−ℎ1ℎ8 − ℎ1ℎ9 + ℎ3ℎ8+ℎ2ℎ7 − ℎ2ℎ9+ℎ3ℎ7)𝑣 ++(−ℎ2ℎ4+ℎ2ℎ6 − ℎ3ℎ4+ℎ1ℎ5 + ℎ1ℎ6−ℎ3ℎ5) = 0 (23) + +and +(ℎ6ℎ7 + ℎ6ℎ8 − ℎ4ℎ9−ℎ5ℎ9)𝑢 ++(ℎ1ℎ9 + ℎ2ℎ9 − ℎ3ℎ7 − ℎ3ℎ8)𝑣 ++(ℎ3ℎ4 + ℎ5ℎ5 − ℎ1ℎ6−ℎ2ℎ6) = 0, +(24) + +respectively, and the outer product in (11) can again be adopted +to find the three components of P𝑣3 as + +𝑀3 = −(ℎ1 + ℎ2)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 ++ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += −(ℎ1 + ℎ2) ∙ det(H), +(25) + +𝑁3 = −(ℎ4 + ℎ5)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +−ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += −(ℎ4 + ℎ5) ∙ det(H), +(26) + +and +𝑘3 = −(ℎ7 + ℎ8)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +−ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += −(ℎ7 + ℎ8) ∙ det(H), +(27) + +or +𝑃𝑣3 = (𝑚3, 𝑛3, 1) = (ℎ1 + ℎ2 +ℎ7 + ℎ8 +, ℎ4 + ℎ5 +ℎ7 + ℎ8 +, 1) . +(28) + +Similar to the derivation of (23) and (24), we can obtain line +equations of 𝐴′𝐸′ +⃡ and 𝐵′𝐷′ +⃡ as + +(ℎ4ℎ9 + ℎ6ℎ8 − ℎ6ℎ7 − ℎ5ℎ9)𝑢 ++(ℎ3ℎ7 + ℎ2ℎ9 − ℎ1ℎ9−ℎ3ℎ8)𝑣 ++(ℎ1ℎ6+ℎ3ℎ5 − ℎ3ℎ4−ℎ2ℎ6) = 0 +(29) + + +and + +(ℎ4ℎ8 + ℎ4ℎ9+ℎ6ℎ8 − ℎ5ℎ7−ℎ5ℎ9 − ℎ6ℎ7)𝑢 ++(ℎ2ℎ7 + ℎ2ℎ9 + ℎ3ℎ7−ℎ1ℎ8 − ℎ1ℎ9−ℎ3ℎ8)𝑣 ++(ℎ1ℎ5+ℎ1ℎ6 + ℎ3ℎ5−ℎ2ℎ4−ℎ2ℎ6−ℎ3ℎ4) = 0, +(30) + +respectively, and the three components of P𝑣4 as + +𝑀4 = (ℎ1 − ℎ2)(ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 − ℎ3ℎ5ℎ7 ++ℎ3ℎ4ℎ8 + ℎ1ℎ5ℎ9 + ℎ2ℎ4ℎ9) += (ℎ1 − ℎ2)(𝑆) = (ℎ1 − ℎ2) ∙ det(H) , +(31) + +𝑁4 = (ℎ4 − ℎ5)(ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 − ℎ1ℎ6ℎ8 ++ℎ2ℎ6ℎ7 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) += (ℎ4 − ℎ4)(𝑆) = (ℎ4 − ℎ5) ∙ det(H) , +(32) + +and + +𝑘4 = (ℎ7 − ℎ8)(ℎ3ℎ4ℎ8 − ℎ3ℎ5ℎ7 − ℎ2ℎ4ℎ9 ++ℎ1ℎ5ℎ9 − ℎ1ℎ6ℎ8 + ℎ2ℎ6ℎ7) += (ℎ7 − ℎ8)(𝑆) = (ℎ7 − ℎ8) ∙ det(H), +(33) + + +4 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +or +𝑃𝑣4 = (𝑚4, 𝑛4, 1) = (ℎ1 − ℎ2 +ℎ7 − ℎ8 +, ℎ4 − ℎ5 +ℎ7 − ℎ8 +, 1). (34) + + +C. The equivalence of (2) and (3) +To find the principal line according to the formulation +elaborated in [9], equations of the four vanishing points, i.e., +(15), (21), (28), and (34), can be substituted into (3) directly, or + +(ℎ2ℎ7 − ℎ1ℎ8 +ℎ7ℎ8 +) 𝑢 + (ℎ5ℎ7 − ℎ4ℎ8 +ℎ7ℎ8 +) 𝑣 + 𝑐 = 0, +(35) + +with +𝑐 = −{(ℎ2 +ℎ8 +− ℎ1 +ℎ7 +) [ +(ℎ1 +ℎ7 ∙ ℎ2 +ℎ8) − (ℎ1 + ℎ2 +ℎ7 + ℎ8) (ℎ1 − ℎ2 +ℎ7 − ℎ8) +ℎ1 +ℎ7 + ℎ2 +ℎ8 − ℎ1 + ℎ2 +ℎ7 + ℎ8 − ℎ1 − ℎ2 +ℎ7 − ℎ8 +] + + (ℎ5 +ℎ8 +− ℎ4 +ℎ7 +) [ +(ℎ4 +ℎ7 ∙ ℎ5 +ℎ8) − (ℎ4 + ℎ5 +ℎ7 + ℎ8) (ℎ4 − ℎ5 +ℎ7 − ℎ8) +ℎ1 +ℎ7 + ℎ2 +ℎ8 − ℎ1 + ℎ2 +ℎ7 + ℎ8 − ℎ1 − ℎ2 +ℎ7 − ℎ8 +]}. + = −{(ℎ2ℎ7 − ℎ1ℎ8 +ℎ7ℎ8 +) + [ℎ1ℎ2ℎ7 +2 − ℎ1ℎ2ℎ8 +2 − ℎ1 +2ℎ7ℎ8 − ℎ2 +2ℎ7ℎ8 +(ℎ2ℎ7 − ℎ1ℎ8)(ℎ7 +2 + ℎ8 +2) +] + + (ℎ5ℎ7 − ℎ4ℎ8 +ℎ7ℎ8 +) + [ℎ4ℎ5ℎ7 +2 − ℎ4ℎ5ℎ8 +2 − ℎ4 +2ℎ7ℎ8 − ℎ5 +2ℎ7ℎ8 +(ℎ5ℎ7 − ℎ4ℎ8)(ℎ7 +2 + ℎ8 +2) +]}. + += − (ℎ2 +2 + ℎ5 +2 − ℎ1 +2 − ℎ4 +2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 +2 − ℎ8 +2) +ℎ7ℎ8(ℎ7 +2 + ℎ8 +2) +. + +(36) + +As ℎ7ℎ8 ≠ 0 is the condition for the existence of both (15) and +(21), (35) is essentially the same as (2). Therefore, we can +conclude that (2) and (3) are mathematically equivalent. +IV. EXTENSION FOR MORE GENERAL SITUATIONS +In this section, the equivalence of Equations (2) and (3) will +be derived for more general situations which include: (i) the +existence of an arbitrary orientation of the two sets of OVPs and +(ii) the existence of infinite vanishing point. To that end, a +simple way of deriving the OVPs is first presented. +A. A simple and intuitive way of deriving OVPs +Assume the plane containing the two squares shown in Fig. +2 is the X-Y plane in the 3D space. As 𝐴𝐵 +⃡ and 𝐷𝐶 +⃡ are parallel, +their intersection can be expressed in homogeneous coordinate +as (1, 0, 0, 0)𝑇 . According to the projective geometry the +projection of (1, 0, 0, 0)𝑇 on the image plane, i.e., 𝑃𝑣1, can be +expressed as 𝜆1𝑃𝑣1 = 𝐾[𝑅 𝑇](1, 0, 0, 0)𝑇 based on a pinhole +camera model with intrinsic matrix K, rotation matrix R and +translation vector T, and for some constant 𝜆1. Therefore, the +corresponding vanishing point 𝑃𝑣1, and the OVP associated +with 𝐴𝐷 +⃡ and 𝐵𝐶 +⃡ , or 𝑃𝑣2, can be expressed as +𝜆1𝑃𝑣1 = 𝐾[𝑅 𝑇] [ +1 +0 +0 +0 +] = 𝐾[𝑟1 𝑟2 𝑟3 𝑇] [ +1 +0 +0 +0 +] = 𝐾[𝑟1 𝑟2 𝑇] [ +1 +0 +0 +] += 𝐻 [ +1 +0 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +1 +0 +0 +] = [ +ℎ1 +ℎ4 +ℎ7 +], (37) +and for some constant 𝜆2, +𝜆2𝑃𝑣2 = 𝐾[𝑅 𝑇] [ +0 +1 +0 +0 +] = 𝐾[𝑟1 𝑟2 𝑟3 𝑇] [ +0 +1 +0 +0 +] = 𝐾[𝑟1 𝑟2 𝑇] [ +0 +1 +0 +] += 𝐻 [ +0 +1 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +0 +1 +0 +] = [ +ℎ2 +ℎ5 +ℎ8 +], (38) +respectively. +Similarly, the following expressions can be obtained for 𝑃𝑣3 +and 𝑃𝑣4 , respectively, as +𝜆3𝑃𝑣3 = 𝐻 [ +1 +1 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +1 +1 +0 +] = [ +ℎ1 + ℎ2 +ℎ4 + ℎ5 +ℎ7 + ℎ8 +] (39) +and +𝜆4𝑃𝑣4 = 𝐻 [ +−1 +1 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +−1 +1 +0 +] = [ +−ℎ1 + ℎ2 +−ℎ4 + ℎ5 +−ℎ7 + ℎ8 +]. (40) +It is easy to see that for finite OVPs, (37) to (40) can be directly +rewritten in form of their counterparts in Sec. III, i.e., (15), (21), +(28), and (34). +B. PL for arbitrarily oriented two sets of OVPs +In Sec. III, OVPs 𝑃𝑣1, 𝑃𝑣2, 𝑃𝑣3 and 𝑃𝑣4 are considered for +equally spaced directions of 0, 90, 45, and 135 obtained the +two unit squares shown in Fig. 2. Without loss of generality, +assume one set of OVPs correspond directions of 0 and 90, +we can use the rectangle shown in Fig. 3, in place of Square I +in Fig. 2, to extend our derivation to the more general case +wherein directions of the second set of OVPs do not correspond +to 45 or 135. While there will be no change to 𝑃𝑣1 and 𝑃𝑣2, +new expressions of for 𝑃𝑣3 and 𝑃𝑣4 can be obtained, similar to +that done in (39) and (40), as + +Figure 3. A rectangle pattern with a new set of OVPs. + + +y +(-b, a, 0) +(0, b, 0) +(a, b, 0) +(0, 0, 0) +(a, 0, 0) +x5 +> REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < + +𝜆3𝑃𝑣3 = 𝐻 [ +𝑎 +𝑏 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +𝑎 +𝑏 +0 +] += [ +𝑎ℎ1 + 𝑏ℎ2 +𝑎ℎ4 + 𝑏ℎ5 +𝑎ℎ7 + 𝑏ℎ8 +] (41) + +𝜆4𝑃𝑣4 = 𝐻 [ +−𝑏 +𝑎 +0 +] = [ +ℎ1 +ℎ2 +ℎ3 +ℎ4 +ℎ5 +ℎ6 +ℎ7 +ℎ8 +ℎ9 +] [ +−𝑏 +𝑎 +0 +] += [ +−𝑏ℎ1 + 𝑎ℎ2 +−𝑏ℎ4 + 𝑎ℎ5 +−𝑏ℎ7 + 𝑎ℎ8 +]. (42) +By substituting 𝑃𝑣1 to 𝑃𝑣4 into (3), with new expressions of +𝑃𝑣3 = (𝑎ℎ1 + 𝑏ℎ2 +𝑎ℎ7 + 𝑏ℎ8 +, 𝑎ℎ4 + 𝑏ℎ5 +𝑎ℎ7 + 𝑏ℎ8 +, 1) +𝑃𝑣4 = (−𝑏ℎ1 + 𝑎ℎ2 +−𝑏ℎ7 + 𝑎ℎ8 +, −𝑏ℎ4 + 𝑎ℎ5 +−𝑏ℎ7 + 𝑎ℎ8 +, 1), (43) +the line equation of PL will have no change in the first two +coefficients, i.e., +(ℎ2ℎ7 − ℎ1ℎ8 +ℎ7ℎ8 +) 𝑢 + (ℎ5ℎ7 − ℎ4ℎ8 +ℎ7ℎ8 +) 𝑣 + 𝑐 = 0, +(44) +with +𝑐 = −{(ℎ2 +ℎ8 +− ℎ1 +ℎ7 +) [ +(ℎ1 +ℎ7 ∙ ℎ2 +ℎ8) − (𝑎ℎ1 + 𝑏ℎ2 +𝑎ℎ7 + 𝑏ℎ8) (−𝑏ℎ1 + 𝑎ℎ2 +−𝑏ℎ7 + 𝑎ℎ8) +ℎ1 +ℎ7 + ℎ2 +ℎ8 − (𝑎ℎ1 + 𝑏ℎ2 +𝑎ℎ7 + 𝑏ℎ8) − (−𝑏ℎ1 + 𝑎ℎ2 +−𝑏ℎ7 + 𝑎ℎ8) +] + + (ℎ5 +ℎ8 +− ℎ4 +ℎ7 +) [ +(ℎ4 +ℎ7 ∙ ℎ5 +ℎ8) − (𝑎ℎ4 + 𝑏ℎ5 +𝑎ℎ7 + 𝑏ℎ8) (−𝑏ℎ4 + 𝑎ℎ5 +−𝑏ℎ7 + 𝑎ℎ8) +ℎ4 +ℎ7 + ℎ5 +ℎ8 − (𝑎ℎ4 + 𝑏ℎ5 +𝑎ℎ7 + 𝑏ℎ8) − (−𝑏ℎ4 + 𝑎ℎ5 +−𝑏ℎ7 + 𝑎ℎ8) +]}. + + = −{(ℎ2ℎ7 − ℎ1ℎ8 +ℎ7ℎ8 +) + [−𝑎𝑏(ℎ1ℎ2ℎ7 +2 − ℎ1ℎ2ℎ8 +2 − ℎ1 +2ℎ7ℎ8 − ℎ2 +2ℎ7ℎ8) +−𝑎𝑏(ℎ2ℎ7 − ℎ1ℎ8)(ℎ7 +2 + ℎ8 +2) +] + + (ℎ5ℎ7 − ℎ4ℎ8 +ℎ7ℎ8 +) + [−𝑎𝑏(ℎ4ℎ5ℎ7 +2 − ℎ4ℎ5ℎ8 +2 − ℎ4 +2ℎ7ℎ8 − ℎ5 +2ℎ7ℎ8) +−𝑎𝑏(ℎ5ℎ7 − ℎ4ℎ8)(ℎ7 +2 + ℎ8 +2) +]}. + += − (ℎ2 +2 + ℎ5 +2 − ℎ1 +2 − ℎ4 +2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 +2 − ℎ8 +2) +ℎ7ℎ8(ℎ7 +2 + ℎ8 +2) +, +which is exactly the same as the constant in (2). +C. PL for infinite vanishing point +While no constraint on location of any member of the two +sets of OVPs is given in [7], location at infinity is implicitly +excluded in the derivation of PL presented in [9]. In this +subsection, a fairly simple extension of the derivation of PL for +𝑃𝑣1 located at infinity is provided, while similar extensions for +other OVPs are omitted for brevity. +For 𝑃𝑣1 located at infinity, e.g., with 𝐴′𝐵′ +⃡ ∥𝐶′𝐷′ +⃡ for the +image of Square I in Fig. 2, its homogeneous coordinate will +become +𝑃𝑣1 = [𝑚1 𝑛1 0]𝑇. (45) +By taking the limit of the PL expression (3) for arbitrarily large +𝑚1 and 𝑛1, we will have +−𝑚1𝑢 + (−𝑛1)𝑣 + 𝑐 = 0, (46) +with +𝑐 = −[−𝑚1𝑚2 − 𝑛1𝑛2] (47) +As the PL expression in (46) is much simpler than that for +(3), a formal proof of its equivalence with (2) is omitted for +brevity. +V. CONCLUSION +In this report, an algebraic proof of the equivalence of two +PL expressions, either derived in [7] in terms of elements of the +homography matrix or established in [9] for two sets of +orthogonal vanishing points, is provided. Accordingly, four +orthogonal vanishing points are firstly derived for a given +homography matrix, wherein two unit square are employed to +simplify the derivation. Then, the PL for these vanishing points, +which has exactly the same line equation as that derived in [7], +is obtained with the procedure outlined in [9]. Moreover, with +another simple and intuitive way of deriving orthogonal +vanishing points, the foregoing equivalence of PL expression is +considered for more general situations which include: (i) the +existence of arbitrary orientations of the two sets of OVPs, i.e., +other than orientations associated with squares and their +diagonals, and (ii) the existence of infinite vanishing point. +REFERENCES +[1] Z. Zhang, “A flexible new technique for camera calibration,” Technical +Report MSR-TR-98-71, Microsoft Research, Dec 1998. +[2] Z. Zhang, “Flexible camera calibration by viewing a plane from unknown +orientations,” in Proc. of the Seventh IEEE International Conference on +Computer Vision, pp. 666-673, 1999. +[3] Z. Zhang, “A flexible new technique for camera calibration,” IEEE +Transactions on Pattern Analysis and Machine Intelligence, vol. 22, pp. +1330–1334, Nov 2000. +[4] Hua-Tsung Chen, “Geometry-Based Camera Calibration Using Five-Point +Correspondences From a Single Image,” IEEE Transactions on Circuits +and Systems for Video Technology, vol. 27, pp. 2555-2566, 2017. +[5] M.-T. Lu and J.-H. Chuang, “Fully automatic camera calibration for +principal point using flat monitors,” in Proc. of 25th IEEE International +Conference on Image Processing (ICIP), pp. 3154–3158, Oct 2018. +[6] J.-H. Chuang, C.-H. Ho, A. Umam, H.-Y. Chen, M.-T. Lu, J.-N. Hwang, +and T.-A. Chen, “A new technique of camera calibration: a geometric +approach based on principal lines,” arXiv:1908.06539v1, Aug 2019. +[7] J.-H. Chuang, C.-H. Ho, A. Umam, H.-Y. Chen, J.-N. Hwang, and T.-A. +Chen, “Geometry-based camera calibration using closed-form solution of +principal line,” IEEE Transactions on Image Processing., vol. 30, pp. +2599-2610, 2021. +[8] X. Wang, L. Wang, and F. Duan, “Calibration for light field cameras based +on fixed point constraint of spatial plane homography,” Optics Express, vol. +30, pp. 24968-24983, 2022. +[9] F. Yang and L. Zhao, “Closed-foam solution of principal line for camera +calibration based on orthogonal vanishing points,” IEEE Transactions on +Circuits and Systems for Video Technology., Early Access, pp. 1-1, 2022. + diff --git a/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/load_file.txt b/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..27147a0f75fe348109646b2e5c9804a9e64123d9 --- /dev/null +++ b/tdE1T4oBgHgl3EQfQQPo/content/tmp_files/load_file.txt @@ -0,0 +1,214 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf,len=213 +page_content='1 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < Equivalence of Two Expressions of Principal Line Cheng Yen Hsu, Hsin Yi Chen and Jen Hui Chuang Abstract—Geometry-based camera calibration using principal line is more precise and robust than calibration using optimization approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' therefore, several researches try to re-derive the principal line from different views of 2D projective geometry to increase alternatives of the calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' In this report, algebraical equivalence of two expressions of principal line, one derived w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='t homography and the other using for two sets of orthogonal vanishing points, is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Moreover, the extension of the second expression to incorporate infinite vanishing point is carried out with simple mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Index Terms—Camera calibration, homography, principal line, vanishing points I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' INTRODUCTION eometry-based camera calibration is more precise and robust than traditional calibration approaches based on algebraic optimization [1-3] due to the consideration of camera parameters individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Chen [4] exploited geometric relationship between a vanishing line on image plane and a plane which contains optical center and principal point (PP), and is perpendicular to object plane, to sequentially derive three rotation angles, three translation distances and a focal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' However, the use of non-planar calibrated target, limit its usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Chuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' [5-7] consider similar geometry but formally define principal line (PL) as the axis of symmetry of the 3D plane containing a checkerboard pattern (CB plane), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', a symmetric image pattern corresponds to a symmetric pattern on the CB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1 illustrates the geometry of a PL, wherein both x-axis and X-axis are parallel to the intersection of the two plane and y-axis correspond to the PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' As each PL thus defined for each camera-pattern pose will pass through the camera principal point (PP), the PP can then be estimated by finding the intersection of linearly independent PLs, possibly from a least square solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Moreover, outliers in checkerboard patterns can also be identified easily in the early stage of calibration and removed from the subsequent estimation of other camera parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' The geometry of a PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' [8] rederive the same closed-from solution of PL for light field camera calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' The same geometric relationship among image plane, a rectangle plane, and PL is considered with a different projection model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', a new homography is derived according to projection model of the standard light field camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Although the derivation of PL starts from line equations of four edges of a rectangular pattern, the final results are exactly the same as [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Higher accuracy is also demonstrated with the final results obtained from their PL- based calibration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' In [9], Yang and Zhao derive a different closed-from expression of PL for orthogonal vanishing points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Without knowing the homography between image plane and CB plane, as in [7], two sets of orthogonal vanishing points projected from the 3D space are used for PL derivation, with experimental results showing comparable accuracy in the calibration results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' However, no proof of the equivalence of two closed-form expressions of PL are elaborated in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Moreover, the expression in [9] is derived for finite orthogonal vanishing points (nonparallel image lines), while there is no such constraint in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' In this report, to address the above issues, the two PL expressions are first reviewed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' II, followed by the proof of their equivalence in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Then, simple extensions of the PL expression in [9] for infinite orthogonal vanishing points are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' IV, before some concluding remarks are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' RELATED WORK The principal line introduced in [7] plays an important role in the camera calibration, as a PL can be obtained for each checkerboard pattern and will pass through the principal point of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Therefore, the PP can be determined as the intersection of a set of linearly independent PLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Based on mathematical derivations provided in [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' if the homographic relationship between a CB plane and the image plane is represented by the homography matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 𝐻 = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] (1) the principal line can be expressed as (−ℎ1ℎ8 + ℎ2ℎ7)𝑢 + (−ℎ4ℎ8 + ℎ5ℎ7)𝑣 + 𝑐 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (2) where 𝑐 = − (ℎ2 2 + ℎ5 2 − ℎ1 2 − ℎ4 2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 2 − ℎ8 2) ℎ7 2 + ℎ8 2 G Aa=45°2 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' a different expression of PL is also derived in [9] based on orthogonal vanishing points (OVPs) of a planar scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' In particular, given the first set of two OVPs (Pv1 = (m1, n1, 1) and Pv2 = (m2, n2, 1)), and the second set (Pv3 = (m3, n3, 1), and Pv4 = (m4, n4, 1)), the PL can be expressed as (𝑚2 − 𝑚1)𝑢 + (𝑛2 − 𝑛1)𝑣 + 𝑐 = 0, (3) where 𝑐 = −[(𝑚2 − 𝑚1) 𝑚1𝑚2 − 𝑚3𝑚4 𝑚1 + 𝑚2 − 𝑚3 − 𝑚4 + (𝑛2 − 𝑛1) 𝑛1𝑛2 − 𝑛3𝑛4 𝑛1 + 𝑛2 − 𝑛3 − 𝑛4 ] In this report, the equivalence of the two expressions of PL, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', Equations (2) and (3), will first be derived in the next section, for two sets of (finite) orthogonal vanishing points of two squares on the CB plane, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2, following procedure outlined in [9] for a homographic transformation between image plane and CB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Similar derivations for more general situations will be considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' A B C D E F (0,0) (1,0) (0,1) (-1,1) (-1,0) (1,1) X Y I II Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Two unit squares used in this report to prove the equivalence of (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' SHOWING THE EQUIVALENCE OF (2) AND (3) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Obtaining 𝑃𝑣1 and 𝑃𝑣2 Consider the homogeneous coordinates of the four vertices of Square I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', A = (0,0,1), B = (1,0,1), C = (1,1,1), and D = (0,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Based on the homography matrix given in (1), the homogeneous coordinates of images of these vertices can be expressed as 𝐴′ = [ ℎ3 ℎ9 ℎ6 ℎ9 1 ] , 𝐵′ = [ ℎ1 + ℎ3 ℎ7 + ℎ9 ℎ4 + ℎ6 ℎ7 + ℎ9 1 ] , 𝐶′ = [ ℎ1 + ℎ2 + ℎ3 ℎ7 + ℎ8 + ℎ9 ℎ4 + ℎ5 + ℎ6 ℎ7 + ℎ8 + ℎ9 1 ] , 𝐷′ = [ ℎ2 + ℎ3 ℎ8 + ℎ9 ℎ5 + ℎ6 ℎ8 + ℎ9 1 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (4) In this subsection, it is assumed that P𝑣1 is the intersection of the 𝐴′𝐵′ ⃡ and 𝐶′𝐷′ ⃡ and P𝑣2 is the intersection of the 𝐴′𝐷′ ⃡ and 𝐵′𝐶′ ⃡ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' For P𝑣1, we first calculate the directional vector 𝐴′𝐵′ = [ ℎ1ℎ9 − ℎ3ℎ7 ℎ9(ℎ7 + ℎ9) ℎ4ℎ9 − ℎ6ℎ7 ℎ9(ℎ7 + ℎ9)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (5) Assume 𝐴′𝐵′ exists, both ℎ9 and (ℎ7+ℎ9) will be nonzero, and we have 𝐴′𝐵′ = [ℎ1ℎ9 − ℎ3ℎ7 ℎ4ℎ9 − ℎ6ℎ7] , (6) and the line equation of 𝐴′𝐵′ ⃡ can be expressed as (ℎ4ℎ9 − ℎ6ℎ7)𝑢 + (−ℎ1ℎ9 + ℎ3ℎ7)𝑣 + 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡1 = 0 (7) By substituting A′ in equation (4) into (7), we will have Constant1 = −ℎ3ℎ4 + ℎ1ℎ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (8) Similarly, the line equation of 𝐶’𝐷’ ⃡ can be expressed as (ℎ5ℎ7 + ℎ6ℎ7 − ℎ4ℎ8 − ℎ4ℎ9)𝑢 +(−ℎ2ℎ7 − ℎ3ℎ7 + ℎ1ℎ8 + ℎ1ℎ9)𝑣 + 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (9) By substituting D′ in equation (4) into (9), we will have Constant2 = ℎ2ℎ4 + ℎ3ℎ4 − ℎ1ℎ5 − ℎ1ℎ6 (10) Therefore, P𝑣1 can be obtained as the intersection of the two lines represented by (7) and (9), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', by evaluating the outer product 𝑃𝑣1 = 𝐴′𝐵′ ⃡ × 𝐶′𝐷′ ⃡ = | 𝑖 𝑗 𝑘 𝑎1 𝑏1 𝑐1 𝑎2 𝑏2 𝑐2 | , (11) where ai, bi, and ci are the coefficients in (7) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Then, the three components in (11) can be evaluated as 𝑀1 = 𝑏1𝑐2 − 𝑏2𝑐1 = ℎ1(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = ℎ1 ∙ det(H), (12) 𝑁1 = 𝑐1𝑎2 − 𝑐2𝑎1 = ℎ4(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = ℎ4 ∙ det(H), (13) and 𝑘1 = 𝑎1𝑏2 − 𝑎2𝑏1 = ℎ7(−ℎ2ℎ4ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +ℎ1ℎ5ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = ℎ7 ∙ det(H), (14) or, if 𝑘1 ≠ 0, 3 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 𝑃𝑣1 = (𝑚1, 𝑛1, 1) = (ℎ1 ℎ7 , ℎ4 ℎ7 , 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (15) Note that if no three vertices of Square I are colinear, we will have det(H)≠0 as the homographic transformation is bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Besides, if 𝑃𝑣1 does not locate at infinity we will have ℎ7≠0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' As for obtaining the location of Pv2, the exact procedure used to derive (7) and (9) can be employed, and line equations of 𝐴′𝐷′ ⃡ and 𝐵′𝐶′ ⃡ can be found as (ℎ4ℎ9 − ℎ6ℎ7)𝑢 + (−ℎ1ℎ9 + ℎ3ℎ7)𝑣 − ℎ3ℎ5 + ℎ2ℎ6 = 0 (16) and (ℎ5ℎ7 + ℎ5ℎ9 − ℎ4ℎ8 − ℎ6ℎ8)𝑢 +(−ℎ2ℎ7 − ℎ2ℎ9 + ℎ1ℎ8 + ℎ3ℎ8)𝑣 −ℎ1ℎ5 − ℎ3ℎ5 + ℎ2ℎ4 + ℎ2ℎ6 = 0, (17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Thus, the three components of the intersection of 𝐴′𝐷′ ⃡ and 𝐵′𝐶′ ⃡ can be obtained, similar to that formulated in (11), as 𝑀2 = ℎ2(ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 + ℎ3ℎ4ℎ8 −ℎ3ℎ5ℎ7 + ℎ2ℎ6ℎ7 + ℎ1ℎ6ℎ8) = ℎ2 ∙ det(H), (18) 𝑁2 = ℎ5(−ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8 + ℎ2ℎ6ℎ7 +ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 − ℎ1ℎ6ℎ8) = ℎ5 ∙ det(H), (19) and 𝑘2 = ℎ8(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 −ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = ℎ8 ∙ det(H), (20) Or, if 𝑘2 ≠ 0, 𝑃𝑣2 = (𝑚2, 𝑛2, 1) = (ℎ2 ℎ8 , ℎ5 ℎ8 , 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (21) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Obtaining 𝑃𝑣3 and 𝑃𝑣4 Consider Square II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2, and homogeneous coordinates of its two vertices E = (−1,1,1) and F = (−1,0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Based on the homography matrix in (1), the homogeneous coordinates of images of these two vertices can be expressed as 𝐸′ = [ −ℎ1 + ℎ2 + ℎ3 −ℎ7 + ℎ8 + ℎ9 −ℎ4 + ℎ5 + ℎ6 −ℎ7 + ℎ8 + ℎ9 1 ] , 𝐹′ = [ −ℎ1 + ℎ3 ℎ7 + ℎ9 −ℎ4 + ℎ6 ℎ7 + ℎ9 1 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (22) Therefore, another pair of orthogonal vanishing points can be identified as the intersection of 𝐴′𝐶′ ⃡ and 𝐷′𝐹′ ⃡ (P𝑣3) and the intersection of 𝐴′𝐸′ ⃡ and 𝐵′𝐷′ ⃡ (P𝑣4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' By procedure adopted earlier to find other line equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' line equations of 𝐷′𝐹′ ⃡ and 𝐴′𝐶′ ⃡ can be found as (−ℎ5ℎ7 + ℎ5ℎ9 − ℎ6ℎ7+ℎ4ℎ8+ℎ4ℎ9 − ℎ6ℎ8)𝑢 +(−ℎ1ℎ8 − ℎ1ℎ9 + ℎ3ℎ8+ℎ2ℎ7 − ℎ2ℎ9+ℎ3ℎ7)𝑣 +(−ℎ2ℎ4+ℎ2ℎ6 − ℎ3ℎ4+ℎ1ℎ5 + ℎ1ℎ6−ℎ3ℎ5) = 0 (23) and (ℎ6ℎ7 + ℎ6ℎ8 − ℎ4ℎ9−ℎ5ℎ9)𝑢 +(ℎ1ℎ9 + ℎ2ℎ9 − ℎ3ℎ7 − ℎ3ℎ8)𝑣 +(ℎ3ℎ4 + ℎ5ℎ5 − ℎ1ℎ6−ℎ2ℎ6) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (24) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' and the outer product in (11) can again be adopted to find the three components of P𝑣3 as 𝑀3 = −(ℎ1 + ℎ2)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 +ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = −(ℎ1 + ℎ2) ∙ det(H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (25) 𝑁3 = −(ℎ4 + ℎ5)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 −ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = −(ℎ4 + ℎ5) ∙ det(H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (26) and 𝑘3 = −(ℎ7 + ℎ8)(ℎ1ℎ5ℎ9 + ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 −ℎ2ℎ4ℎ9 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = −(ℎ7 + ℎ8) ∙ det(H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (27) or 𝑃𝑣3 = (𝑚3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 𝑛3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1) = (ℎ1 + ℎ2 ℎ7 + ℎ8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' ℎ4 + ℎ5 ℎ7 + ℎ8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (28) Similar to the derivation of (23) and (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' we can obtain line equations of 𝐴′𝐸′ ⃡ and 𝐵′𝐷′ ⃡ as (ℎ4ℎ9 + ℎ6ℎ8 − ℎ6ℎ7 − ℎ5ℎ9)𝑢 +(ℎ3ℎ7 + ℎ2ℎ9 − ℎ1ℎ9−ℎ3ℎ8)𝑣 +(ℎ1ℎ6+ℎ3ℎ5 − ℎ3ℎ4−ℎ2ℎ6) = 0 (29) and (ℎ4ℎ8 + ℎ4ℎ9+ℎ6ℎ8 − ℎ5ℎ7−ℎ5ℎ9 − ℎ6ℎ7)𝑢 +(ℎ2ℎ7 + ℎ2ℎ9 + ℎ3ℎ7−ℎ1ℎ8 − ℎ1ℎ9−ℎ3ℎ8)𝑣 +(ℎ1ℎ5+ℎ1ℎ6 + ℎ3ℎ5−ℎ2ℎ4−ℎ2ℎ6−ℎ3ℎ4) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (30) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' and the three components of P𝑣4 as 𝑀4 = (ℎ1 − ℎ2)(ℎ2ℎ6ℎ7 − ℎ1ℎ6ℎ8 − ℎ3ℎ5ℎ7 +ℎ3ℎ4ℎ8 + ℎ1ℎ5ℎ9 + ℎ2ℎ4ℎ9) = (ℎ1 − ℎ2)(𝑆) = (ℎ1 − ℎ2) ∙ det(H) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (31) 𝑁4 = (ℎ4 − ℎ5)(ℎ1ℎ5ℎ9 − ℎ2ℎ4ℎ9 − ℎ1ℎ6ℎ8 +ℎ2ℎ6ℎ7 − ℎ3ℎ5ℎ7 + ℎ3ℎ4ℎ8) = (ℎ4 − ℎ4)(𝑆) = (ℎ4 − ℎ5) ∙ det(H) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (32) and 𝑘4 = (ℎ7 − ℎ8)(ℎ3ℎ4ℎ8 − ℎ3ℎ5ℎ7 − ℎ2ℎ4ℎ9 +ℎ1ℎ5ℎ9 − ℎ1ℎ6ℎ8 + ℎ2ℎ6ℎ7) = (ℎ7 − ℎ8)(𝑆) = (ℎ7 − ℎ8) ∙ det(H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (33) 4 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < or 𝑃𝑣4 = (𝑚4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 𝑛4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1) = (ℎ1 − ℎ2 ℎ7 − ℎ8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' ℎ4 − ℎ5 ℎ7 − ℎ8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (34) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' The equivalence of (2) and (3) To find the principal line according to the formulation elaborated in [9], equations of the four vanishing points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', (15), (21), (28), and (34), can be substituted into (3) directly, or (ℎ2ℎ7 − ℎ1ℎ8 ℎ7ℎ8 ) 𝑢 + (ℎ5ℎ7 − ℎ4ℎ8 ℎ7ℎ8 ) 𝑣 + 𝑐 = 0, (35) with 𝑐 = −{(ℎ2 ℎ8 − ℎ1 ℎ7 ) [ (ℎ1 ℎ7 ∙ ℎ2 ℎ8) − (ℎ1 + ℎ2 ℎ7 + ℎ8) (ℎ1 − ℎ2 ℎ7 − ℎ8) ℎ1 ℎ7 + ℎ2 ℎ8 − ℎ1 + ℎ2 ℎ7 + ℎ8 − ℎ1 − ℎ2 ℎ7 − ℎ8 ] + (ℎ5 ℎ8 − ℎ4 ℎ7 ) [ (ℎ4 ℎ7 ∙ ℎ5 ℎ8) − (ℎ4 + ℎ5 ℎ7 + ℎ8) (ℎ4 − ℎ5 ℎ7 − ℎ8) ℎ1 ℎ7 + ℎ2 ℎ8 − ℎ1 + ℎ2 ℎ7 + ℎ8 − ℎ1 − ℎ2 ℎ7 − ℎ8 ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' = −{(ℎ2ℎ7 − ℎ1ℎ8 ℎ7ℎ8 ) [ℎ1ℎ2ℎ7 2 − ℎ1ℎ2ℎ8 2 − ℎ1 2ℎ7ℎ8 − ℎ2 2ℎ7ℎ8 (ℎ2ℎ7 − ℎ1ℎ8)(ℎ7 2 + ℎ8 2) ] + (ℎ5ℎ7 − ℎ4ℎ8 ℎ7ℎ8 ) [ℎ4ℎ5ℎ7 2 − ℎ4ℎ5ℎ8 2 − ℎ4 2ℎ7ℎ8 − ℎ5 2ℎ7ℎ8 (ℎ5ℎ7 − ℎ4ℎ8)(ℎ7 2 + ℎ8 2) ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' = − (ℎ2 2 + ℎ5 2 − ℎ1 2 − ℎ4 2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 2 − ℎ8 2) ℎ7ℎ8(ℎ7 2 + ℎ8 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (36) As ℎ7ℎ8 ≠ 0 is the condition for the existence of both (15) and (21), (35) is essentially the same as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Therefore, we can conclude that (2) and (3) are mathematically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' EXTENSION FOR MORE GENERAL SITUATIONS In this section, the equivalence of Equations (2) and (3) will be derived for more general situations which include: (i) the existence of an arbitrary orientation of the two sets of OVPs and (ii) the existence of infinite vanishing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' To that end, a simple way of deriving the OVPs is first presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' A simple and intuitive way of deriving OVPs Assume the plane containing the two squares shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2 is the X-Y plane in the 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' As 𝐴𝐵 ⃡ and 𝐷𝐶 ⃡ are parallel, their intersection can be expressed in homogeneous coordinate as (1, 0, 0, 0)𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' According to the projective geometry the projection of (1, 0, 0, 0)𝑇 on the image plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', 𝑃𝑣1, can be expressed as 𝜆1𝑃𝑣1 = 𝐾[𝑅 𝑇](1, 0, 0, 0)𝑇 based on a pinhole camera model with intrinsic matrix K, rotation matrix R and translation vector T, and for some constant 𝜆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Therefore, the corresponding vanishing point 𝑃𝑣1, and the OVP associated with 𝐴𝐷 ⃡ and 𝐵𝐶 ⃡ , or 𝑃𝑣2, can be expressed as 𝜆1𝑃𝑣1 = 𝐾[𝑅 𝑇] [ 1 0 0 0 ] = 𝐾[𝑟1 𝑟2 𝑟3 𝑇] [ 1 0 0 0 ] = 𝐾[𝑟1 𝑟2 𝑇] [ 1 0 0 ] = 𝐻 [ 1 0 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ 1 0 0 ] = [ ℎ1 ℎ4 ℎ7 ], (37) and for some constant 𝜆2, 𝜆2𝑃𝑣2 = 𝐾[𝑅 𝑇] [ 0 1 0 0 ] = 𝐾[𝑟1 𝑟2 𝑟3 𝑇] [ 0 1 0 0 ] = 𝐾[𝑟1 𝑟2 𝑇] [ 0 1 0 ] = 𝐻 [ 0 1 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ 0 1 0 ] = [ ℎ2 ℎ5 ℎ8 ], (38) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Similarly, the following expressions can be obtained for 𝑃𝑣3 and 𝑃𝑣4 , respectively, as 𝜆3𝑃𝑣3 = 𝐻 [ 1 1 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ 1 1 0 ] = [ ℎ1 + ℎ2 ℎ4 + ℎ5 ℎ7 + ℎ8 ] (39) and 𝜆4𝑃𝑣4 = 𝐻 [ −1 1 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ −1 1 0 ] = [ −ℎ1 + ℎ2 −ℎ4 + ℎ5 −ℎ7 + ℎ8 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (40) It is easy to see that for finite OVPs, (37) to (40) can be directly rewritten in form of their counterparts in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' III, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', (15), (21), (28), and (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' PL for arbitrarily oriented two sets of OVPs In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' III, OVPs 𝑃𝑣1, 𝑃𝑣2, 𝑃𝑣3 and 𝑃𝑣4 are considered for equally spaced directions of 0\uf0b0, 90\uf0b0, 45\uf0b0, and 135\uf0b0 obtained the two unit squares shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Without loss of generality, assume one set of OVPs correspond directions of 0\uf0b0 and 90\uf0b0, we can use the rectangle shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 3, in place of Square I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2, to extend our derivation to the more general case wherein directions of the second set of OVPs do not correspond to 45\uf0b0 or 135\uf0b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' While there will be no change to 𝑃𝑣1 and 𝑃𝑣2, new expressions of for 𝑃𝑣3 and 𝑃𝑣4 can be obtained, similar to that done in (39) and (40), as Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' A rectangle pattern with a new set of OVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' y (-b, a, 0) (0, b, 0) (a, b, 0) (0, 0, 0) (a, 0, 0) x5 > REPLACE THIS LINE WITH YOUR MANUSCRIPT ID NUMBER (DOUBLE-CLICK HERE TO EDIT) < 𝜆3𝑃𝑣3 = 𝐻 [ 𝑎 𝑏 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ 𝑎 𝑏 0 ] = [ 𝑎ℎ1 + 𝑏ℎ2 𝑎ℎ4 + 𝑏ℎ5 𝑎ℎ7 + 𝑏ℎ8 ] (41) 𝜆4𝑃𝑣4 = 𝐻 [ −𝑏 𝑎 0 ] = [ ℎ1 ℎ2 ℎ3 ℎ4 ℎ5 ℎ6 ℎ7 ℎ8 ℎ9 ] [ −𝑏 𝑎 0 ] = [ −𝑏ℎ1 + 𝑎ℎ2 −𝑏ℎ4 + 𝑎ℎ5 −𝑏ℎ7 + 𝑎ℎ8 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (42) By substituting 𝑃𝑣1 to 𝑃𝑣4 into (3), with new expressions of 𝑃𝑣3 = (𝑎ℎ1 + 𝑏ℎ2 𝑎ℎ7 + 𝑏ℎ8 , 𝑎ℎ4 + 𝑏ℎ5 𝑎ℎ7 + 𝑏ℎ8 , 1) 𝑃𝑣4 = (−𝑏ℎ1 + 𝑎ℎ2 −𝑏ℎ7 + 𝑎ℎ8 , −𝑏ℎ4 + 𝑎ℎ5 −𝑏ℎ7 + 𝑎ℎ8 , 1), (43) the line equation of PL will have no change in the first two coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', (ℎ2ℎ7 − ℎ1ℎ8 ℎ7ℎ8 ) 𝑢 + (ℎ5ℎ7 − ℎ4ℎ8 ℎ7ℎ8 ) 𝑣 + 𝑐 = 0, (44) with 𝑐 = −{(ℎ2 ℎ8 − ℎ1 ℎ7 ) [ (ℎ1 ℎ7 ∙ ℎ2 ℎ8) − (𝑎ℎ1 + 𝑏ℎ2 𝑎ℎ7 + 𝑏ℎ8) (−𝑏ℎ1 + 𝑎ℎ2 −𝑏ℎ7 + 𝑎ℎ8) ℎ1 ℎ7 + ℎ2 ℎ8 − (𝑎ℎ1 + 𝑏ℎ2 𝑎ℎ7 + 𝑏ℎ8) − (−𝑏ℎ1 + 𝑎ℎ2 −𝑏ℎ7 + 𝑎ℎ8) ] + (ℎ5 ℎ8 − ℎ4 ℎ7 ) [ (ℎ4 ℎ7 ∙ ℎ5 ℎ8) − (𝑎ℎ4 + 𝑏ℎ5 𝑎ℎ7 + 𝑏ℎ8) (−𝑏ℎ4 + 𝑎ℎ5 −𝑏ℎ7 + 𝑎ℎ8) ℎ4 ℎ7 + ℎ5 ℎ8 − (𝑎ℎ4 + 𝑏ℎ5 𝑎ℎ7 + 𝑏ℎ8) − (−𝑏ℎ4 + 𝑎ℎ5 −𝑏ℎ7 + 𝑎ℎ8) ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' = −{(ℎ2ℎ7 − ℎ1ℎ8 ℎ7ℎ8 ) [−𝑎𝑏(ℎ1ℎ2ℎ7 2 − ℎ1ℎ2ℎ8 2 − ℎ1 2ℎ7ℎ8 − ℎ2 2ℎ7ℎ8) −𝑎𝑏(ℎ2ℎ7 − ℎ1ℎ8)(ℎ7 2 + ℎ8 2) ] + (ℎ5ℎ7 − ℎ4ℎ8 ℎ7ℎ8 ) [−𝑎𝑏(ℎ4ℎ5ℎ7 2 − ℎ4ℎ5ℎ8 2 − ℎ4 2ℎ7ℎ8 − ℎ5 2ℎ7ℎ8) −𝑎𝑏(ℎ5ℎ7 − ℎ4ℎ8)(ℎ7 2 + ℎ8 2) ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' = − (ℎ2 2 + ℎ5 2 − ℎ1 2 − ℎ4 2)ℎ7ℎ8 + (ℎ1ℎ2 + ℎ4ℎ5)(ℎ7 2 − ℎ8 2) ℎ7ℎ8(ℎ7 2 + ℎ8 2) , which is exactly the same as the constant in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' PL for infinite vanishing point While no constraint on location of any member of the two sets of OVPs is given in [7], location at infinity is implicitly excluded in the derivation of PL presented in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' In this subsection, a fairly simple extension of the derivation of PL for 𝑃𝑣1 located at infinity is provided, while similar extensions for other OVPs are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' For 𝑃𝑣1 located at infinity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', with 𝐴′𝐵′ ⃡ ∥𝐶′𝐷′ ⃡ for the image of Square I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2, its homogeneous coordinate will become 𝑃𝑣1 = [𝑚1 𝑛1 0]𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' (45) By taking the limit of the PL expression (3) for arbitrarily large 𝑚1 and 𝑛1, we will have −𝑚1𝑢 + (−𝑛1)𝑣 + 𝑐 = 0, (46) with 𝑐 = −[−𝑚1𝑚2 − 𝑛1𝑛2] (47) As the PL expression in (46) is much simpler than that for (3), a formal proof of its equivalence with (2) is omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' CONCLUSION In this report, an algebraic proof of the equivalence of two PL expressions, either derived in [7] in terms of elements of the homography matrix or established in [9] for two sets of orthogonal vanishing points, is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Accordingly, four orthogonal vanishing points are firstly derived for a given homography matrix, wherein two unit square are employed to simplify the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Then, the PL for these vanishing points, which has exactly the same line equation as that derived in [7], is obtained with the procedure outlined in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Moreover, with another simple and intuitive way of deriving orthogonal vanishing points, the foregoing equivalence of PL expression is considered for more general situations which include: (i) the existence of arbitrary orientations of the two sets of OVPs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', other than orientations associated with squares 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Hwang, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Chen, “Geometry-based camera calibration using closed-form solution of principal line,” IEEE Transactions on Image Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 30, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 2599-2610, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Wang, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' Duan, “Calibration for light field cameras based on fixed point constraint of spatial plane homography,” Optics Express, vol.' metadata={'source': 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+page_content=', Early Access, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} +page_content=' 1-1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdE1T4oBgHgl3EQfQQPo/content/2301.03039v1.pdf'} diff --git a/vNAyT4oBgHgl3EQfafc7/content/tmp_files/2301.00242v1.pdf.txt b/vNAyT4oBgHgl3EQfafc7/content/tmp_files/2301.00242v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd93a8a19d7747910f30c239a73e91ddcc9dab65 --- /dev/null +++ b/vNAyT4oBgHgl3EQfafc7/content/tmp_files/2301.00242v1.pdf.txt @@ -0,0 +1,1537 @@ +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +C. VIGNAT +Abstract. This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion +theorem. On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and +other mysterious objects. This study was inspired by some results from the fascinating article [1]. +1. Introduction +We study here the suspiciously simple identity +(1.1) +log +� +�1 − +� +n≥1 +(an)n−1 +wn +n! +� +� = − +� +n≥1 +(an + 1)n−1 +wn +n! , +for a parameter a > 0 and over the range |w| ≤ (a+1)a+1 +aa +, and where (a)n is the notation for the Pochhammer +symbol Γ(a+n) +Γ(a) . +It will be shown that this identity can be obtained as a special case of the classical Rothe-Hagen identity. But +there is more: substituting the variable w with xa − xa+1 produces, now for |x| ≤ 1, the identity +log +� +�1 − +� +n≥1 +(an)n−1 +� +xa − xa+1�n +n! +� +� = − +� +n≥1 +(an + 1)n−1 +� +xa − xa+1�n +n! +as expected, with the subtlety that over the interval x ∈ +� +a +a+1, 1 +� +, this identity reduces to +x = x, +meaning that, for x ∈ +� +a +a+1, 1 +� +, +log +� +�1 − +� +n≥1 +(an)n−1 +� +xa − xa+1�n +n! +� +� = x +and +− +� +n≥1 +(an + 1)n−1 +� +xa − xa+1�n +n! += x. +How can such functions be related by a simple logarithm ? And how can they coincide with the identity function +over a whole interval ? +Before we start explaining these strange identities, here is another piece of the puzzle: a two parameters 0 < a < b +version of (1.1) is +− log +� +�1 − +� +n≥1 +� +a +b − an +� +n−1 +� +xa − xb�n +n! +� +� = +� +n≥1 +� +a +b − an + 1 +� +n−1 +� +xa − xb�n +n! +, |x| < 1. +It does not seem much deeper than (1.1); however, the limit case ϵ → 0 of this identity with parameters a = 1, b = +1 + ϵ produces +− log +� +�1 − +� +n≥1 +(n − 1)n−1 +n! +zn +� +� = +� +n≥1 +nn−1 +n! zn, +an identity known to anyone who is familiar with the Lambert and the Cayley tree functions. +The remaining of this article is dedicated to providing insight into these identities, the functions involved and +some of their integrals. +1 +arXiv:2301.00242v1 [math.FA] 31 Dec 2022 + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +2 +2. Involutions +An easy way to build involutions over [0, 1], i.e. functions that satisfy +f (f (x)) = x, 0 ≤ x ≤ 1, +is as follows: start from a continuous unimodal function φ : [0, 1] → R, say increasing from 0 to φ (x0) on [0, x0] +and decreasing from φ (x0) to 0 on [x0, 1] , and define f (x) as the unique solution to the equation +φ ◦ f (x) = φ (x) , x ̸= x0 +and +f (x0) = x0. +A typical example of this construction appears in [1] in the case φ (x) = −x log x, x ∈ [0, 1]. +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.2 +0.3 +Back to the general case, let us denote l and r the invertible functions defined as the restrictions of φ on the intervals +[0, x0] and [x0, 1] respectively. Then the function f satisfies +(2.1) +f (x) = +� +r−1 ◦ φ (x) , +0 ≤ x ≤ x0, +l−1 ◦ φ (x) , +x0 ≤ x ≤ 1. +In the next sections, we’ll study the involution f in the case where the inverse of the functions r and l are computed +using Lagrange inversion theorem. This technique provides the inverse r−1 of a function r in the neighborhood of +a point x = a, assuming that r′ (a) ̸= 0, in the form of the power series +r−1 (x) = a + +� +n≥1 +cn +(x − r (a))n +n! +with the coefficients +cn = lim +x→a +dn−1 +dxn−1 +� +x − a +r (x) − r (a) +�n +. +In the case where r′ (1) ̸= 0 and where the power series expansion for r−1 at z = 1 converges on the domain [0, x0], +we will make use of the auxiliary function g (x) defined as 1 +(2.2) +g (x) = r−1 ◦ φ (x) , 0 ≤ x ≤ 1, +noticing that +(2.3) +g (x) = +� +f (x) +0 ≤ x ≤ x0 +x +x0 ≤ x ≤ 1. +The fact that the identity function appears in this construction suggests a possible hint to the mysterious linear +behavior described in Section 1. Before we can confirm that this construction is related to our problem - it obviously +is - let us extend this study to the computation of integrals. +3. Integrals +Assume that we want to evaluate the integral +I = +� 1 +0 +f (x) dx +1in the case l′ (0) ̸= 0, we would choose an expansion of l−1 at a = 0 and define rather g (x) = l−1 ◦ r (x) , 0 ≤ x ≤ 1. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +3 +with the function f an involution as defined in the previous section. Instead of computing both inverses r−1 and +l−1 as required by (2.1), we prefer to exploit the symmetry due to the involutive property of f : as illustrated in +the examples below, the integral +J = +� 1 +0 +g (x) dx +often turns out to be easier to evaluate than I itself. +Our main result in this section is the following proposition that provides an identity between both integrals, +allowing to compute the integral I - and in fact a generalization of it - in terms of the easier J. +Proposition 1. For h a monotone, differentiable function defined over [0, 1], and with the notation h (f (1)) = +limx→1 h (f (x)) , +(3.1) +� 1 +0 +h (f (x)) h′ (x) dx = 2 +� 1 +0 +h (g (x)) h′ (x) dx − h (1) (h (1) − h (f (1))) . +Moreover, +(3.2) +� x0 +0 +h (f (x)) h′ (x) dx = +� 1 +0 +h (g (x)) h′ (x) dx + 1 +2 +� +h2 (x0) − h2 (1) +� +. +In the simple case h (x) = x, identities (3.1) and (3.2) have a simple geometric interpretation: the following figure +shows the graphs of both functions f (the quarter circle shaped curve) and g (the top eighth circle shaped curve +over +� +0, 3 +4 +� +followed by the linear curve over +� 3 +4, 1 +� +) in the special case φ (x) = x3 − x4, for which x0 = 3 +4. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Visualizing on this graph the different areas I1, I2, J1 and J2 that appear in the proof provides a straightforward +geometric interpretation of Proposition 1. +The application of this result, together with the computation of functions f and g in the special case φ (x) = +xa − xb, 0 < a < b are addressed in the following sections; the corresponding solutions are denoted as fa,b and ga,b; +in the particular case b = a + 1, we use the shortcuts fa for fa,a+1 and ga for ga,a+1. +4. Some special cases +In this section we apply the construction of involutions using the Lagrange inversion technique as described in +Section 2 to the parametrized function φ (x) = xa − xb with 0 < a < b. +4.1. the case b = a+1. In the case φ (x) = xa−xa+1, we have the following results, with the notation (a)n = Γ(a+n) +Γ(a) +for the Pochhammer symbol. +Proposition 2. The function ga defined by (2.2) has the power series expansion +(4.1) +ga (x) = 1 − +� +n≥1 +(an)n−1 +(xa (1 − x))n +n! +, 0 ≤ x ≤ 1 +and coincides with the function fa over the interval [0, x0] . It coincides with the identity function on the comple- +mentary interval [x0, 1] . +Remark. For integer values of a, the function ga is a generalized hypergeometric function +ga (x) = +1 +a + 1 + +a +a + 1 aFa−1 +� +− +1 +a+1, +1 +a+1, +2 +a+1, . . . , a−1 +a+1 +1 +a, 2 +a, . . . , a−1 +a +; (a + 1)a+1 +aa +xa (1 − x) +� +. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +4 +Its linear behavior on the interval on [x0, 1] may thus appear as a surprise. However, this linear behavior appears +in disguise in the specialization ν = a + 1, µ = −1, z = 1 +x of Entry 5.2.13.30 in [5] (a may not be integer here, and +notice that (0)−1 = −1 ) +� +n≥0 +Γ (nν + µ − 1) +Γ (nν − n + µ) +zn +n! = yµ−1 +µ − 1, z = y − 1 +yν +, |z| < |(ν − 1)ν−1 +νν +|. +Let us now apply Lagrange inversion formula to the function +log ◦r−1 +a += (ra ◦ exp)−1 ; +this produces the unexpected identity +Proposition 3. For a > 0, +(4.2) +− log +� +�1 − +� +n≥1 +(an)n−1 +wn +n! +� +� = +� +n≥1 +(an + 1)n−1 +wn +n! , |w| ≤ (a + 1)a+1 +aa +. +Not surprisingly, this identity can be identified as the specialization of a more classical identity. +Proposition 4. Identity (4.2) is equivalent to a special case of the Rothe-Hagen identity [2, 3.146] +(4.3) +n +� +k=0 +�x + kz +k +��y − kz +n − k +� +p + kq +(x + kz) (y − kz) = +p (x + y − nz) +x (x + y) (y − nz) +�x + y +n +� +with the specialization x = a, y = (a + 1) n − 1, z = a + 1, p = a and q = a + 1. +4.2. the general case 0 < a < b. We now address the case φa,b (x) = xa − xb. The function ga,b can be computed +as follows. +Proposition 5. The function ga,b (x) satisfies +(4.4) +gb−a +a,b (x) = 1 − +� +n≥1 +� +a +b − an +� +n−1 +� +xa − xb�n +n! +, 0 ≤ x ≤ 1. +Applying now Lagrange inversion formula to +log r−1 +a,b = (ra,b ◦ exp)−1 +produces a power series expansion for the function − log ga,b. +Proposition 6. With b − a > 0, we have +(4.5) +− log ga,b (x) = +1 +b − a +� +n≥1 +� +a +b − an + 1 +� +n−1 +� +xa − xb�n +n! +, 0 ≤ x ≤ 1. +This extends identity (4.2) to +− log +� +�1 − +� +n≥1 +� +a +b − an +� +n−1 +wn +n! +� +� = +� +n≥1 +� +a +b − an + 1 +� +n−1 +wn +n! , |w| < +�aa +bb +� +1 +b−a +. +5. Integrals +Proposition 7. The integral of fa over [0, 1] is evaluated as +� 1 +0 +fa (x) dx = +1 +1 + a +� +1 − +aπ +a + 1 cot +aπ +a + 1 +� +(5.1) +whereas +� x0 +0 +fa (x) dx = +1 +2 (1 + a)2 +� +1 + a + a2 − aπ cot +aπ +a + 1 +� +. +We provide two proofs of the first result: one as a straightforward application of our main result Proposition 1, +and another based on the expression of the integral (5.1) as a double integral, a clever technique that we borrow +from [4]. +In the general case, the inversion technique used above in the case b = a + 1 extends to the general case using +the following observation. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +5 +Lemma 8. Assume that b′ +a′ = b +a then +gb−a +a,b +� +y +1 +a +� += gb′−a′ +a′,b′ +� +y +1 +a′ +� +. +As a consequence, with a′ = +a +b−a and b′ = +b +b−a, +ga′,a′+1 (x) = gb−a +a,b +� +x +1 +b−a +� +. +Proof. These identities can be checked directly from the expression (4.4), or deduced from the remark (see [1]) that +the function s (x) = (fa,b (xγ)) +1 +γ satisfies the equation +saγ (x) − sbγ (x) = xaγ − xbγ +so that it coincides with faγ,bγ. +From these identities, we deduce +□ +Proposition 9. We have the following evaluations +� 1 +0 +gb−a +a,b (x) xb−a−1dx = +1 +b − a +� +1 − a +2b2 +� +b + (b − a) π cot +�πa +b +��� +, +� 1 +0 +f b−a +a,b (x) xb−a−1dx = 2 − b + a − a +b +b − a +− a +b2 π cot +�πa +b +� +. +6. The limit Lambert case +The function φ (x) = −x log x, x ∈ [0, 1] is shown to be the limit case of φ1,1+ϵ above as ϵ → 0. It is a famous +case of Lagrange inversion and is related to the Lambert and the Cayley tree functions. +Proposition 10. In the limit Lambert case, we have +g (w) = 1 − +� +n≥1 +(n − 1)n−1 +n! +(−w log w)n , 0 ≤ w ≤ 1 +and +log g (w) = +� +n≥1 +(−n)n−1 +n! +(w log w)n = W0 (w log w) , +where W0 is the principal branch of the Lambert function. +The identity +− log +� +�1 − +� +n≥1 +(n − 1)n−1 +n! +zn +� +� = +� +n≥1 +nn−1 +n! zn +is the limit case of (4.2). +Notice that we recover the identity +log g (w) = log w, w ≥ 1 +e, +as a well-known property of the Lambert function W0. +Proposition 11. We have the following integrals +� 1 +0 +g (w) dw = 1 − +� +n≥1 +(n − 1)n−1 +(n + 1)n+1 = 0.659495 . . . +� 1 +0 +f (w) dw = 2 − 2 +� +n≥1 +(n − 1)n−1 +(n + 1)n+1 = 1.31899 . . . +� 1 +0 +− log g (w) +w +dw = π2 +6 +and +� 1 +0 +− log f (x) +x +dx = π2 +3 . + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +6 +7. Proofs +7.1. Proof of Proposition 1. Denote +I1 = +� x0 +0 +h (f (x)) h′ (x) dx, I2 = +� 1 +x0 +h (f (x)) h′ (x) dx. +Changing variable y = f (x) and integrating by parts in I1 produces +I1 = − [h (y) h (f (y))]1 +x0 + +� 1 +x0 +h′ (y) h (f (y)) dy += h2 (x0) − h (f (1)) h (1) + I2 +(7.1) +Considering now +J1 = +� x0 +0 +h (g (x)) h′ (x) dx, J2 = +� 1 +x0 +h (g (x)) h′ (x) dx, +we have, by (2.3), J1 = I1 and +J2 = +� 1 +x0 +h (x) h′ (x) dx = 1 +2 +� +h2 (1) − h2 (x0) +� +. +Moreover, define +J = J1 + J2 = +� 1 +0 +h (g (x)) h′ (x) dx. +We deduce +h2 (x0) = I1 − I2 + ¯h (1) h (1) = h2 (1) − 2J2 = h2 (1) − 2 (J − I1) +so that +(7.2) +I1 + I2 = 2J − h2 (1) + h (1) h (f (1)) . +Next, I1 is deduced by solving the linear system that consists of the two equations (7.1) and (7.2). +7.2. Proof of Proposition 2. Let us compute the inverse function r−1 +a +using Lagrange inversion formula: at +w = 1, the function +w �→ ra (w) = z = wa (1 − w) +can be inverted since dra +dw |w=1 ̸= 0. The coefficients of the series expansion of its inverse +w = r−1 +a +(z) = 1 + +� +k≥1 +cn +zn +n! +are computed as +cn = lim +w→1 +dn−1 +dwn−1 +� +w − 1 +φa,2 (w) − φa,2 (1) +�n += lim +w→1 +dn−1 +dwn−1 +� +w − 1 +wa (1 − w) +�n += (−1)n lim +w→1 +dn−1 +dwn−1 +� +w−an� += − (an)n−1 +with (a)n = Γ(a+n) +Γ(a) . We deduce +(7.3) +r−1 +a +(z) = 1 − +� +k≥1 +(an)n−1 +zn +n! , 0 ≤ z ≤ φa (x0) +and +(7.4) +fa (x) = 1 − +� +n≥1 +(an)n−1 +(xa (1 − x))n +n! +, 0 ≤ x ≤ x0. +The series in (7.4) is convergent over R : let us define the function +(7.5) +ga (x) = 1 − +� +n≥1 +(an)n−1 +(xa (1 − x))n +n! +, 0 ≤ x ≤ 1. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +7 +7.3. Proof of Proposition 4. Noticing that (0)−1 = −1 allows to rewrite the desired identity as +− log +� +�− +� +n≥0 +(an)n−1 +wn +n! +� +� = +� +n≥1 +(an + 1)n−1 +wn +n! . +Taking the derivative with respect to w on both sides produces +� +n≥0 +(an + a + 1)n +wn +n! = − +� +n≥0 (an + a)n +wn +n! +� +n≥0 (an)n−1 +wn +n! +so that we need to check the convolution identity +� +�� +n≥0 +(an + a + 1)n +wn +n! +� +� +� +�� +n≥0 +(an)n−1 +wn +n! +� +� = − +� +n≥0 +(an + a)n +wn +n! +or equivalently +� +k≥0 +(ak + a + 1)k +k! +(an − ak)n−k−1 +(n − k)! += −(an + a)n +n! +. +This is indeed identity (4.3) with the specialization x = a, y = (a + 1) n − 1, z = a + 1, p = a and q = a + 1. +7.4. Proof of Proposition 3. The coefficients of the expansion are now computed as +cn = lim +w→0 +dn−1 +dwn−1 +� +wn +enaw (1 − ew)n +� +. +Since +� +w +ew − 1 +�n +e−naw = +� +k≥0 +B(n) +k +(−na) +k! +wk +is the generating function for the higher-order Bernoulli polynomials B(n) +k +(z), we deduce +cn = B(n) +n−1 (−na) . +These special values of the higher-order Bernoulli polynomials appear in [3, Ch.6] as +B(n) +n−1 (x) = (x − 1) . . . (x − n + 1) +so that +cn = (−1)n (−na − 1) . . . (−na − n + 1) = − (an + 1)n−1 +and we obtain the power series expansion +log r−1 +a += +� +k≥1 +(an + 1)n−1 +zn +n! , 0 ≤ z ≤ φa (x0) . +7.5. Proof of Proposition 7. Consider h (x) = x in identity (3.1). The right-hand side integral is computed using +(7.5) as +� 1 +0 +ga (x) dx = +� 1 +0 +� +�1 − +� +n≥1 +(an)n−1 +(xa (1 − x))n +n! +� +� dx = 1 − +� +n≥1 +(an)n−1 +n! +� 1 +0 +(xa (1 − x))n dx +and evaluating the beta integral produces +� 1 +0 +ga (x) dx = 1 − +� +n≥1 +(an)n−1 +n! +Γ (an + 1) Γ (n + 1) +Γ (an + n + 2) += += 1 − +� +n≥1 +Γ (an + 1) Γ (an + n − 1) +Γ (an + n + 2) Γ (an) += 1 − +a +2 (1 + a) +� +1 − +π +a + 1 cot +π +a + 1 +� +. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +8 +Finally, as h (1) +� +h (1) − ¯h (1) +� += 1, +� 1 +0 +fa (x) dx = 2 +� 1 +0 +ga (x) dx − 1 = 1 − +a +1 + a +� +1 − +π +a + 1 cot +π +a + 1 +� += +1 +a + 1 − +aπ +(a + 1)2 cot +aπ +a + 1. +Moreover +� x0 +0 +fa (x) dx = 1 − +a +2 (1 + a) +� +1 − +π +a + 1 cot +π +a + 1 +� ++ 1 +2 +� +a2 +(1 + a)2 − 1 +� += +1 +2 (1 + a)2 +� +1 + a + a2 − aπ cot +aπ +a + 1 +� +. +7.6. A proof of Proposition 7 borrowed from [4]. We first restate, for completeness, Andrews, Eriksson, +Petrov and Romik’s elegant solution [4] to the evaluation of +Ia,b = +� 1 +0 +−log fa,b (x) +x +dx = π2 +3ab. +Rewrite Ia,b as the double integral +Ia,b = +� 1 +0 +dx +x +� 1 +fa,b(x) +dy +y = +�� +D +dxdy +xy +over the domain D = {0 ≤ x ≤ 1, fa,b (x) ≤ y ≤ 1} . Dividing this domain into two equal subdomains and change +variables +� x +y +� +→ +� +x +t = y +x +� +so that dxdy → xdxdt and +Ia,b = 2 +�� +D′ +dxdt +tx +over the new domain D′ = +� +0 ≤ t ≤ 1, +� +1−ta +1−tb +� +1 +b−a ≤ x ≤ 1 +� +produces +Ia,b = 2 +� 1 +0 +� 1 +� +1−ta +1−tb +� +1 +b−a +dx +x +dt +t = −2 +� 1 +0 +log +�1 − ta +1 − tb +� +1 +b−a +dt += +2 +b − a +�� 1 +0 +log +� +1 − tb� +dt − +� 1 +0 +log (1 − ta) dt +� +. +Substituting x = tb in the first integral and x = ta in the second provides the desired result. +This approach is now used to produce another proof of (5.1) as follows: denote +Ia = +� 1 +0 +fa,a+1 (x) dx +and rewrite it as the double integral +Ia = +� 1 +0 +� fa,a+1(x) +0 +dydx = 2 +�� +D′ xdxdt +with +D′ = +� +0 ≤ t ≤ 1, 0 ≤ x ≤ +1 − ta +1 − ta+1 +� +so that +Ia = 2 +� 1 +0 +1 +2x2 (t) dt = +� 1 +0 +� 1 − ta +1 − ta+1 +�2 +dt. +This integral is evaluated using the change of variable t = ez producing +Ia = +� ∞ +0 +� +sinh +� az +2 +� +sinh +� a+1 +2 z +� +�2 +dz. + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +9 +This is the special case b = a +2, c = a+1 +2 +of Entry 2.4.4.2 in [5] +� ∞ +0 +�sinh bx +sinh cx +�2 +dx = 1 +2c − πb +2c2 cot +�πb +c +� +so that +Ia = +1 +1 + a − +aπ +(a + 1)2 cot +aπ +a + 1. +7.7. Proof of Proposition 6. The coefficients of the series expansions of r−1 +a,b are +cn = lim +w→0 +dn−1 +dwn−1 +� +wn +enaw � +1 − ew(b−a)�n +� +. +Denoting c = b − a and expanding +wn +enaw (1 − ewc)n = +� +−1 +c +�n +(cw)n +(ewc − 1)n e−naw = +� +−1 +c +�n � +k≥0 +B(n) +k +� +−n a +c +� +k! +(wc)k +produces +cn = (−1)n +c +B(n) +n−1 +� +−na +c +� += − +� +n +a +b−a + 1 +� +n−1 +b − a +, +so that, for 0 ≤ x ≤ x0, +1 +b − a +� +n≥1 +� +a +b − an + 1 +� +n−1 +� +xa − xb�n +n! += − log fa,b (x) . +7.8. Proof of Proposition 5. Replacing a with +a +b−a in (4.2) produces +� +n≥1 +� +a +b − an + 1 +� +n−1 +� +x +a +b−a − x +b +b−a +�n +n! += − log +� +�1 − +� +n≥1 +� +a +b − an +� +n−1 +� +x +a +b−a − x +b +b−a +�n +n! +� +� . +The left hand-side is identified from (4.5) as (b − a) ga,b +� +x +1 +b−a +� +and we deduce +(b − a) ga,b +� +x +1 +b−a +� += − log +� +�1 − +� +n≥1 +� +a +b − an +� +n−1 +� +x +a +b−a − x +b +b−a +�n +n! +� +� +or equivalently +gb−a +a,b (x) = 1 − +� +n≥1 +� +a +b − an +� +n−1 +� +xa − xb�n +n! +. +7.9. Proof of Proposition 9. Start from +� 1 +0 +gb−a +a,b (x) xb−a−1dx = +� 1 +0 +xb−a−1 +� +�1 − +� +n≥1 +� +a +b − an +� +n−1 +� +xa − xb�n +n! +� +� += +1 +b − a − +� +n≥1 +� +a +b−an +� +n−1 +n! +� 1 +0 +xb−a−1 � +xa − xb�n dx. +The integral is a beta integral evaluated as +� 1 +0 +xb−a−1 � +xa − xb�n dx = +1 +b − a +Γ +� +a +b−an + 1 +� +Γ (n + 1) +Γ +� +a +b−an + n + 2 +� + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +10 +so that the sum is +� +n≥1 +� +a +b − an +� +n−1 +� 1 +0 +xb−a−1 +� +xa − xb�n +n! +dx = +1 +b − a +� +n≥1 +Γ +� +a +b−an + n − 1 +� +Γ +� +a +b−an +� +Γ +� +a +b−an + 1 +� +Γ (n + 1) +n!Γ +� +a +b−an + n + 2 +� += +1 +b − a +a +b − a +� +n≥1 +n +Γ +� +a +b−an + n − 1 +� +Γ +� +a +b−an + n + 2 +� +and the latest sum is evaluated as +� +n≥1 +n +Γ +� +a +b−an + n − 1 +� +Γ +� +a +b−an + n + 2 +� = (b − a) +� +b + (b − a) π cot +� πa +b +�� +2b2 +so that the desired integral is +1 +b − a +� +1 − a +2b2 +� +b + (b − a) π cot +�πa +b +��� +. +7.10. Proof of Proposition 10. The critical point is now x0 = e−1 and the inverse of the right-hand function is +computed as +r−1 (z) = 1 + +� +n≥1 +cn +zn +n! +with the coefficients +cn = lim +w→1 +dn−1 +dwn−1 +� +w − 1 +−w log w +�n += − (n − 1)n−1 , n ≥ 1 +with the convention 00 = 1 so that c1 = −1. We deduce +g (w) = r−1 (l (w)) = 1 − +� +n≥1 +(n − 1)n−1 +n! +(−w log w)n , 0 ≤ w ≤ 1. +If we now apply Lagrange’s inversion theorem to (log ◦r)−1 = r−1 ◦ exp, we obtain the new coefficients +cn = lim +w→0 +dn−1 +dwn +� +w +−eww +�n += −nn−1 +so that +log g (w) = − +� +n≥1 +nn−1 +n! +(−w log w)n = +� +n≥1 +(−n)n−1 +n! +(w log w)n = W0 (w log w) , +the principal branch of the Lambert function. +7.11. Proof of Proposition 11. Using +� 1 +0 +(w log w)n dw = (−1)n Γ (n + 1) +(n + 1)n+1 , n ≥ 0, +produces +� 1 +0 +g (w) dw = 1 − +� +n≥1 +(n − 1)n−1 +(n + 1)n+1 = 0.728466. +Moreover, since +� 1 +0 wn−1 (log w)n dw = (−1)n +nn +(n − 1)!, we deduce +� 1 +0 +− log g (w) +w +dw = +� 1 +0 +−W0 (w log w) +w +dw = +� +n≥1 +(−n)n−1 +n! +� 1 +0 +wn−1 (log w)n dw += +� +n≥1 +(−n)n +n! +(−1)n +nn +(n − 1)! = +� +n≥1 +1 +n2 = ζ (2) . +We now use the formula +� 1 +0 +h (fa (x)) h′ (x) dx = 2 +� 1 +0 +h (ga (x)) h′ (x) dx − h (1) (h (1) − h (f (1))) + +UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES +11 +to deduce +� 1 +0 +− log f (x) +x +dx = 2 +� 1 +0 +− log (g (x)) +x +− h2 (1) + h (1) h (f (1)) +with h (x) = −x log x and h (1) = 0, h (f (1)) = limx→1 h (f (x)) = limx→0 −x log x = 0 so that +� 1 +0 +− log f (x) +x +dx = 2 +� 1 +0 +− log (g (x)) +x += 2ζ (2) = π2 +3 . +References +[1] A.E. Holroyd, T.M. Liggett and D. Romik, Integrals, Partitions, and Cellular Automata, Transactions of the American Mathematical +Society, 356-8, 3349-3368, 2004 +[2] H.W. Gould, Combinatorial identities; a standardized set of tables listing 500 binomial coefficient summations, 1972 +[3] N.E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin (1924) +[4] G. Andrews, H. Eriksson, F. Petrov, D. Romik, Integrals, partitions and MacMahon’s Theorem, Journal of Combinatorial Theory, +Series A 114 (2007) 545–554 +[5] A.P. Prudnikov, Integrals and Series, Volume I. Gordon and Breach Science Publishers, 1990 +Department of Mathematics, Tulane University, New Orleans, USA and LSS, CentraleSupelec, Université Paris- +Sud Orsay, France + diff --git a/vNAyT4oBgHgl3EQfafc7/content/tmp_files/load_file.txt b/vNAyT4oBgHgl3EQfafc7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d857dd249fe30be493854b14d0132f7f885b2519 --- /dev/null +++ b/vNAyT4oBgHgl3EQfafc7/content/tmp_files/load_file.txt @@ -0,0 +1,334 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf,len=333 +page_content='UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' VIGNAT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This is a journey through integrals of involutions and surprising consequences of the Lagrange inversion theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' On the way, we meet unexpected logarithmic identities, hypergeometric functions with a linear regime and other mysterious objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This study was inspired by some results from the fascinating article [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Introduction We study here the suspiciously simple identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) log � �1 − � n≥1 (an)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = − � n≥1 (an + 1)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , for a parameter a > 0 and over the range |w| ≤ (a+1)a+1 aa , and where (a)n is the notation for the Pochhammer symbol Γ(a+n) Γ(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' It will be shown that this identity can be obtained as a special case of the classical Rothe-Hagen identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' But there is more: substituting the variable w with xa − xa+1 produces, now for |x| ≤ 1, the identity log � �1 − � n≥1 (an)n−1 � xa − xa+1�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = − � n≥1 (an + 1)n−1 � xa − xa+1�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' as expected, with the subtlety that over the interval x ∈ � a a+1, 1 � , this identity reduces to x = x, meaning that, for x ∈ � a a+1, 1 � , log � �1 − � n≥1 (an)n−1 � xa − xa+1�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = x and − � n≥1 (an + 1)n−1 � xa − xa+1�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' How can such functions be related by a simple logarithm ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' And how can they coincide with the identity function over a whole interval ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Before we start explaining these strange identities, here is another piece of the puzzle: a two parameters 0 < a < b version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) is − log � �1 − � n≥1 � a b − an � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = � n≥1 � a b − an + 1 � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , |x| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' It does not seem much deeper than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' however, the limit case ϵ → 0 of this identity with parameters a = 1, b = 1 + ϵ produces − log � �1 − � n≥1 (n − 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' zn � � = � n≥1 nn−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' zn, an identity known to anyone who is familiar with the Lambert and the Cayley tree functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The remaining of this article is dedicated to providing insight into these identities, the functions involved and some of their integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='00242v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='FA] 31 Dec 2022 UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Involutions An easy way to build involutions over [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' functions that satisfy f (f (x)) = x, 0 ≤ x ≤ 1, is as follows: start from a continuous unimodal function φ : [0, 1] → R, say increasing from 0 to φ (x0) on [0, x0] and decreasing from φ (x0) to 0 on [x0, 1] , and define f (x) as the unique solution to the equation φ ◦ f (x) = φ (x) , x ̸= x0 and f (x0) = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' A typical example of this construction appears in [1] in the case φ (x) = −x log x, x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3 Back to the general case, let us denote l and r the invertible functions defined as the restrictions of φ on the intervals [0, x0] and [x0, 1] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Then the function f satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) f (x) = � r−1 ◦ φ (x) , 0 ≤ x ≤ x0, l−1 ◦ φ (x) , x0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the next sections, we’ll study the involution f in the case where the inverse of the functions r and l are computed using Lagrange inversion theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This technique provides the inverse r−1 of a function r in the neighborhood of a point x = a, assuming that r′ (a) ̸= 0, in the form of the power series r−1 (x) = a + � n≥1 cn (x − r (a))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' with the coefficients cn = lim x→a dn−1 dxn−1 � x − a r (x) − r (a) �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the case where r′ (1) ̸= 0 and where the power series expansion for r−1 at z = 1 converges on the domain [0, x0], we will make use of the auxiliary function g (x) defined as 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) g (x) = r−1 ◦ φ (x) , 0 ≤ x ≤ 1, noticing that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3) g (x) = � f (x) 0 ≤ x ≤ x0 x x0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The fact that the identity function appears in this construction suggests a possible hint to the mysterious linear behavior described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Before we can confirm that this construction is related to our problem - it obviously is - let us extend this study to the computation of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Integrals Assume that we want to evaluate the integral I = � 1 0 f (x) dx 1in the case l′ (0) ̸= 0, we would choose an expansion of l−1 at a = 0 and define rather g (x) = l−1 ◦ r (x) , 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 3 with the function f an involution as defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Instead of computing both inverses r−1 and l−1 as required by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1), we prefer to exploit the symmetry due to the involutive property of f : as illustrated in the examples below, the integral J = � 1 0 g (x) dx often turns out to be easier to evaluate than I itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Our main result in this section is the following proposition that provides an identity between both integrals, allowing to compute the integral I - and in fact a generalization of it - in terms of the easier J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' For h a monotone, differentiable function defined over [0, 1], and with the notation h (f (1)) = limx→1 h (f (x)) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) � 1 0 h (f (x)) h′ (x) dx = 2 � 1 0 h (g (x)) h′ (x) dx − h (1) (h (1) − h (f (1))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Moreover, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) � x0 0 h (f (x)) h′ (x) dx = � 1 0 h (g (x)) h′ (x) dx + 1 2 � h2 (x0) − h2 (1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the simple case h (x) = x, identities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) have a simple geometric interpretation: the following figure shows the graphs of both functions f (the quarter circle shaped curve) and g (the top eighth circle shaped curve over � 0, 3 4 � followed by the linear curve over � 3 4, 1 � ) in the special case φ (x) = x3 − x4, for which x0 = 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='0 Visualizing on this graph the different areas I1, I2, J1 and J2 that appear in the proof provides a straightforward geometric interpretation of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The application of this result, together with the computation of functions f and g in the special case φ (x) = xa − xb, 0 < a < b are addressed in the following sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' the corresponding solutions are denoted as fa,b and ga,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' in the particular case b = a + 1, we use the shortcuts fa for fa,a+1 and ga for ga,a+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Some special cases In this section we apply the construction of involutions using the Lagrange inversion technique as described in Section 2 to the parametrized function φ (x) = xa − xb with 0 < a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' the case b = a+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the case φ (x) = xa−xa+1, we have the following results, with the notation (a)n = Γ(a+n) Γ(a) for the Pochhammer symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The function ga defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) has the power series expansion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) ga (x) = 1 − � n≥1 (an)n−1 (xa (1 − x))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ x ≤ 1 and coincides with the function fa over the interval [0, x0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' It coincides with the identity function on the comple- mentary interval [x0, 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' For integer values of a, the function ga is a generalized hypergeometric function ga (x) = 1 a + 1 + a a + 1 aFa−1 � − 1 a+1, 1 a+1, 2 a+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , a−1 a+1 1 a, 2 a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , a−1 a ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (a + 1)a+1 aa xa (1 − x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 4 Its linear behavior on the interval on [x0, 1] may thus appear as a surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' However, this linear behavior appears in disguise in the specialization ν = a + 1, µ = −1, z = 1 x of Entry 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='30 in [5] (a may not be integer here, and notice that (0)−1 = −1 ) � n≥0 Γ (nν + µ − 1) Γ (nν − n + µ) zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = yµ−1 µ − 1, z = y − 1 yν , |z| < |(ν − 1)ν−1 νν |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Let us now apply Lagrange inversion formula to the function log ◦r−1 a = (ra ◦ exp)−1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' this produces the unexpected identity Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' For a > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) − log � �1 − � n≥1 (an)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = � n≥1 (an + 1)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , |w| ≤ (a + 1)a+1 aa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Not surprisingly, this identity can be identified as the specialization of a more classical identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) is equivalent to a special case of the Rothe-Hagen identity [2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='146] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3) n � k=0 �x + kz k ��y − kz n − k � p + kq (x + kz) (y − kz) = p (x + y − nz) x (x + y) (y − nz) �x + y n � with the specialization x = a, y = (a + 1) n − 1, z = a + 1, p = a and q = a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' the general case 0 < a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We now address the case φa,b (x) = xa − xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The function ga,b can be computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The function ga,b (x) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4) gb−a a,b (x) = 1 − � n≥1 � a b − an � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Applying now Lagrange inversion formula to log r−1 a,b = (ra,b ◦ exp)−1 produces a power series expansion for the function − log ga,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' With b − a > 0, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='5) − log ga,b (x) = 1 b − a � n≥1 � a b − an + 1 � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This extends identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) to − log � �1 − � n≥1 � a b − an � n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = � n≥1 � a b − an + 1 � n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , |w| < �aa bb � 1 b−a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Integrals Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The integral of fa over [0, 1] is evaluated as � 1 0 fa (x) dx = 1 1 + a � 1 − aπ a + 1 cot aπ a + 1 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) whereas � x0 0 fa (x) dx = 1 2 (1 + a)2 � 1 + a + a2 − aπ cot aπ a + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We provide two proofs of the first result: one as a straightforward application of our main result Proposition 1, and another based on the expression of the integral (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) as a double integral, a clever technique that we borrow from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the general case, the inversion technique used above in the case b = a + 1 extends to the general case using the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 5 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Assume that b′ a′ = b a then gb−a a,b � y 1 a � = gb′−a′ a′,b′ � y 1 a′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' As a consequence, with a′ = a b−a and b′ = b b−a, ga′,a′+1 (x) = gb−a a,b � x 1 b−a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' These identities can be checked directly from the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4), or deduced from the remark (see [1]) that the function s (x) = (fa,b (xγ)) 1 γ satisfies the equation saγ (x) − sbγ (x) = xaγ − xbγ so that it coincides with faγ,bγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' From these identities, we deduce □ Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We have the following evaluations � 1 0 gb−a a,b (x) xb−a−1dx = 1 b − a � 1 − a 2b2 � b + (b − a) π cot �πa b ��� , � 1 0 f b−a a,b (x) xb−a−1dx = 2 − b + a − a b b − a − a b2 π cot �πa b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The limit Lambert case The function φ (x) = −x log x, x ∈ [0, 1] is shown to be the limit case of φ1,1+ϵ above as ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' It is a famous case of Lagrange inversion and is related to the Lambert and the Cayley tree functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' In the limit Lambert case, we have g (w) = 1 − � n≥1 (n − 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (−w log w)n , 0 ≤ w ≤ 1 and log g (w) = � n≥1 (−n)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (w log w)n = W0 (w log w) , where W0 is the principal branch of the Lambert function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The identity − log � �1 − � n≥1 (n − 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' zn � � = � n≥1 nn−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' zn is the limit case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Notice that we recover the identity log g (w) = log w, w ≥ 1 e, as a well-known property of the Lambert function W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We have the following integrals � 1 0 g (w) dw = 1 − � n≥1 (n − 1)n−1 (n + 1)n+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='659495 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � 1 0 f (w) dw = 2 − 2 � n≥1 (n − 1)n−1 (n + 1)n+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='31899 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � 1 0 − log g (w) w dw = π2 6 and � 1 0 − log f (x) x dx = π2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proofs 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Denote I1 = � x0 0 h (f (x)) h′ (x) dx, I2 = � 1 x0 h (f (x)) h′ (x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Changing variable y = f (x) and integrating by parts in I1 produces I1 = − [h (y) h (f (y))]1 x0 + � 1 x0 h′ (y) h (f (y)) dy = h2 (x0) − h (f (1)) h (1) + I2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) Considering now J1 = � x0 0 h (g (x)) h′ (x) dx, J2 = � 1 x0 h (g (x)) h′ (x) dx, we have, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3), J1 = I1 and J2 = � 1 x0 h (x) h′ (x) dx = 1 2 � h2 (1) − h2 (x0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Moreover, define J = J1 + J2 = � 1 0 h (g (x)) h′ (x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We deduce h2 (x0) = I1 − I2 + ¯h (1) h (1) = h2 (1) − 2J2 = h2 (1) − 2 (J − I1) so that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) I1 + I2 = 2J − h2 (1) + h (1) h (f (1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Next, I1 is deduced by solving the linear system that consists of the two equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Let us compute the inverse function r−1 a using Lagrange inversion formula: at w = 1, the function w �→ ra (w) = z = wa (1 − w) can be inverted since dra dw |w=1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The coefficients of the series expansion of its inverse w = r−1 a (z) = 1 + � k≥1 cn zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' are computed as cn = lim w→1 dn−1 dwn−1 � w − 1 φa,2 (w) − φa,2 (1) �n = lim w→1 dn−1 dwn−1 � w − 1 wa (1 − w) �n = (−1)n lim w→1 dn−1 dwn−1 � w−an� = − (an)n−1 with (a)n = Γ(a+n) Γ(a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We deduce (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3) r−1 a (z) = 1 − � k≥1 (an)n−1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ z ≤ φa (x0) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4) fa (x) = 1 − � n≥1 (an)n−1 (xa (1 − x))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ x ≤ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The series in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4) is convergent over R : let us define the function (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='5) ga (x) = 1 − � n≥1 (an)n−1 (xa (1 − x))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Noticing that (0)−1 = −1 allows to rewrite the desired identity as − log � �− � n≥0 (an)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = � n≥1 (an + 1)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Taking the derivative with respect to w on both sides produces � n≥0 (an + a + 1)n wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = − � n≥0 (an + a)n wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � n≥0 (an)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' so that we need to check the convolution identity � �� n≥0 (an + a + 1)n wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � � �� n≥0 (an)n−1 wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = − � n≥0 (an + a)n wn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' or equivalently � k≥0 (ak + a + 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (an − ak)n−k−1 (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = −(an + a)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This is indeed identity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='3) with the specialization x = a, y = (a + 1) n − 1, z = a + 1, p = a and q = a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The coefficients of the expansion are now computed as cn = lim w→0 dn−1 dwn−1 � wn enaw (1 − ew)n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Since � w ew − 1 �n e−naw = � k≥0 B(n) k (−na) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' wk is the generating function for the higher-order Bernoulli polynomials B(n) k (z), we deduce cn = B(n) n−1 (−na) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' These special values of the higher-order Bernoulli polynomials appear in [3, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='6] as B(n) n−1 (x) = (x − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (x − n + 1) so that cn = (−1)n (−na − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (−na − n + 1) = − (an + 1)n−1 and we obtain the power series expansion log r−1 a = � k≥1 (an + 1)n−1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' , 0 ≤ z ≤ φa (x0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Consider h (x) = x in identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The right-hand side integral is computed using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='5) as � 1 0 ga (x) dx = � 1 0 � �1 − � n≥1 (an)n−1 (xa (1 − x))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � dx = 1 − � n≥1 (an)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � 1 0 (xa (1 − x))n dx and evaluating the beta integral produces � 1 0 ga (x) dx = 1 − � n≥1 (an)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Γ (an + 1) Γ (n + 1) Γ (an + n + 2) = = 1 − � n≥1 Γ (an + 1) Γ (an + n − 1) Γ (an + n + 2) Γ (an) = 1 − a 2 (1 + a) � 1 − π a + 1 cot π a + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 8 Finally, as h (1) � h (1) − ¯h (1) � = 1, � 1 0 fa (x) dx = 2 � 1 0 ga (x) dx − 1 = 1 − a 1 + a � 1 − π a + 1 cot π a + 1 � = 1 a + 1 − aπ (a + 1)2 cot aπ a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Moreover � x0 0 fa (x) dx = 1 − a 2 (1 + a) � 1 − π a + 1 cot π a + 1 � + 1 2 � a2 (1 + a)2 − 1 � = 1 2 (1 + a)2 � 1 + a + a2 − aπ cot aπ a + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' A proof of Proposition 7 borrowed from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We first restate, for completeness, Andrews, Eriksson, Petrov and Romik’s elegant solution [4] to the evaluation of Ia,b = � 1 0 −log fa,b (x) x dx = π2 3ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Rewrite Ia,b as the double integral Ia,b = � 1 0 dx x � 1 fa,b(x) dy y = �� D dxdy xy over the domain D = {0 ≤ x ≤ 1, fa,b (x) ≤ y ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Dividing this domain into two equal subdomains and change variables � x y � → � x t = y x � so that dxdy → xdxdt and Ia,b = 2 �� D′ dxdt tx over the new domain D′ = � 0 ≤ t ≤ 1, � 1−ta 1−tb � 1 b−a ≤ x ≤ 1 � produces Ia,b = 2 � 1 0 � 1 � 1−ta 1−tb � 1 b−a dx x dt t = −2 � 1 0 log �1 − ta 1 − tb � 1 b−a dt = 2 b − a �� 1 0 log � 1 − tb� dt − � 1 0 log (1 − ta) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Substituting x = tb in the first integral and x = ta in the second provides the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This approach is now used to produce another proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='1) as follows: denote Ia = � 1 0 fa,a+1 (x) dx and rewrite it as the double integral Ia = � 1 0 � fa,a+1(x) 0 dydx = 2 �� D′ xdxdt with D′ = � 0 ≤ t ≤ 1, 0 ≤ x ≤ 1 − ta 1 − ta+1 � so that Ia = 2 � 1 0 1 2x2 (t) dt = � 1 0 � 1 − ta 1 − ta+1 �2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' This integral is evaluated using the change of variable t = ez producing Ia = � ∞ 0 � sinh � az 2 � sinh � a+1 2 z � �2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 9 This is the special case b = a 2, c = a+1 2 of Entry 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2 in [5] � ∞ 0 �sinh bx sinh cx �2 dx = 1 2c − πb 2c2 cot �πb c � so that Ia = 1 1 + a − aπ (a + 1)2 cot aπ a + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The coefficients of the series expansions of r−1 a,b are cn = lim w→0 dn−1 dwn−1 � wn enaw � 1 − ew(b−a)�n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Denoting c = b − a and expanding wn enaw (1 − ewc)n = � −1 c �n (cw)n (ewc − 1)n e−naw = � −1 c �n � k≥0 B(n) k � −n a c � k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (wc)k produces cn = (−1)n c B(n) n−1 � −na c � = − � n a b−a + 1 � n−1 b − a , so that, for 0 ≤ x ≤ x0, 1 b − a � n≥1 � a b − an + 1 � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = − log fa,b (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Replacing a with a b−a in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='2) produces � n≥1 � a b − an + 1 � n−1 � x a b−a − x b b−a �n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = − log � �1 − � n≥1 � a b − an � n−1 � x a b−a − x b b−a �n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The left hand-side is identified from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='5) as (b − a) ga,b � x 1 b−a � and we deduce (b − a) ga,b � x 1 b−a � = − log � �1 − � n≥1 � a b − an � n−1 � x a b−a − x b b−a �n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � or equivalently gb−a a,b (x) = 1 − � n≥1 � a b − an � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Start from � 1 0 gb−a a,b (x) xb−a−1dx = � 1 0 xb−a−1 � �1 − � n≥1 � a b − an � n−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � � = 1 b − a − � n≥1 � a b−an � n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � 1 0 xb−a−1 � xa − xb�n dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The integral is a beta integral evaluated as � 1 0 xb−a−1 � xa − xb�n dx = 1 b − a Γ � a b−an + 1 � Γ (n + 1) Γ � a b−an + n + 2 � UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 10 so that the sum is � n≥1 � a b − an � n−1 � 1 0 xb−a−1 � xa − xb�n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' dx = 1 b − a � n≥1 Γ � a b−an + n − 1 � Γ � a b−an � Γ � a b−an + 1 � Γ (n + 1) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='Γ � a b−an + n + 2 � = 1 b − a a b − a � n≥1 n Γ � a b−an + n − 1 � Γ � a b−an + n + 2 � and the latest sum is evaluated as � n≥1 n Γ � a b−an + n − 1 � Γ � a b−an + n + 2 � = (b − a) � b + (b − a) π cot � πa b �� 2b2 so that the desired integral is 1 b − a � 1 − a 2b2 � b + (b − a) π cot �πa b ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' The critical point is now x0 = e−1 and the inverse of the right-hand function is computed as r−1 (z) = 1 + � n≥1 cn zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' with the coefficients cn = lim w→1 dn−1 dwn−1 � w − 1 −w log w �n = − (n − 1)n−1 , n ≥ 1 with the convention 00 = 1 so that c1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We deduce g (w) = r−1 (l (w)) = 1 − � n≥1 (n − 1)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (−w log w)n , 0 ≤ w ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' If we now apply Lagrange’s inversion theorem to (log ◦r)−1 = r−1 ◦ exp, we obtain the new coefficients cn = lim w→0 dn−1 dwn � w −eww �n = −nn−1 so that log g (w) = − � n≥1 nn−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (−w log w)n = � n≥1 (−n)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (w log w)n = W0 (w log w) , the principal branch of the Lambert function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Proof of Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Using � 1 0 (w log w)n dw = (−1)n Γ (n + 1) (n + 1)n+1 , n ≥ 0, produces � 1 0 g (w) dw = 1 − � n≥1 (n − 1)n−1 (n + 1)n+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='728466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Moreover, since � 1 0 wn−1 (log w)n dw = (−1)n nn (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=', we deduce � 1 0 − log g (w) w dw = � 1 0 −W0 (w log w) w dw = � n≥1 (−n)n−1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' � 1 0 wn−1 (log w)n dw = � n≥1 (−n)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' (−1)n nn (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' = � n≥1 1 n2 = ζ (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' We now use the formula � 1 0 h (fa (x)) h′ (x) dx = 2 � 1 0 h (ga (x)) h′ (x) dx − h (1) (h (1) − h (f (1))) UNEXPECTED LOGARITHMIC IDENTITIES AND OTHER SURPRISES 11 to deduce � 1 0 − log f (x) x dx = 2 � 1 0 − log (g (x)) x − h2 (1) + h (1) h (f (1)) with h (x) = −x log x and h (1) = 0, h (f (1)) = limx→1 h (f (x)) = limx→0 −x log x = 0 so that � 1 0 − log f (x) x dx = 2 � 1 0 − log (g (x)) x = 2ζ (2) = π2 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Holroyd, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Liggett and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Romik, Integrals, Partitions, and Cellular Automata, Transactions of the American Mathematical Society, 356-8, 3349-3368, 2004 [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Gould, Combinatorial identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' a standardized set of tables listing 500 binomial coefficient summations, 1972 [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Nörlund, Vorlesungen über Differenzenrechnung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Springer-Verlag, Berlin (1924) [4] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Andrews, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Eriksson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Petrov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Romik, Integrals, partitions and MacMahon’s Theorem, Journal of Combinatorial Theory, Series A 114 (2007) 545–554 [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Prudnikov, Integrals and Series, Volume I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} +page_content=' Gordon and Breach Science Publishers, 1990 Department of Mathematics, Tulane University, New Orleans, USA and LSS, CentraleSupelec, Université Paris- Sud Orsay, France' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf'} diff --git a/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/2301.05037v1.pdf.txt b/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/2301.05037v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3d3ae91af67d3805d5a0e84d634683b995d66e09 --- /dev/null +++ b/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/2301.05037v1.pdf.txt @@ -0,0 +1,981 @@ +Dymnikova-Schwinger traversable wormholes +Milko Estrada∗1 and C. R. Muniz†2 +1Facultad de Ingenier´ıa, Ciencia y Tecnolog´ıa, +Universidad Bernardo O’Higgins, Santiago, Chile. +2Universidade Estadual do Cear´a (UECE), Faculdade de Educa¸c˜ao, +Ciˆencias e Letras de Iguatu, 63500-000, Iguatu, CE, Brazil. +Abstract +In this paper, we obtain new d-dimensional and asymptotically flat wormhole solutions with the +presence of determined matter fields in the energy-momentum tensor. This is made by generalizing +and adjusting to our purposes the so-called Dymnikova model, originally studied in the context +of regular black holes. Thus, we find the constraints of the involved parameters to the formation +of those wormholes. +Following, we study the properties of such solutions, namely, embedding +diagrams, Weak and Null Energy Conditions (WEC and NEC), as well as position-dependent state +parameters obeying a linear EoS. We show that the larger the dimension, the larger the flatness +of the wormhole and the more pronounced the violation of the energy conditions. We also show +that the corresponding fluid behaves as phantom-like in all the space for d ≥ 4. Furthermore, we +specialize the employed model for d = 4 spacetime, associating it to the gravitational analogue of +the Schwinger effect in a vacuum and correcting the model by introducing a fundamental minimal +length via Generalized Uncertainty Principle (GUP). Considering a very small minimal length, we +obtain a novel traversable and asymptotically flat wormhole solution. The embedding diagram +shows that the presence of that length increases the slope of the wormhole towards its throat +compared with the case without it. The correction also attenuates the WEC (and NEC) violations +nearby the throat, with the fluid ceasing to be a phantom-type at the Planck scale. +Keywords: Dymnikova vacuum; Wormholes; Energy conditions. +∗ E-mail: milko.estrada@gmail.com +† E-mail: celio.muniz@uece.br +1 +arXiv:2301.05037v1 [gr-qc] 12 Jan 2023 + +I. +INTRODUCTION +Wormholes are hypothetical objects with non-trivial geometry and topology predicted by +general relativity (GR), representing a kind of tunnel in the spacetime that connects two +remote regions of the same universe or two different universes (see [1, 2], and references +therein). +These theoretical structures have been recently investigated in a fundamental +level from the J. Maldacena works [3–5] and also in more applied contexts of condensed +matter systems [6, 7]. Usually, it is required some type of exotic matter sourcing traversable +wormholes. However, in scenarios of modified theories of gravity, such a feature can change +with non-exotic matter working as a source for the wormhole geometry [8–14]. Furthermore, +it is worth mentioning that, because there are several branches of theoretical physics that +have predicted the existence of extra dimensions (as for example the string theory), the +study of higher dimensional wormholes also has been of physical interest in the last years. +See [15, 16], for instance. +An example of a source of exotic matter that can sustain a traversable wormhole is the +Casimir vacuum energy. Thus, wormholes with Casimir-type energy density profiles have +been studied in the last years [16–20]. In particular, [18] considered models which introduce +a fundamental minimal length via Generalized Uncertainty Principle (GUP) in order to +correct that energy and study wormholes solutions in the GR context. +On the other hand, finding analytic wormhole solutions to the Einstein field equations, +with the presence of matter sources in the energy-momentum tensor, is not an easy task +because of the highly nonlinear behavior of the equations of motion. One very common +strategy consists in using different state equations in order to obtain those solutions. Some +examples can be found in references [21, 22]. Among other examples of wormhole solutions +with energy density profiles we have the Class one approach [23], the dark matter energy +profile [24, 25], Yukawa–Casimir wormholes [26], etc. +In this work, we will study new traversable and asymptotically flat wormhole solutions +with local sources of matter. Thus, we will employ the d-dimensional generalization [27] of +the Dymnikova energy density [28]. This model has been used to generate regular black hole +solutions (RBH), such that its energy is quasi-localized at infinity. Such a model of RBH +is restricted to the case where ρ = −pr and gtt = g−1 +rr . It is worth mentioning that, clearly, +both the geometric and the physical characteristics of the wormhole solutions differ from the +2 + +RBH solutions. In the wormhole context, the concept of quasi–localized energy at infinity +seems to have no physical meaning. In this sense, our wormhole model will differ from the +Dymnikova RBH model. +Therefore, we will obtain a novel family of d-dimensional and asymptotically flat worm- +hole solutions, showing that the employed material source is capable of forming traversable +wormholes under specific conditions (principally the flaring-out one). We will study their +principal properties, such as embedding diagrams, Weak and Null Energy Conditions (WEC +and NEC), and the position-dependent state parameter. Following, we will specialize this +analysis for a 4-dimensional universe and, by formally associating the Dyminikova density +profile with the Schwinger particle-antiparticle pair production in a vacuum according to +[29, 30], we will correct the model introducing a fundamental minimal length via Generalized +Uncertainty Principle (GUP) following [31, 32]. With this, we find a novel traversable and +asymptotically flat wormhole solution and also study those properties, remarking on the role +of the minimal length in them. +The paper is structured as follows: In Section II we review the basic features of d- +dimensional traversable wormholes and their sources. In section III we enunciate a list of +conditions that must satisfy our model. In Section IV, we present our model and build new +d-dimensional wormhole solutions, studying the corresponding properties. In Section V we +present and study Dyminikova-Schwinger GUP-corrected wormhole solutions. Finally, in +Section VI, we conclude the paper. +II. +A SHORT REVIEW OF d-DIMENSIONAL WORMHOLES +The following line element describes the geometry of a static and spherically symmetric +Lorentzian traversable wormhole in a spacetime d-dimensional [16]: +ds2 = −e2Φ(r)dt2 + +dr2 +1 − b(r)/r + r2dΩ2 +d−2, +(1) +where Φ(r) and b(r) are arbitrary functions of the radial coordinate, r, denoted as the redshift +function, and the shape function, respectively. That coordinate decreases from infinity to a +minimum value r0, the radius of the throat, where b(r0) = r0. The quantity dΩd−2 is the +solid angle element in d − 2 spacetime dimensions. +For this spacetime to represent a wormhole solution, the following conditions must be +3 + +satisfied[21] +1. The function Φ(r) must be finite for all value of r ≥ r0 in order to avoid singularities +and horizons. +2. The minimum value of r at the throat of the wormhole, corresponds to the point +r = r0 = b(r0), where the function g−1 +rr = 1 − b(r)/r vanishes +3. The proper radial distance +l(r) = ± +ˆ r +r0 +dr +� +1 − b(r) +r +(2) +must be finite. For this it is necessary that +1 − b(r) +r +≥ 0 +(3) +for all value of r ≥ r0. +4. For the asymptotic flatness condition it must be satisfied that +lim +r→∞ +b(r) +r +→ 0 +(4) +This ensures that at infinity the proper radial distance l → ±∞. +III. +OUR MODEL +Now we provide a list of constraints that the energy density as well as both the redshift and +shape functions must satisfy for our model to represent a d-dimensional and asymptotically +flat wormhole solution: +(a) The energy density must be a continuous, positive and decreasing function such that +lim +r→∞ ρ(r) → 0 +(5) +Furthermore, as we will see below, through the equations of motion, this energy density +leads to a function b(r) of the form +b(r) = +¯b(r) +rd−4 +(6) +4 + +where ¯b(r) must be an increasing function, such that, due to the condition (5), it must +be satisfied that: +lim +r→∞ +¯b(r) = Constant +(7) +This latter ensures condition 4 of introduction. +(b) Relating with condition 2 above, the largest solution of the equation g−1 +rr = 1−b/r = 0 +corresponds to the minimum value of r = r0 at the wormhole throat. +(c) The function b(r), equation (6), must be such that the function g−1 +rr = 1 − b/r be an +increasing function from g−1 +rr (r = r0) = 0 up to g−1 +rr (r → ∞) → 1 (value provided by +equation 7). So, it must be satisfied that +(g−1 +rr )′|r≥r0 > 0 +(8) +This item ensures condition 3 of introduction. +(d) In order to avoid singularities, with a finite value of e2Φ(r), condition 1 of introduction, +and in order to have an asymptotically flat behavior, we impose that +e2Φ(r0) = A +(9) +with 0 < A ≤ 1, and +lim +r→∞ e2Φ(r) = 1 +(10) +Furthermore the derivative +� +e2Φ(r)�′ ≥ 0 +(11) +for r ≥ r0. This ensures that the function e2Φ(r) can be either a constant (for A = 1 +and +� +e2Φ(r)�′ = 0, i.e the zero tidal case) or an increasing function. Furthermore we +impose that : +lim +r→∞ Φ′ = 0 +(12) +For the line element (1), the (t, t) component of the d− dimensional field equations is: +� +rd−4b(r) +�′ = +2 +d − 2rd−2ρ(r) +(13) +5 + +where we have made Einstein’s constant κ = 1 and where ρ is constructed such that the +conditions (a) of section III must be satisfied. This energy density gives rise to the following +solution +b(r) = +¯b(r) +rd−4 +(14) +where b and ¯b must also satisfy the conditions (a) of section III and where +¯b(r) = +2 +d − 2 +ˆ +rd−2ρ(r)dr +(15) +Using b(r) and Φ(r) (which are constructed such that the conditions (d) must be satisfied), +we can determinate the radial pressure using the (r, r) component +pr = d − 2 +2r2 +�� +1 − b +r +� +(2rΦ′ + (d − 3)) − (d − 3) +� +(16) +Furthermore, from equations (7) and (12) : +lim +r→∞ pr → 0 +(17) +The energy-momentum tensor has the form T µ +ν = diag(−ρ, pr, pθ, pφ, ...). From spherical +symmetry we have for all the (d − 2) angular coordinates pt = pθ = pφ = ... and, the +conservation law T AB +;B += 0 gives: +(ρ + pr)Φ′ + p′ +r + d − 2 +r +(pr − pt) = 0. +(18) +So, using the last equation can be determined the tangential pressure values. It is direct to +check that +lim +r→∞ pt = 0 +(19) +IV. +THE NEW FAMILY OF WORMHOLE SOLUTIONS +So far it has been shown a suitable model to describe an higher dimensional and asymp- +totically flat wormhole solution. Thus, we have not proposed a specific form for the energy +density ρ and the Φ function, and for the further values of b(r) , pr and pt. Thus, we will +employ here the d-dimensional generalization [27] of the Dymnikova energy density [28]. In +four dimensions this model of energy density has give rise to regular black hole solutions, +as it was already mentioned in Introduction, as well as to wormhole magnetic monopoles, +6 + +under the constraints ρ = −pr, studied in reference [33] in a different framework from what +is made here. +The energy density profile is given by +ρ(r) = d − 2 +2 +ρ0 exp +� +−rd−1 +ad−1 +� +(20) +where ρ0, a > 0 are constants. It is direct to check that this model of energy density is +consistent with the conditions (a). +Replacing in equation (15): +¯b(r) = ad−1ρ0 +(d − 1) +� +1 − exp +� +−r +a +�d−1� +. +(21) +where has been used the value +ad−1ρ0 +(d−1) as integration constant. It is direct to check that +condition (7) is satisfied. +Thus, it is worth mentioning that the conditions (a) of section III are satisfied. +From equation (14) +b(r) = +ad−1ρ0 +(d − 1)rd−4 +� +1 − exp +� +−r +a +�d−1� +. +(22) +In order to satisfy the condition (b) of section III, one has b(r0) = r0, where r0 is the +throat radius, so we must find a relationship between ρ0, d, a, and r0, which is given by +ρ0 = (d − 1)a1−drd−3 +0 +1 − exp +� +− r0 +a +�d−1. +(23) +Thus, the g−1 +rr metric component is +g−1 +rr = 1 − rd−3 +0 +rd−3 +� +1 − exp +� +− r +a +�d−1 +1 − exp +� +− r0 +a +�d−1 +� +(24) +As it was mentioned in conditions (b) and (c), r0 must represent the largest solution of +the equation g−1 +rr = 0 and furthermore the derivative (g−1 +rr )′|r=r0 must be positive. It is easy +to check that, for r ∈ [0, ∞], there is a critical value r = r∗, where r∗ represents a point +where the function g−1 +rr reaches a local minimum. So, r0, such that g−1 +rr (r = r0) = 0, must +be located after the critical value, i.e r0 > r∗. This occurs provided r0/a > β(d), where +β(d) is a number of order of unit and that can be calculated numerically from the Lambert +function, W−1(z). As for example for d = 4, β(4) ≈ 1.24, for d = 5, β(5) ≈ 1.06, and for +d = 8, β(8) = 0.94. So, the conditions (b) and (c) are satisfied. Fig. 1 shows us the behavior +7 + +of the function g−1 +rr = F(r) = 1 − b(r)/r. Notice that the obtained wormhole solutions are +asymptotically flat, since when r → ∞, b(r)/r → 0, for all d. +As redshift function Φ(r) we choose in arbitrary way the following form: +e2Φ(r) = 1 − C exp +� +−r − r0 +a +� +(25) +where it is easy to check that C = 0 represents the zero-tidal case. It is easily noted that +the condition (10) is satisfied. The function evaluated at r = r0 is: +e2Φ(r = r0) = 1 − C = A +(26) +which in order to satisfy (9) must satisfy that +0 ≤ C < 1 +(27) +Furthermore it is direct to check that the conditions (11) and (12) are satisfied. So, this +function serves as test of prove in order to satisfy the condition (d). +The value of the radial pressure is obtained easily replacing equations (23), 22 and (25) +into the equation (16). The tangential pressure is computed from equation (18). +A. +Flaring-out condition +A fundamental property of a wormhole is that a flaring-out condition of the throat, given +by (b−b′r)/b2 > 0 [1], and at the throat b(r0) = r = r0, such that the condition b′(r0) < 1 is +imposed to have wormhole solutions. It is precisely these restrictions that impose the WEC +and NEC violations in classical general relativity. For our model, using the condition (8), +i.e. (1 − b/r)′|r=r0 > 0 it is direct to check that the condition b′(r0) < 1 is automatically +satisfied, and thus, the flaring out condition is satisfied for r = r0 . Furthermore, using the +same condition for (1 − b/r)′|r>r0 > 0 it is also direct to check that the flaring out condition +is satisfied for r > r0. +B. +Embedding diagram +This diagram is usually obtained by comparing the spatial three-dimensional flat met- +ric written in cylindrical coordinates (r, φ, z) with the spatial sector of the Morris-Thorne +8 + +d = 4 +d = 5 +d = 8 +20 +30 +40 +50 +r +0.2 +0.4 +0.6 +0.8 +1.0 +F +FIG. 1: Behavior of the function F(r) = 1−b(r)/r, for some spacetime dimensions. The parameter +set is r0 = 10.0 and a = 8.0. +metric (fixing the polar angle at θ = π/2). The same reasoning can be generalized for a d- +dimensional spacetime, with the flat spatial sector being compared with the d−1-dimensional +spatial sector of the spherical wormhole (fixing also the (d − 3) polar angular coordinates at +π/2), and identifying the azimuthal angles, arriving at +dz = ±dr +� +b(r)/r +1 − b(r)/r. +(28) +. By means of equation (7) it is easy to check that +lim +r→∞ +dz +dr = 0 +(29) +We can see the generic behavior in figure 4. We can note that in higher dimensions the +flatness of the wormhole is bigger. +C. +Energy conditions +In figures 3 we can test the behavior of the Null Energy Conditions (NEC ρ + pi ≥ 0), +Weak Energy Conditions (WEC ρ ≥ 0, ρ + pi ≥ 0). We can test that these behaviors are +generic for other values of d. In the second figure we can note that WEC (and NEC) are +violated because the condition ρ + pr ≥ 0 is only satisfied for r → ∞. +In figure IV C we analyze the exoticity of this matter by considering the state parameter +of a perfect fluid, which is dependent on the radial coordinate, according to EoS ω(r) = pr/ρ. +We can notice that the fluid behaves like a phantom fluid since ω(r) < −1 in all the space, +9 + +0 +20 +40 +60 +80 +100 +-50 +0 +50 +r +z +d = 4 +d = 5 +d = 6 +FIG. 2: Embedding diagram. The parameter settings are a = 8 and r0 = 10, in Planck units for +d = 4, 5, 6. +in any dimension. Below, we will see that, in the 4–dimensional scenario, the presence of a +fundamental length modifies this behavior near the throat. +10 +11 +12 +13 +14 +15 +16 +17 +0.000 +0.002 +0.004 +0.006 +0.008 +0.010 +r +ρ +ρ(d = 4) +ρ(d = 5) +ρ(d = 6) +10 +15 +20 +25 +30 +-0.04 +-0.03 +-0.02 +-0.01 +0.00 +r +ρ+pr +ρ(d = 4)+ pr(d = 4) +ρ(d = 5)+ pr(d = 5) +ρ(d = 6)+ pr(d = 6) +10 +12 +14 +16 +18 +20 +22 +24 +-0.01 +0.00 +0.01 +0.02 +0.03 +r +ρ+pt +ρ(d = 4)+ pt(d = 4) +ρ(d = 5)+ pt(d = 5) +ρ(d = 6)+ pt(d = 6) +FIG. 3: Fist figure : ρ. Second figure : ρ+pr. Third figure : ρ+pr. The parameter set is r0 = 10.0, +a = 8.0, C = 0.6 and d = 4, 5, 6. +10 + +1.00 +1.05 +1.10 +1.15 +1.20 +1.25 +1.30 +1.35 +1.40 +-8 +-6 +-4 +-2 +0 +r +pr/ρ +pr (d=4) +ρ(d=4) +pr (d=5) +ρ(d=5) +pr (d=6) +ρ(d=6) +FIG. 4: The parameter state, with parameter settings a = 0.8, r0 = 1.0, and C = 0.6, in Planck +units, for d = 4, 5, 6. +V. +DYMNIKOVA-SCHWINGER 4D WORMHOLE AND THE INFLUENCE OF +A MINIMAL LENGTH +According to [29, 30], the d = 4 Dymnikova density profile can be seen as the gravi- +tational analogue of the electron-positron pair production rate, Γ ∼ exp (−Ec/E), in the +vacuum – the so-called Schwinger effect. This QED phenomenon is associated with the ap- +plication of an intense uniform electric field (E) that results in vacuum polarization and the +corresponding production of particle pairs. The critical electric field necessary for abundant +pair production is given by Ec = πℏm2 +e/e, where me and e are the electron mass and charge, +respectively. The gravitational equivalent is considered when one makes the association of +the electric field with the gravity tension characterized by a curvature term, namely E ∼ r−3 +and Ec ∼ a−3, so that we obtain the d = 4 Dymnikova-Schwinger density profile of Eq. (20). +The correction to the Schwinger effect associated to the existence of a minimal length +was obtained in [31, 32] by means of the Generalized Uncertainty Principle (GUP). In this +case, the electron-positron pair production rate becomes +Γ ∼ exp +� +−πℏm2 +e +eE ++ απ3eE +� +, +(30) +where α comes from GUP via +∆x∆p ∼ ℏ +2 +� +1 + α(∆p)2 +ℏ +� +, +(31) +with α = ℓ2. Here ℓ is the minimal length. Identifying the electrical field with the grav- +itational tension as it was previously discussed, the GUP correction to the Dymnikova- +11 + +Schwinger density profile will be, therefore +ρ(r) = ρ0 exp +� +−r3 +a3 + αa +r3 +� +≈ ρ0 exp +� +−r3 +a3 +� � +1 + αa +r3 +� +, +(32) +where we have considered in the second relation that the minimal length is very small, +α/r2 ≪ 1, and r ≥ r0, which is valid for any wormhole. +The shape function of Dymnikova-Schwinger GUP-corrected wormhole can be obtained +from the (t, t) component of Einstein’s equations, (13), for d = 4, with the energy density +profile (32), and it can be written as +b(r) = r0 +� +1 − exp +� +− r3 +a3 +� ++ α +a2Ei +� +− r3 +a3 +�� +� +1 − exp +� +− r3 +0 +a3 +� ++ α +a2Ei +� +− r3 +0 +a3 +��, +(33) +so that the integration constant was chosen in order to do b(r0) = r0, with Ei(z) being the +exponential integral function. The metric of the wormhole under consideration is, therefore, +ds2 = − +� +1 − C exp +� +−r − r0 +a +�� +dt2 + +dr2 +1 − r0 +r +� +1−exp +� +− r3 +a3 +� ++ α +a2 Ei +� +− r3 +a3 +�� +� +1−exp +� +− +r3 +0 +a3 +� ++ α +a2 Ei +� +− +r3 +0 +a3 +�� ++ r2dΩ2, +(34) +where we have taken into account the redshift function given in Eq. (25). +r +z +10 +20 +30 +40 +-60 +-40 +-20 +20 +40 +60 +a = 0 +a = 1 +r +z +10 +15 +20 +25 +30 +10 +20 +30 +40 +50 +FIG. 5: Embedding diagram of the Dymnikova-Schwinger GUP-corrected wormhole profile, in the +left panel. The right panel exhibits the greater slope of the GUP-corrected wormhole (α ̸= 0). The +parameter settings are a = 8.0 and r0 = 10.0, in Planck units. +In Fig. 5 we depict the embedding diagram profiles of the Dymnikova-Schwinger GUP- +corrected wormhole. Replacing into the above equation the expression (33) and integrating +it, we obtain the corresponding embedding diagrams, for both α = 0 and α ̸= 0 Dymnikova- +Schwinger wormholes. +By considering these diagrams, we note that the presence of the +12 + +minimal length increases the slope of the wormhole, which is more accentuated the greater +this length. +A. +Energy conditions and state parameter +Null energy conditions (NEC) are not obeyed by wormholes, at least in the context of +GR, provided the flaring-out conditions are valid. Thus, Fig. 6 shows us the behaviors of +Sr = ρ+pr (density with radial pressure - left panel) and St = ρ+pt (density with transversal +pressure - right panel), for Dymnikova-Schwinger GUP corrected wormholes. Considering +the behavior of Sr, the violation is more pronounced for the zero-tidal wormhole (C = 0). +C = 0.0 +C = 0.6 +Sr +r +12 +14 +16 +18 +20 +22 +24 +-0.004 +-0.003 +-0.002 +-0.001 +C = 0.0 +C = 0.6 +St +r +12 +14 +16 +18 +20 +22 +24 +0.002 +0.004 +0.006 +0.008 +0.010 +FIG. 6: Sum of density with radial (transversal) pressure, in left (right) panel, as a function of the +radial coordinate, r, for D-S GUP corrected wormholes, considering both zero tidal (C = 0) and +non-zero tidal cases. The parameter settings are a = 8.0, r0 = 10.0, α = 1.0, in Planck units. +Now let us analyze the influence of the minimal length on NECs. Although these condi- +tions remain still unfulfilled demanding the presence of exotic matter, the occurrence of a +fundamental minimal length attenuates this violation. The left panel of Fig. 7 reveals this +feature, on exhibiting that Sα=0 +r +< Sα̸=0 +r +, nearby the wormhole throat. +We can also analyze the exoticity of the material source by considering the state parameter +of a perfect fluid, which is dependent on the radial coordinate, according to EoS ω(r) = pr/ρ. +In the right panel of Fig. 7, we can notice that the fluid is quintessential nearby the throat +(−1 < ω < −1/3) and tends to be a phantom fluid far from the throat, since ω(r) < −1. +This panel reveals therefore that the presence of the minimal length increases the state +parameter value associated to the source. +Thus, for Planckian wormholes, the minimal +length reduces the exoticity of the source in such a manner that it is no longer phantom-like +13 + +a = 0 +a = 1 +r +sr +10.1 +10.2 +10.3 +10.4 +10.5 +10.6 +10.7 +-0.0020 +-0.0015 +-0.0010 +-0.0005 +wHrL +r +a = 0.0 +a = 0.1 +1.00 +1.05 +1.10 +1.15 +1.20 +-2.0 +-1.5 +-1.0 +FIG. 7: In the left panel, the sum of density with the radial pressure, Sr, as a function of the radial +coordinate, for a = 8.0, r0 = 10, and C = 0.6, for both the Dymnikova-Schwinger (α = 0.0) and +D-S GUP corrected (α = 1.0) wormholes. In the right panel, the parameter state of D-S (α = 0.0) +and D-S GUP corrected (α = 0.1) wormholes, with parameter settings a = 0.8, r0 = 1.0, and +C = 0.6, in Planck units. +nearby the throat. These finds corroborate the analysis of NEC made above. +VI. +CONCLUSION +In this work, we have found new d-dimensional and asymptotically flat wormhole solutions +with the presence of matter fields in the energy-momentum tensor. The energy density and +the pressure components have local values for different values of the radial coordinate, i.e +are the so-called localized sources of matter. For this latter, in section III we have proposed +a list of constraints that the energy density as well as both the redshift and shape functions +must satisfy in such a manner that the new wormhole solutions fulfill the corresponding +criteria. Thus, we have initially used the d-dimensional generalization [27] of the Dymnikova +energy density [28]. Then, from this matter distribution and solving the time component +of Einstein’s equation, we have found wormhole exact solutions in d spacetime dimensions, +which depend on the throat radius, r0, and on a characteristic length, a. This latter is +a typical scale at which the source scatters through space. We have also defined a suited +logarithmic redshift function dependent on these parameters and a factor that, on vanishing, +one obtains a zero-tidal simplest traversable wormhole. Remarkably the energy density, the +redshift function, and the pressure components satisfy the criteria to the building of a +wormhole. +14 + +The detailed study of the conditions for the formation of these objects (as the flaring-out +one) has shown that there has to be a relationship between r0, a, and the spacetime dimen- +sion d in order to guarantee their existence. Following, we have generated the embedding +diagram for some of these wormholes, showing that on increasing the spacetime dimension +the asymptotic flatness is more quickly reached. In other words, the greater d the smaller +the slope towards the wormhole throat. +With respect still to d-dimensional Dymnikova wormholes, we have analyzed Weak and +Null Energy Conditions (WEC and NEC) concerning the material source. As expected, this +latter is considered exotic, since the radial part of those conditions is violated – while the +transversal one is not – and such a violation is more pronounced the larger the spacetime +dimension, in regions nearby the wormhole throat. +This feature is corroborated by the +behavior of the position-dependent state parameter ω(r) = pr/ρ, namely, the quantity that +relates the radial pressure with the matter density, when one supposes that the source is an +ideal fluid. Its behavior indicates that such a fluid is phantom-like since ω(r) < −1 in all +space. +In sequence, we specialized the previous study for 4d wormholes, which we have called +Dymnikova-Schwinger since the material source under consideration is the gravitational ana- +logue of the pair density produced in a vacuum by means of the application of an intense +(electric) field – the so-called Schwinger effect, predicted by QED. The gravitational coun- +terpart of this a phenomenon holds some similarity with that one associated to the Casimir +effect, which has been widely studied in the context of wormholes [16–20]. Thus, on con- +sidering the correction to the Schwinger effect due to the introduction of a minimal length +(ℓ = √α) via GUP [31, 32], we have found the corresponding correction to the gravitational +analogue of Dymnikova’s matter distribution. Supposing that the minimal length is very +small, we obtained a novel traversable and asymptotically flat wormhole solution that re- +duces to the one previously studied when α = 0 and d = 4, for the same redshift function +employed before. +The corresponding embedding diagram shows that the presence of the minimal length +increases the slope of the wormhole towards its throat compared with the case without such +a length – in other words, it lessens the wormhole flatness. On the other hand, the WEC +and NEC analysis have shown that this quantity, although still yields the non-fulfillment +of those conditions, attenuates their violation since ρ + pr increases with it. Finally, the +15 + +study of the state parameter ω(r) has revealed that, in presence of the fundamental length, +the material source ceases to be a phantom-type nearby the wormhole throat, at least at +Planck’s scale. +Acknowledgments +Celio R. 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C80, 8, 777 (2020), doi: 10.1140/epjc/s10052-020-8363-2, [arXiv: +2005.12075 [gr-qc]]. +[33] J. +M. +Romero +and +M. +Bellini, +Eur. +Phys. +J. +Plus +134, +no.11, +579 +(2019) +doi:10.1140/epjp/i2019-12926-1 [arXiv:1906.00062 [gr-qc]]. +18 + diff --git a/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/load_file.txt b/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5143c15adf150db86a9d952288760d6875970c9b --- /dev/null +++ b/vdE4T4oBgHgl3EQfXAw5/content/tmp_files/load_file.txt @@ -0,0 +1,625 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf,len=624 +page_content='Dymnikova-Schwinger traversable wormholes Milko Estrada∗1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Muniz†2 1Facultad de Ingenier´ıa, Ciencia y Tecnolog´ıa, Universidad Bernardo O’Higgins, Santiago, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 2Universidade Estadual do Cear´a (UECE), Faculdade de Educa¸c˜ao, Ciˆencias e Letras de Iguatu, 63500-000, Iguatu, CE, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Abstract In this paper, we obtain new d-dimensional and asymptotically flat wormhole solutions with the presence of determined matter fields in the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This is made by generalizing and adjusting to our purposes the so-called Dymnikova model, originally studied in the context of regular black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, we find the constraints of the involved parameters to the formation of those wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Following, we study the properties of such solutions, namely, embedding diagrams, Weak and Null Energy Conditions (WEC and NEC), as well as position-dependent state parameters obeying a linear EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We show that the larger the dimension, the larger the flatness of the wormhole and the more pronounced the violation of the energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We also show that the corresponding fluid behaves as phantom-like in all the space for d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Furthermore, we specialize the employed model for d = 4 spacetime, associating it to the gravitational analogue of the Schwinger effect in a vacuum and correcting the model by introducing a fundamental minimal length via Generalized Uncertainty Principle (GUP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Considering a very small minimal length, we obtain a novel traversable and asymptotically flat wormhole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The embedding diagram shows that the presence of that length increases the slope of the wormhole towards its throat compared with the case without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The correction also attenuates the WEC (and NEC) violations nearby the throat, with the fluid ceasing to be a phantom-type at the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Keywords: Dymnikova vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Wormholes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Energy conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' ∗ E-mail: milko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='estrada@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='com † E-mail: celio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='muniz@uece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='br 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='05037v1 [gr-qc] 12 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' INTRODUCTION Wormholes are hypothetical objects with non-trivial geometry and topology predicted by general relativity (GR), representing a kind of tunnel in the spacetime that connects two remote regions of the same universe or two different universes (see [1, 2], and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' These theoretical structures have been recently investigated in a fundamental level from the J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Maldacena works [3–5] and also in more applied contexts of condensed matter systems [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Usually, it is required some type of exotic matter sourcing traversable wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' However, in scenarios of modified theories of gravity, such a feature can change with non-exotic matter working as a source for the wormhole geometry [8–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Furthermore, it is worth mentioning that, because there are several branches of theoretical physics that have predicted the existence of extra dimensions (as for example the string theory), the study of higher dimensional wormholes also has been of physical interest in the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' See [15, 16], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' An example of a source of exotic matter that can sustain a traversable wormhole is the Casimir vacuum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, wormholes with Casimir-type energy density profiles have been studied in the last years [16–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In particular, [18] considered models which introduce a fundamental minimal length via Generalized Uncertainty Principle (GUP) in order to correct that energy and study wormholes solutions in the GR context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' On the other hand, finding analytic wormhole solutions to the Einstein field equations, with the presence of matter sources in the energy-momentum tensor, is not an easy task because of the highly nonlinear behavior of the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' One very common strategy consists in using different state equations in order to obtain those solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Some examples can be found in references [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Among other examples of wormhole solutions with energy density profiles we have the Class one approach [23], the dark matter energy profile [24, 25], Yukawa–Casimir wormholes [26], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In this work, we will study new traversable and asymptotically flat wormhole solutions with local sources of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, we will employ the d-dimensional generalization [27] of the Dymnikova energy density [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This model has been used to generate regular black hole solutions (RBH), such that its energy is quasi-localized at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Such a model of RBH is restricted to the case where ρ = −pr and gtt = g−1 rr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is worth mentioning that, clearly, both the geometric and the physical characteristics of the wormhole solutions differ from the 2 RBH solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In the wormhole context, the concept of quasi–localized energy at infinity seems to have no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In this sense, our wormhole model will differ from the Dymnikova RBH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Therefore, we will obtain a novel family of d-dimensional and asymptotically flat worm- hole solutions, showing that the employed material source is capable of forming traversable wormholes under specific conditions (principally the flaring-out one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We will study their principal properties, such as embedding diagrams, Weak and Null Energy Conditions (WEC and NEC), and the position-dependent state parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Following, we will specialize this analysis for a 4-dimensional universe and, by formally associating the Dyminikova density profile with the Schwinger particle-antiparticle pair production in a vacuum according to [29, 30], we will correct the model introducing a fundamental minimal length via Generalized Uncertainty Principle (GUP) following [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' With this, we find a novel traversable and asymptotically flat wormhole solution and also study those properties, remarking on the role of the minimal length in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The paper is structured as follows: In Section II we review the basic features of d- dimensional traversable wormholes and their sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In section III we enunciate a list of conditions that must satisfy our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In Section IV, we present our model and build new d-dimensional wormhole solutions, studying the corresponding properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In Section V we present and study Dyminikova-Schwinger GUP-corrected wormhole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Finally, in Section VI, we conclude the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' A SHORT REVIEW OF d-DIMENSIONAL WORMHOLES The following line element describes the geometry of a static and spherically symmetric Lorentzian traversable wormhole in a spacetime d-dimensional [16]: ds2 = −e2Φ(r)dt2 + dr2 1 − b(r)/r + r2dΩ2 d−2, (1) where Φ(r) and b(r) are arbitrary functions of the radial coordinate, r, denoted as the redshift function, and the shape function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' That coordinate decreases from infinity to a minimum value r0, the radius of the throat, where b(r0) = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The quantity dΩd−2 is the solid angle element in d − 2 spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' For this spacetime to represent a wormhole solution, the following conditions must be 3 satisfied[21] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The function Φ(r) must be finite for all value of r ≥ r0 in order to avoid singularities and horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The minimum value of r at the throat of the wormhole, corresponds to the point r = r0 = b(r0), where the function g−1 rr = 1 − b(r)/r vanishes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The proper radial distance l(r) = ± ˆ r r0 dr � 1 − b(r) r (2) must be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' For this it is necessary that 1 − b(r) r ≥ 0 (3) for all value of r ≥ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' For the asymptotic flatness condition it must be satisfied that lim r→∞ b(r) r → 0 (4) This ensures that at infinity the proper radial distance l → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' OUR MODEL Now we provide a list of constraints that the energy density as well as both the redshift and shape functions must satisfy for our model to represent a d-dimensional and asymptotically flat wormhole solution: (a) The energy density must be a continuous,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' positive and decreasing function such that lim r→∞ ρ(r) → 0 (5) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' as we will see below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' through the equations of motion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' this energy density leads to a function b(r) of the form b(r) = ¯b(r) rd−4 (6) 4 where ¯b(r) must be an increasing function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' due to the condition (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' it must be satisfied that: lim r→∞ ¯b(r) = Constant (7) This latter ensures condition 4 of introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (b) Relating with condition 2 above, the largest solution of the equation g−1 rr = 1−b/r = 0 corresponds to the minimum value of r = r0 at the wormhole throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (c) The function b(r), equation (6), must be such that the function g−1 rr = 1 − b/r be an increasing function from g−1 rr (r = r0) = 0 up to g−1 rr (r → ∞) → 1 (value provided by equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' So, it must be satisfied that (g−1 rr )′|r≥r0 > 0 (8) This item ensures condition 3 of introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (d) In order to avoid singularities, with a finite value of e2Φ(r), condition 1 of introduction, and in order to have an asymptotically flat behavior, we impose that e2Φ(r0) = A (9) with 0 < A ≤ 1, and lim r→∞ e2Φ(r) = 1 (10) Furthermore the derivative � e2Φ(r)�′ ≥ 0 (11) for r ≥ r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This ensures that the function e2Φ(r) can be either a constant (for A = 1 and � e2Φ(r)�′ = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='e the zero tidal case) or an increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Furthermore we impose that : lim r→∞ Φ′ = 0 (12) For the line element (1), the (t, t) component of the d− dimensional field equations is: � rd−4b(r) �′ = 2 d − 2rd−2ρ(r) (13) 5 where we have made Einstein’s constant κ = 1 and where ρ is constructed such that the conditions (a) of section III must be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This energy density gives rise to the following solution b(r) = ¯b(r) rd−4 (14) where b and ¯b must also satisfy the conditions (a) of section III and where ¯b(r) = 2 d − 2 ˆ rd−2ρ(r)dr (15) Using b(r) and Φ(r) (which are constructed such that the conditions (d) must be satisfied),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' we can determinate the radial pressure using the (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' r) component pr = d − 2 2r2 �� 1 − b r � (2rΦ′ + (d − 3)) − (d − 3) � (16) Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' from equations (7) and (12) : lim r→∞ pr → 0 (17) The energy-momentum tensor has the form T µ ν = diag(−ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' pr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' pφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' From spherical symmetry we have for all the (d − 2) angular coordinates pt = pθ = pφ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' and, the conservation law T AB ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='B = 0 gives: (ρ + pr)Φ′ + p′ r + d − 2 r (pr − pt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (18) So, using the last equation can be determined the tangential pressure values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is direct to check that lim r→∞ pt = 0 (19) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' THE NEW FAMILY OF WORMHOLE SOLUTIONS So far it has been shown a suitable model to describe an higher dimensional and asymp- totically flat wormhole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, we have not proposed a specific form for the energy density ρ and the Φ function, and for the further values of b(r) , pr and pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, we will employ here the d-dimensional generalization [27] of the Dymnikova energy density [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In four dimensions this model of energy density has give rise to regular black hole solutions, as it was already mentioned in Introduction, as well as to wormhole magnetic monopoles, 6 under the constraints ρ = −pr, studied in reference [33] in a different framework from what is made here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The energy density profile is given by ρ(r) = d − 2 2 ρ0 exp � −rd−1 ad−1 � (20) where ρ0, a > 0 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is direct to check that this model of energy density is consistent with the conditions (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Replacing in equation (15): ¯b(r) = ad−1ρ0 (d − 1) � 1 − exp � −r a �d−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (21) where has been used the value ad−1ρ0 (d−1) as integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is direct to check that condition (7) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, it is worth mentioning that the conditions (a) of section III are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' From equation (14) b(r) = ad−1ρ0 (d − 1)rd−4 � 1 − exp � −r a �d−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (22) In order to satisfy the condition (b) of section III, one has b(r0) = r0, where r0 is the throat radius, so we must find a relationship between ρ0, d, a, and r0, which is given by ρ0 = (d − 1)a1−drd−3 0 1 − exp � − r0 a �d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (23) Thus, the g−1 rr metric component is g−1 rr = 1 − rd−3 0 rd−3 � 1 − exp � − r a �d−1 1 − exp � − r0 a �d−1 � (24) As it was mentioned in conditions (b) and (c), r0 must represent the largest solution of the equation g−1 rr = 0 and furthermore the derivative (g−1 rr )′|r=r0 must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is easy to check that, for r ∈ [0, ∞], there is a critical value r = r∗, where r∗ represents a point where the function g−1 rr reaches a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' So, r0, such that g−1 rr (r = r0) = 0, must be located after the critical value, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='e r0 > r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This occurs provided r0/a > β(d), where β(d) is a number of order of unit and that can be calculated numerically from the Lambert function, W−1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' As for example for d = 4, β(4) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='24, for d = 5, β(5) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='06, and for d = 8, β(8) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' So, the conditions (b) and (c) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 1 shows us the behavior 7 of the function g−1 rr = F(r) = 1 − b(r)/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Notice that the obtained wormhole solutions are asymptotically flat, since when r → ∞, b(r)/r → 0, for all d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' As redshift function Φ(r) we choose in arbitrary way the following form: e2Φ(r) = 1 − C exp � −r − r0 a � (25) where it is easy to check that C = 0 represents the zero-tidal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is easily noted that the condition (10) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The function evaluated at r = r0 is: e2Φ(r = r0) = 1 − C = A (26) which in order to satisfy (9) must satisfy that 0 ≤ C < 1 (27) Furthermore it is direct to check that the conditions (11) and (12) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' So, this function serves as test of prove in order to satisfy the condition (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The value of the radial pressure is obtained easily replacing equations (23), 22 and (25) into the equation (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The tangential pressure is computed from equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Flaring-out condition A fundamental property of a wormhole is that a flaring-out condition of the throat, given by (b−b′r)/b2 > 0 [1], and at the throat b(r0) = r = r0, such that the condition b′(r0) < 1 is imposed to have wormhole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' It is precisely these restrictions that impose the WEC and NEC violations in classical general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' For our model, using the condition (8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (1 − b/r)′|r=r0 > 0 it is direct to check that the condition b′(r0) < 1 is automatically satisfied, and thus, the flaring out condition is satisfied for r = r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Furthermore, using the same condition for (1 − b/r)′|r>r0 > 0 it is also direct to check that the flaring out condition is satisfied for r > r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Embedding diagram This diagram is usually obtained by comparing the spatial three-dimensional flat met- ric written in cylindrical coordinates (r, φ, z) with the spatial sector of the Morris-Thorne 8 d = 4 d = 5 d = 8 20 30 40 50 r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 F FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 1: Behavior of the function F(r) = 1−b(r)/r, for some spacetime dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The parameter set is r0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 and a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' metric (fixing the polar angle at θ = π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The same reasoning can be generalized for a d- dimensional spacetime, with the flat spatial sector being compared with the d−1-dimensional spatial sector of the spherical wormhole (fixing also the (d − 3) polar angular coordinates at π/2), and identifying the azimuthal angles, arriving at dz = ±dr � b(r)/r 1 − b(r)/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (28) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' By means of equation (7) it is easy to check that lim r→∞ dz dr = 0 (29) We can see the generic behavior in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We can note that in higher dimensions the flatness of the wormhole is bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Energy conditions In figures 3 we can test the behavior of the Null Energy Conditions (NEC ρ + pi ≥ 0), Weak Energy Conditions (WEC ρ ≥ 0, ρ + pi ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We can test that these behaviors are generic for other values of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In the second figure we can note that WEC (and NEC) are violated because the condition ρ + pr ≥ 0 is only satisfied for r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In figure IV C we analyze the exoticity of this matter by considering the state parameter of a perfect fluid, which is dependent on the radial coordinate, according to EoS ω(r) = pr/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We can notice that the fluid behaves like a phantom fluid since ω(r) < −1 in all the space, 9 0 20 40 60 80 100 50 0 50 r z d = 4 d = 5 d = 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 2: Embedding diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The parameter settings are a = 8 and r0 = 10, in Planck units for d = 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Below, we will see that, in the 4–dimensional scenario, the presence of a fundamental length modifies this behavior near the throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 10 11 12 13 14 15 16 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='010 r ρ ρ(d = 4) ρ(d = 5) ρ(d = 6) 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='00 r ρ+pr ρ(d = 4)+ pr(d = 4) ρ(d = 5)+ pr(d = 5) ρ(d = 6)+ pr(d = 6) 10 12 14 16 18 20 22 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='03 r ρ+pt ρ(d = 4)+ pt(d = 4) ρ(d = 5)+ pt(d = 5) ρ(d = 6)+ pt(d = 6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 3: Fist figure : ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Second figure : ρ+pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Third figure : ρ+pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The parameter set is r0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6 and d = 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='40 8 6 4 2 0 r pr/ρ pr (d=4) ρ(d=4) pr (d=5) ρ(d=5) pr (d=6) ρ(d=6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 4: The parameter state, with parameter settings a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='8, r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6, in Planck units, for d = 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' DYMNIKOVA-SCHWINGER 4D WORMHOLE AND THE INFLUENCE OF A MINIMAL LENGTH According to [29, 30], the d = 4 Dymnikova density profile can be seen as the gravi- tational analogue of the electron-positron pair production rate, Γ ∼ exp (−Ec/E), in the vacuum – the so-called Schwinger effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This QED phenomenon is associated with the ap- plication of an intense uniform electric field (E) that results in vacuum polarization and the corresponding production of particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The critical electric field necessary for abundant pair production is given by Ec = πℏm2 e/e, where me and e are the electron mass and charge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The gravitational equivalent is considered when one makes the association of the electric field with the gravity tension characterized by a curvature term, namely E ∼ r−3 and Ec ∼ a−3, so that we obtain the d = 4 Dymnikova-Schwinger density profile of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The correction to the Schwinger effect associated to the existence of a minimal length was obtained in [31, 32] by means of the Generalized Uncertainty Principle (GUP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In this case, the electron-positron pair production rate becomes Γ ∼ exp � −πℏm2 e eE + απ3eE � , (30) where α comes from GUP via ∆x∆p ∼ ℏ 2 � 1 + α(∆p)2 ℏ � , (31) with α = ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Here ℓ is the minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Identifying the electrical field with the grav- itational tension as it was previously discussed, the GUP correction to the Dymnikova- 11 Schwinger density profile will be, therefore ρ(r) = ρ0 exp � −r3 a3 + αa r3 � ≈ ρ0 exp � −r3 a3 � � 1 + αa r3 � , (32) where we have considered in the second relation that the minimal length is very small, α/r2 ≪ 1, and r ≥ r0, which is valid for any wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The shape function of Dymnikova-Schwinger GUP-corrected wormhole can be obtained from the (t, t) component of Einstein’s equations, (13), for d = 4, with the energy density profile (32), and it can be written as b(r) = r0 � 1 − exp � − r3 a3 � + α a2Ei � − r3 a3 �� � 1 − exp � − r3 0 a3 � + α a2Ei � − r3 0 a3 ��, (33) so that the integration constant was chosen in order to do b(r0) = r0, with Ei(z) being the exponential integral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The metric of the wormhole under consideration is, therefore, ds2 = − � 1 − C exp � −r − r0 a �� dt2 + dr2 1 − r0 r � 1−exp � − r3 a3 � + α a2 Ei � − r3 a3 �� � 1−exp � − r3 0 a3 � + α a2 Ei � − r3 0 a3 �� + r2dΩ2, (34) where we have taken into account the redshift function given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' r z 10 20 30 40 60 40 20 20 40 60 a = 0 a = 1 r z 10 15 20 25 30 10 20 30 40 50 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 5: Embedding diagram of the Dymnikova-Schwinger GUP-corrected wormhole profile, in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The right panel exhibits the greater slope of the GUP-corrected wormhole (α ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The parameter settings are a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 and r0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, in Planck units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 5 we depict the embedding diagram profiles of the Dymnikova-Schwinger GUP- corrected wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Replacing into the above equation the expression (33) and integrating it, we obtain the corresponding embedding diagrams, for both α = 0 and α ̸= 0 Dymnikova- Schwinger wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' By considering these diagrams, we note that the presence of the 12 minimal length increases the slope of the wormhole, which is more accentuated the greater this length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Energy conditions and state parameter Null energy conditions (NEC) are not obeyed by wormholes, at least in the context of GR, provided the flaring-out conditions are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 6 shows us the behaviors of Sr = ρ+pr (density with radial pressure - left panel) and St = ρ+pt (density with transversal pressure - right panel), for Dymnikova-Schwinger GUP corrected wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Considering the behavior of Sr, the violation is more pronounced for the zero-tidal wormhole (C = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6 Sr r 12 14 16 18 20 22 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='001 C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6 St r 12 14 16 18 20 22 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='010 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 6: Sum of density with radial (transversal) pressure, in left (right) panel, as a function of the radial coordinate, r, for D-S GUP corrected wormholes, considering both zero tidal (C = 0) and non-zero tidal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The parameter settings are a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, r0 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, in Planck units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Now let us analyze the influence of the minimal length on NECs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Although these condi- tions remain still unfulfilled demanding the presence of exotic matter, the occurrence of a fundamental minimal length attenuates this violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 7 reveals this feature, on exhibiting that Sα=0 r < Sα̸=0 r , nearby the wormhole throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We can also analyze the exoticity of the material source by considering the state parameter of a perfect fluid, which is dependent on the radial coordinate, according to EoS ω(r) = pr/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 7, we can notice that the fluid is quintessential nearby the throat (−1 < ω < −1/3) and tends to be a phantom fluid far from the throat, since ω(r) < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This panel reveals therefore that the presence of the minimal length increases the state parameter value associated to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, for Planckian wormholes, the minimal length reduces the exoticity of the source in such a manner that it is no longer phantom-like 13 a = 0 a = 1 r sr 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='2 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0005 wHrL r a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 7: In the left panel, the sum of density with the radial pressure, Sr, as a function of the radial coordinate, for a = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, r0 = 10, and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6, for both the Dymnikova-Schwinger (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0) and D-S GUP corrected (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0) wormholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In the right panel, the parameter state of D-S (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0) and D-S GUP corrected (α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='1) wormholes, with parameter settings a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='8, r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='0, and C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='6, in Planck units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' nearby the throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' These finds corroborate the analysis of NEC made above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' CONCLUSION In this work, we have found new d-dimensional and asymptotically flat wormhole solutions with the presence of matter fields in the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The energy density and the pressure components have local values for different values of the radial coordinate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content='e are the so-called localized sources of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' For this latter, in section III we have proposed a list of constraints that the energy density as well as both the redshift and shape functions must satisfy in such a manner that the new wormhole solutions fulfill the corresponding criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, we have initially used the d-dimensional generalization [27] of the Dymnikova energy density [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Then, from this matter distribution and solving the time component of Einstein’s equation, we have found wormhole exact solutions in d spacetime dimensions, which depend on the throat radius, r0, and on a characteristic length, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This latter is a typical scale at which the source scatters through space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' We have also defined a suited logarithmic redshift function dependent on these parameters and a factor that, on vanishing, one obtains a zero-tidal simplest traversable wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Remarkably the energy density, the redshift function, and the pressure components satisfy the criteria to the building of a wormhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' 14 The detailed study of the conditions for the formation of these objects (as the flaring-out one) has shown that there has to be a relationship between r0, a, and the spacetime dimen- sion d in order to guarantee their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Following, we have generated the embedding diagram for some of these wormholes, showing that on increasing the spacetime dimension the asymptotic flatness is more quickly reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In other words, the greater d the smaller the slope towards the wormhole throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' With respect still to d-dimensional Dymnikova wormholes, we have analyzed Weak and Null Energy Conditions (WEC and NEC) concerning the material source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' As expected, this latter is considered exotic, since the radial part of those conditions is violated – while the transversal one is not – and such a violation is more pronounced the larger the spacetime dimension, in regions nearby the wormhole throat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' This feature is corroborated by the behavior of the position-dependent state parameter ω(r) = pr/ρ, namely, the quantity that relates the radial pressure with the matter density, when one supposes that the source is an ideal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Its behavior indicates that such a fluid is phantom-like since ω(r) < −1 in all space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' In sequence, we specialized the previous study for 4d wormholes, which we have called Dymnikova-Schwinger since the material source under consideration is the gravitational ana- logue of the pair density produced in a vacuum by means of the application of an intense (electric) field – the so-called Schwinger effect, predicted by QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The gravitational coun- terpart of this a phenomenon holds some similarity with that one associated to the Casimir effect, which has been widely studied in the context of wormholes [16–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Thus, on con- sidering the correction to the Schwinger effect due to the introduction of a minimal length (ℓ = √α) via GUP [31, 32], we have found the corresponding correction to the gravitational analogue of Dymnikova’s matter distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Supposing that the minimal length is very small, we obtained a novel traversable and asymptotically flat wormhole solution that re- duces to the one previously studied when α = 0 and d = 4, for the same redshift function employed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' The corresponding embedding diagram shows that the presence of the minimal length increases the slope of the wormhole towards its throat compared with the case without such a length – in other words, it lessens the wormhole flatness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' On the other hand, the WEC and NEC analysis have shown that this quantity, although still yields the non-fulfillment of those conditions, attenuates their violation since ρ + pr increases with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Finally, the 15 study of the state parameter ω(r) has revealed that, in presence of the fundamental length, the material source ceases to be a phantom-type nearby the wormhole throat, at least at Planck’s scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Acknowledgments Celio R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Muniz thanks the Conselho Nacional de Desenvolvimento Cient´ıfico e Tec- nol´oogico (CNPq), grant no 308268/2021-6 for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE4T4oBgHgl3EQfXAw5/content/2301.05037v1.pdf'} +page_content=' Milko Estrada is funded by the 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/dev/null +++ b/wdE3T4oBgHgl3EQflQpC/content/tmp_files/2301.04604v1.pdf.txt @@ -0,0 +1,983 @@ +LinkGAN: Linking GAN Latents to Pixels for Controllable Image Synthesis +Jiapeng Zhu†*1 +Ceyuan Yang†2 +Yujun Shen†3 +Zifan Shi*1 +Deli Zhao4 +Qifeng Chen1 +1HKUST 2Shanghai AI Laboratory 3Ant Group 4Alibaba Group +Figure 1. Precise local control achieved by LinkGAN, where we can manipulate the image content within a spatial region (e.g., single +eye or right half of the image) or a semantic category (e.g., car) simply by resampling the latent code on same sparse axes. Our approach +works well with both 2D image synthesis, like StyleGAN2 [20] (left three columns), and 3D-aware image synthesis, like EG3D [4] (right +two columns). It is noteworthy that, under the 3D-aware case, we can control both the appearance and the underlying geometry. +Abstract +This work presents an easy-to-use regularizer for GAN +training, which helps explicitly link some axes of the latent +space to an image region or a semantic category (e.g., sky) +in the synthesis. Establishing such a connection facilitates +a more convenient local control of GAN generation, where +users can alter image content only within a spatial area +simply by partially resampling the latent codes. +Exper- +imental results confirm four appealing properties of our +regularizer, which we call LinkGAN. (1) Any image region +can be linked to the latent space, even if the region is +pre-selected before training and fixed for all instances. +(2) Two or multiple regions can be independently linked +to different latent axes, surprisingly allowing tokenized +control of synthesized images. +(3) Our regularizer can +improve the spatial controllability of both 2D and 3D GAN +models, barely sacrificing the synthesis performance. (4) +The models trained with our regularizer are compatible +with GAN inversion techniques and maintain editability on +real images. Project page can be found here. +† indicates equal contribution. +* This work was done during an internship at Ant Group. +1. Introduction +Generative adversarial networks (GANs) [10] have been +shown to produce photo-realistic and highly diverse images, +facilitating a wide range of real world applications [9, 15, +26, 31, 32]. The generator in a GAN is formulated to take +a randomly sampled latent code as the input and output +an image with a feed forward network. +Given a well- +learned GAN model, it is generally accepted that a variety +of semantics and visual concepts automatically emerge in +the latent space [12,16,33,45,51], which naturally support +image manipulation. +Some recent work also reveals the +potential of GANs in local editing by steering the latent +code along a plausible trajectory in the latent space [22,50]. +However, most studies on the relationship between the +latent codes and their corresponding images depend on a +posterior discovery, which usually suffers from three major +drawbacks. (1) Instability: The identification of emerging +latent semantics is very sensitive to the samples used for +analysis, such that different samples may lead to different +results [12,32]. (2) Inaccuracy: Given the high-dimensional +latent space (e.g., 512d in the popular StyleGAN family [19, +20]), finding a semantically meaningful subspace can be +challenging. (3) Inflexibility: Existing manipulation models +arXiv:2301.04604v1 [cs.CV] 11 Jan 2023 + +are usually linear (i.e., based on vector arithmetic [16,32]), +limiting the editing diversity. +This work offers a new perspective on learning control- +lable image synthesis. Instead of discovering the semantics +from pre-trained GAN models, we introduce an efficient +regularizer into the training of GANs, which is able to +explicitly link some latent axes with an image region or a +semantic category (e.g., sky) in the synthesis, as shown in +Fig. 2. In this way, the selected axes and the remaining +axes are related to the in-region pixels and out-region +pixels, respectively, with little cross-influence. +Such a +design, termed as LinkGAN, enables a more accurate and +more convenient control of the generation, where we can +alter the image content within the linked region simply by +resampling on the corresponding axes. +We conduct experiments on various datasets to evaluate +the efficacy of our approach and demonstrate its four +appealing properties. (1) It is possible to link an arbitrary +image region to the latent axes, no matter the region is pre- +selected before training and fixed for all instances, or refers +to a semantic category and varies across instances (see +Sec. 4.2.1). (2) Our regularizer is capable of linking multi- +ple regions to different sets of latent axes independently and +simultaneously, and achieves joint manipulation of these +regions. We even push the limit to tokenized image control +(see Sec. 4.2.2). +(3) Our approach lends itself well to +both 2D image synthesis models [20] and 3D-aware image +synthesis models [4], by sufficiently improving the con- +trollability yet barely harming the synthesis performance +(see Fig. 1). (4) The models trained with our regularizer +are compatible with GAN inversion techniques [52] and +maintain the editability on real images (see Sec. 4.3). We +believe that this work makes a big step towards the spatial +controllability of GANs as well as the explicit disentangle- +ment of GAN latent space. +2. Related Work +Generative adversarial networks. Generative adversarial +networks (GANs) are composited by a generator and a +discriminator, which are trained simultaneously by playing +a two-player minimax game [10], have made tremendous +progress in generating high quality and diversity images [3, +18–20, 44]. In turn, there are widely used in a variety of +tasks, such as representation learning [17, 43], image-to- +image translation [6, 15], image segmentation [48], image +editing [5,22,32], 3D generation [4,34,42] etc. +Semantic discovery in GANs. Interpreting the pre-trained +GANs [1, 2, 32] has drawn lots of attention recently since +it will help us to understand how an image in GANs is +rendered, which in turn, enables us to control the generation +process of the GANs. Many recent works [7, 12, 32, 38, +50] have demonstrated the latent space of GANs encodes +rich semantic information that can be used to edit the +Latent code +Synthesis +Pixels of interest +Explicit link +Figure 2. Concept diagram of LinkGAN, where some axes of the +latent space are explicitly linked to the image pixels of a spatial +area. In this way, we can alter the image content within the linked +region simply by resampling the latent code on these axes. +output images globally or locally by merely shifting the +corresponding latent codes. For the methods that control +the output images globally [5, 12, 16, 21, 29, 32, 33, 36, +38, 39, 45, 47], they either use some off-shell classifiers to +find the variation of factors supervised or just find them +unsupervised. +For the methods that control the output +images locally [1, 2, 7, 22, 37, 41, 43, 50, 51], they either +control from the feature maps +[2, 37, 43], control from +weights of the generator [1], or control from the latent +space [7, 22, 41, 50, 51]. However, all of these works try +to interpret the pre-trained GANs, but few of them apply +those findings during GAN training. +Regularizers for GAN training. +Many attempts have +been made to regularize GANs during training [11, 13, +20, 23, 28, 35, 40, 44]. Some of them try to improve the +training stability of GANs by regularizing the gradients +of the discriminator [11, 23], the spectral norm of each +layer [24], or the singular values of the generator [25]. +Besides, some of them [8,13,20,28,35,40] aim to improve +the disentanglement property of GANs. +For example, +[28, 40] try to disentangle each component in the latent +vectors so that each dimension in the latent codes can +only affect one attribute on the output images by adding +some regularizers (e.g., Hessian Penalty or Orthogonal +Jacobian Regularization). In StyleGAN2 [20], a path length +regularization is added to encourage the learned generator +to become much smoother. And [8, 35] try to disentangle +different semantics in the latent space by borrowing some +labels. But none of them tries to relate an arbitrary region +to a specific latent part. +3. Method +In this section, we first give some background knowledge +regarding GANs as well as the conventional manipulation +model using GANs in Sec. 3.1. Our method that bridges +specified subspaces of the whole latent space with a parti- + +tion of a synthesized image is presented in Sec. 3.2. +3.1. Preliminaries +A generative adversarial network (GAN) consists of a +generator G(·) that maps latent vectors z ∼ p(z) to fake +images, i.e. ˜x = G(z), and a discriminator D(·) that tries +to differentiate fake images from real ones. They are trained +in an adversarial manner, in the sense that the generator tries +to fool the discriminator while the discriminator learns to +measure the realness of generated images. The training loss +can be formulated as follows: +LG = Ep(˜x)[f(1 − D(˜x))], +(1) +LD = Ep(x)[f(D(x))] − Ep(˜x)[f(1 − D(˜x))], +(2) +where p(x) and p(˜x) are the distributions of real images and +synthesized images, respectively. Besides, f is a model- +specific function that varies between different GANs. +Conventional manipulation leverages a pre-trained GAN +for content editing since many prior arts [30, 32] have +shown that the latent space of a GAN is correlated with +the property of semantic arithmetic. For instance, adding +or removing the glass from an image can be fulfilled by +performing simple additive or subtractive operations in the +latent space. The manipulation model can be formulated as +follows: +xedit = G(z + αn), +(3) +where n is the attribute vector in the latent space and α +is the step for the manipulation. +In other words, when +a latent code is moved toward the attribute vector, the +corresponding attribute contained in the output image will +vary accordingly. +Notably, such a manipulation strategy +tends to modify global attributes of synthesis. +3.2. Linking Latents to Pixels +With the rapid development of manipulation technique, +several works [22, 50] have shown that some subspaces of +the latent space (i.e., the w space in StyleGAN [19]) can +control local semantics over output images. Specifically, +traversing a latent code within those subspaces results in a +local modification in the synthesis. However, there lacks +an explicit connection between the local regions and the +specified axes of latent spaces. To this end, we propose +a new regularizer explicitly linking the axes to arbitrary +partitions of synthesized images. +Partition of latent codes and images. In order to set up +the explicit link between some axes of latent space and +local regions of an image, we first introduce some notations +for the corresponding partition. Taking StyleGAN [19] as +an example, w ∈ Rdw is the intermediate latent vector of +dimension dw derived from the mapping network. Through +a generator G(·), an image ˜x ∈ Rdx with dimension dx +is produced, i.e., ˜x = G(w). Note that here the image +in editing is reshaped as a vector for brevity. Namely, the +dimension dx is equal to the multiplication between the +width and height of images (the color channels are also +neglected here). +We first divide the latent space into several subspaces. +Namely, a latent code w could be divided into K fragments +i.e., w += +[w1, w2, . . . , wK], where one partition wi +consists of multiple channels wi = [wi1, wi2, . . . , wini] +and ΣK +i=1ni = dw. +Similarly, an image ˜x could also +produce several partitions i.e., ˜x += +[˜x1, ˜x2, . . . , ˜xK], +where ˜xi ∈ Rmi and ΣK +i=1mi = dx. +Importantly, we +further define wc +i and ˜xc +i are the complements of wi and ˜xi, +respectively, i.e., wc +i = [w1, . . . , wi−1, wi+1, . . . , wK], +˜xc +i = [˜x1, . . . , ˜xi−1, ˜xi+1, . . . , ˜xK]. +Fig. 2 presents an +example (K is equal to 2) where the blue part of the latent +code and pixels within the blue bounding box denote the +partitions wi and ˜xi, respectively. Now, our goal is that the +latent fragment wi only controls the pixels in ˜xi and wc +i +controls the pixels in ˜xc +i, namely, building an explicit link +illustrated in Fig. 2. +Learning objectives. To our surprise, we find in practice +that a simple regularizer combined with the StyleGAN +framework is sufficient to achieve this goal. Formally, we +can randomly perturb wi and wc +i and then minimize the +variations on ˜xc +i and ˜xi, respectively, expecting that wi +merely controls ˜xi and hardly affects ˜xc +i and vice versa. +To be specific, we use two vectors p and p to perturb +wi and wc +i , respectively, where p = [0, . . . , 0, pi, 0, . . . , 0], +p = [p1, . . . , pi−1, 0, pi+1, . . . , pK], and each non-zero +sub-vector in p and p is sampled from a standard Gaussian +distribution N(0, I). Thus we can obtain two perturbed +images, i.e., ˜x1 = G(w + αp) and ˜x2 = G(w + αp) +regarding the original one ˜x. The pixel change in ˜xc +i after +the perturbation by p can be computed as +L1 = ||M1⊙(˜x1−˜x)||2 +2 = ||M1⊙(G(w+αp)−G(w))||2 +2, +(4) +where M1 is the binary mask denoting the region out of +interest, ||·||2 denotes the ℓ2 norm, and α is the perturbation +strength. We want the pixels to change in the region ˜xc +i as +minimally as possible after the perturbation by p. Similarly, +the pixels change in ˜xi after the perturbation by p can be +written as +L2 = ||M2⊙(˜x2−˜x)||2 +2 = ||M2⊙(G(w+αp)−G(w))||2 +2, +(5) +where M2 is the binary mask indicating the chosen pixels +of interest. These two losses L1 and L2 can be integrated as +a regularizer in the StyleGAN framework. +Lreg = λ1L1 + λ2L2, +(6) +where λ1 and λ2 are the weights to balance these two terms. +Therefore, the total loss to train the generator in StyleGAN + +can be formulated as +L = LG + Lreg. +(7) +Practically, we could apply the new regularization Lreg +in a lazy way, in the sense that Lreg is calculated once +every several iterations (8 iterations in this paper), greatly +improving the training efficiency. +Additionally, the per- +turbed images would be also fed into the discriminator +during training. Noticeably, Eq. (6) gives the regularization +loss on how to build one explicit link, if we want to build +multiple links between image regions and latent fragments, +we need to compute Lreg on each link and sum them all, +i.e., Σk +j=1Lj +reg, where k (1 ≤ k ≤ K) is the number of +links we want to build. +4. Experiments +4.1. Experimental Setup +We conduct extensive experiments to evaluate our pro- +posed method. We mainly conduct our experiment on Style- +GAN2 [20] and EG3D [4] models. The datasets we use +are FFHQ [19], AFHQ [6], LSUN-Church, LSUN-Car [46]. +We also use a segmentation model [49] to select pixels with +the same semantic (e.g., all the pixels in the sky on LSUN- +Church). The main metrics we use to qualify our method +are Fr´echet Inception Distance (FID) [14] and the masked +Mean Squared Error (MSE) [50]. +The experiments are +organized as follows. First, Sec. 4.2 shows the properties +of LinkGAN, which can relate an arbitrary region in the +image to the latent fragment. Second, Sec. 4.3 gives some +applications of our method, such as local control on the +3D generative model, real images, and some comparisons +with the baselines regarding the editing precision. At last, +Sec. 4.4 presents an ablation study on the size of the link +latent subspace. +4.2. Properties of LinkGAN +In this section, we mainly demonstrate the effectiveness +of the proposed approach by explicitly linking the pixels +in any region (both the single region or multi-regions) to +a partition of the corresponding latent codes, while seldom +deteriorating the quality of synthesis. Tab. 1 reports FID on +different datasets when our regularizer is added, from which +we can see our regularizer only has a minor influence on the +synthesized quality. Empirically, we find that it would more +stable if the proposed regularizer is incorporated after the +convergence of the generator. Therefore, we start training +from a relatively well-trained generator and equipping it +with our approach. +4.2.1 +Linking Latents to Single Region +Regarding the partition of latent codes, we could easily +choose the first several channels as one group. +Accord- +Table 1. +Performance change after introducing our proposed +regularizer into 2D and 3D baselines, where the synthesis quality +slightly drops but the controllability significantly improves (see +Figs. 3 to 6 for details). +StyleGAN2 [20] +EG3D [4] +Dataset +FFHQ AFHQ +Car +Church +FFHQ +Baseline +3.98 +7.92 +2.95 +3.82 +4.28 +LinkGAN (ours) +5.11 +10.2 +3.50 +4.59 +4.67 +ingly, the remaining ones become the complementary code. +Therefore, the goal of the proposed regularizer is to enable +the explicit control of certain regions of interest through the +chosen channels. Note that the number of first channels +that would be grouped usually depends on the area ratio of +the chosen region over the entire image. In the following +context, we will show different ways of choosing pixels out +of images and building explicit links between the chosen +channels and pixels. +Region-based control. One general way of grouping pixels +is to use a bounding box that could cover a rectangle region. +Fig. 3 presents the qualitative results of choosing different +regions randomly. Red bounding boxes in Fig. 3 denote +the chosen regions of interest. In terms of human faces on +FFHQ, we randomly select two spatial patches that usually +contain either complicated or non-special semantics (e.g., +half of faces or just a cube of background). +Obviously, +after building the explicit link, we could merely change the +chosen regions by perturbing the corresponding partition of +latent codes, while maintaining the rest regions untouched. +Besides, perturbing the complementary latent codes results +in substantial change for regions out of interest, demonstrat- +ing that the spatial controlling is well-built by the proposed +explicit link. Additionally, we also verify the effectiveness +of our regularizer on various datasets. +For instance, the +connection between a partition of latent code and two eyes +or one ear of animal faces could be also easily set up, +causing appealing editing results. +Moreover, not only a +relatively small region but also the larger one could be well +linked to several axes of latent space. Results on LSUN +Church suggest that even half of the entire image is also +controllable with the aid of the proposed approach. The +difference maps further present how well such an explicit +link could control a region of interest. +Semantic-based control. +Prior experimental results +demonstrate the control on a rectangle region that seems to +be irrelevant to a certain visual concept. Namely, this link is +semantic-agnostic since it merely bridges several channels +with spatial locations rather than semantics. +Therefore, +we further conduct experiments on semantic controlling. +To be specific, by leveraging an off-the-shelf segmentation +model [49], we could easily obtain mask annotations that +specify various semantics. + +Original +FFHQ +AFHQ +Church +Input +In-Region +Out-Region +In-Region +Out-Region +Figure 3. Linking latents to single fixed region, which is pre-selected before training and shared by all instances. Linked regions are +highlighted with red boxes, and the heatmaps reflect the change of pixel values after in-region resampling and out-region resampling. We +find that LinkGAN can robustly link the latents to an arbitrary image region, even semantically meaningless ones (e.g., the second row). +Fig. 4 presents the semantic control on two datasets, +LSUN-Church and LSUN-Car [46]. In particular, churches +and cars are chosen as the semantics that we would like to +build a link between latent space to, no matter where the +chosen semantics are. +Similarly, we manage to connect +several channels of latent space with a given semantic +such that perturbing the chosen channels will result in the +obvious change of semantics. For instance, the color and +shape of a church vary while the sky keeps the same and +vice versa. Regarding the experiments on cars, the color +could be modified no matter what cars face and how many +pixels cars occupy. +All these results together with the +rectangle region control demonstrate the arbitrary region +control enabled by our approach. + +Y90.?Original +Resample +Heatmap +Figure 4. Linking latents to semantic region (i.e., church, sky, and car) which dynamically varies across instances. Our LinkGAN +manages to precisely control a particular semantic category simply by resampling on some sparse latent axes. +Original +Resample +Heatmap +Resample +Heatmap +Resample +Heatmap +Original +Resample +Heatmap +Resample +Heatmap +Resample +Heatmap +Resample +Heatmap +Figure 5. Linking latents to multiple regions, where the regions highlighted by red boxes are simultaneously linked to some non- +overlapping sets of latent axes and can be independently controlled by partially resampling the latent codes. We even achieve tokenized +control of the synthesis as shown below. +4.2.2 +Linking Latents to Multiple Regions +After checking the effectiveness of our approach on build- +ing one explicit link, a natural question then arises: is +it possible to link multiple regions of interest to multiple +partitions of latent codes? +Joint control. Fig. 5 presents the corresponding results. +On top of that, we link three subspaces that contain 64 +channels to three image regions i.e., eye, nose, and mouth, +respectively. Even though we could remain to manipulate +semantics individually. +Tokenized control. +The bottom one moves forward to +a more challenging setting that both latent spaces and +images are equally divided into four groups and four corners +without any overlap. To this end, we could tell that such a +regularizer could build a full explicit link between the entire +latent space and the whole synthesis in a disentangled way. +Namely, we can even tokenize an image and assign one +subspace to each token. + +Original +Original +Figure 6. Controllability on 3D-aware generative model, i.e., EG3D [4], under the cases of mouth and nose. We find that LinkGAN is +well compatible with 3D-aware image synthesis and allows controlling both the appearance and the underlying geometry. +4.3. Applications of LinkGAN +In this part, we show that our proposed method can +be used in various applications, such as controlling 3D +generative models, real image manipulation, and precise +local image editing, etc. +Towards 3D-aware generation. We implement our regu- +larizer on the 3D generative model EG3D [4]. Surprisingly, +our regularizer performs well not only in controlling the +RGB images but also in controlling the geometry of the +corresponding image, showing the good generalization +ability of our regularizer. +Fig. 6 shows the results of +controlling the mouth and nose region by perturbing the first +64 channels of latent codes. Importantly, controlling the +linked subspace simultaneously changes the RGB images +and their geometry, i.e., mouth is opening for both RGB +and corresponding 3D geometry. +Real image editing. +After the generator is trained, we +can use the property of the trained generator to control +real images locally by inversion [20, 52]. Fig. 7 shows the +editing results on the real image, in which the eyes can be +independently controlled, i.e., we can only open one eye +yet keep another eye untouched. In this case, we need to +explicitly link two eye regions to two latent subspaces, i.e., +one subspace controls one eye. And when the generator is +well-learned, we can edit the eye region by controlling the +corresponding subspace on the inverted latent code. +Comparison with existing methods. +Now we compare +our method with some state-of-the-art algorithms (e.g., +ReSeFa [51]) regarding local control precision. As shown +in Fig. 8, we can observe that our method can reach more +precise control on the local regions than ReSeFa. +For +instance, when modifying eyes, ReSeFa also results in a +change of face color. On the contrary, when editing the +specific region, our method has negligible changes in the +other regions. Tab. 2 reports the masked MSE between our +method and ReSeFa when controlling those three regions. +Input +Inversion +Left Eye +Right Eye +Figure 7. Real image editing achieved by LinkGAN via borrow- +ing the GAN inversion technique [20]. We manage to edit the +two eyes of human independently in a very convenient way, i.e., +partially resampling the inverted code. +Namely, when editing a specific region, we want the change +in this region to be as larger as possible (the higher MSE, +the better) and the change in the remaining region as small +as possible (the smaller MSE, the better). We can observe +that the MSEs within the edited regions are comparable. +However, regarding the MSEs out of the edited regions, our +method significantly outperforms ReSeFa. +4.4. Ablation Study on Linking Dimensionality +In this part, we conduct an ablation study on how +many axes are required to build an explicit link. +Eyes + +LinkGAN +ReSeFa +Original +Eyes +Nose +Mouth +Figure 8. Qualitative comparison with ReSeFa [51], which posteriorly discovers semantics from a pre-trained model, on the task of local +editing. LinkGAN achieves more precise control within the regions of interest. See Tab. 2 for quantitative results. +Table 2. Quantitative comparison with ReSeFa [51] on the task +of local editing. Pixel-wise mean square error (MSE) within/out +of the region of interest (scaled by 1e−3 for better readability) is +used as the metric. Lower MSEo and higher MSEi are better. +Region +Eyes +Nose +Mouth +Metrics +MSEi MSEo MSEi MSEo MSEi MSEo +ReSeFa [51] +5.90 +61.14 +1.12 +60.4 +2.02 +50.55 +LinkGAN (ours) +5.25 +2.24 +1.82 +2.25 +3.10 +2.21 +of faces are chosen as regions of interest. +Tab. 3 gives +the quantitative results of changing in/out eye regions with +the same perturbation strength. In Tab. 3, all the training +configurations are the same except for the number of axes +during training. MSEi and MSEo are computed in and out +of the eye region when perturbing on their complementary +latent space, respectively. +Take axes number 8 as an +example, and the MSEi is computed within the eye region +when perturbing on axes from 8 to 512, while MSEo is +computed out of the eye region perturbing on axes from 0 to +8. In such a way, precise control could be obtained since the +perturbing on the complementary latent space should barely +influence the regions of interest. Hence, in this situation, +both MSEi and MSEo are the smaller, the better. Obviously, +when occupying the first 64 axes, we can get satisfying +results since the sum of them is the smallest. In practice, +we set the number of axes in latent code to 64 in most cases, +such as when controlling on eyes, nose, mouth, etc. And the +detailed axes for other datasets can be found in Appendix. +Table 3. Ablation study on the linking dimensionality. MSEi +measures the effect of unlinked axes on the linked region, while +MSEo measures the effect of linked axes on the unlinked region, +both of with enjoy a small value. All numbers are scaled by 1e−3 +for better readability. +# Linked axes +8 +16 +32 +64 +128 +256 +MSEi +17.45 +16.70 +3.29 +0.95 +0.78 +0.43 +MSEo +0.86 +1.53 +7.41 +8.20 +8.71 +24.78 +5. Discussion and Conclusion +After linking an arbitrary region to some latent axes with +the size of n, any perturbation with randomly sampled n +dimension vector on the linked subspace results in the con- +tent change only in the linked image region, which can be +viewed as a local semantic direction since it only influences +the linked region. However, some of the sampled latent +vectors can not generate realistic manipulation, and some of +them can. 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We first give the implementation details +of our method in Appendix A. Second, we give an ablation +study on whether or not to use the discriminator on the +perturbed images in Appendix B. Third, more comparison +results with other methods are given in Appendix C to show +the advantages of our method. +A. Implementation Details +We use the official Pytorch implementation of Style- +GAN2 +[20] +and +official +Pytorch +implementation +of +EG3D [4] to validate our method. We keep all the param- +eters untouched except our newly added regularizer during +training. For FID, we directly follow the original codebase, +and for the masked MSE, we computed it on 10,000 +images for each edit. For the update frequency of our lazy +regularization, we calculate once every 8 minibatches. For +how many axes we use to control the specific region, we list +as below: 1). For small regions, we use 64 axes, such as the +eye, nose, mouth, and ear region on FFHQ or AFHQ. 2). +For the larger region, such as the left region of the human +face and the bottom part of the church in Fig.3 of the main +text, we use 128 axes. Also, for tokenized control in Fig.5 +of the main text, each part has a size of 128 since we evenly +spilt the latent codes. 3). When the partition size becomes +bigger, such as half of the image, we use 256 axes, and for +the semantic control (church, sky, and car) in Fig.4 of the +main text, we use 256 axes as well. For the loss weight λ1 +and λ2, we list as below: 1). For the latent segment with 64 +axes, we set λ1 equal to 0.01 and λ2 equal to 0.04. 2). For +the latent segment with 128 axes, we set λ1 equal to 0.01 +and λ2 equal to 0.03. 3). For the latent segment with 256 +axes, we set λ1 equal to 0.01 and λ2 equal to 0.01. +B. Ablation Study +Here we conduct an ablation study, that is, whether or not +to use the discriminator on the perturbed images. And we +do the study on the eye region of AFHQ [6] dataset. Fig. 9 +gives some abnormal random perturbed results on the eyes +region without involving the discriminator on the perturbed +images during training, from which we can discover some +unrealistic perturbation. For instance, the eyes of the first +cat are distorted, which can not be in the real image, and the +eyes of the second cat turn to be dog eyes. Also, the leopard +feature appears when perturbing the dog’s eyes. Hence, in +such cases, we need to involve the discriminator in those +perturbed images to hinder the generator from synthesizing +those unrealistic perturbed images. +Original +Resample +Figure 9. Ablation study on AFHQ [6] when not to use discrimi- +nator on the perturbed images. +C. More results +Besides the method we compare in the main text, here +we compare our method with the baseline StyleCLIP [27], +which can use text to control the synthesized images. We +also provide the comparison results with ReSeFa [51]. +Fig. 10 gives the comparison results on opening eyes, +for LinkGAN and ReSeFa [51], we posteriorly discover +semantics that can change eye size. For StyleCLIP [27], +we use the text “extremely big eyes” for the optimization. +As we can see from Fig. 10, StyleCLIP can successfully +make the eyes bigger. +However, the global color of the +edited images is easily changed, such as the first column +shows in Fig. 10. Fig. 11 shows the comparison results +on the opening mouth, for LinkGAN and ReSeFa [51], +we also posteriorly discover semantics that can open the +mouth of a face image. +For StyleCLIP, we use the text +“open mouth” for the optimization. As shown in Fig. 11, +StyleCLIP also suffers from the global change of the image +when editing a specific region. For instance, the identity of +the man changed in the third column, and the background +of the man in the six column varied. On the contrary, our +LinkGAN achieves much more precise control of the local +region thanks to our explicit link. + +LinkGAN +ReSeFa +Original +StyleCLIP +Figure 10. Qualitative comparison on eyes with ReSeFa [51] and StyleCLIP [27]. LinkGAN achieves more precise control within the +regions of interest. + +LinkGAN +ReSeFa +Original +StyleCLIP +Figure 11. Qualitative comparison on mouth with ReSeFa [51] and StyleCLIP [27]. LinkGAN achieves more precise control within the +regions of interest. + diff --git a/wdE3T4oBgHgl3EQflQpC/content/tmp_files/load_file.txt b/wdE3T4oBgHgl3EQflQpC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..626f50d2d0aae3dd3fa23330bf11a8fe57c0e233 --- /dev/null +++ b/wdE3T4oBgHgl3EQflQpC/content/tmp_files/load_file.txt @@ -0,0 +1,810 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf,len=809 +page_content='LinkGAN: Linking GAN Latents to Pixels for Controllable Image Synthesis Jiapeng Zhu†*1 Ceyuan Yang†2 Yujun Shen†3 Zifan Shi*1 Deli Zhao4 Qifeng Chen1 1HKUST 2Shanghai AI Laboratory 3Ant Group 4Alibaba Group Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Precise local control achieved by LinkGAN, where we can manipulate the image content within a spatial region (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', single eye or right half of the image) or a semantic category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', car) simply by resampling the latent code on same sparse axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Our approach works well with both 2D image synthesis, like StyleGAN2 [20] (left three columns), and 3D-aware image synthesis, like EG3D [4] (right two columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' It is noteworthy that, under the 3D-aware case, we can control both the appearance and the underlying geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Abstract This work presents an easy-to-use regularizer for GAN training, which helps explicitly link some axes of the latent space to an image region or a semantic category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', sky) in the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Establishing such a connection facilitates a more convenient local control of GAN generation, where users can alter image content only within a spatial area simply by partially resampling the latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Exper- imental results confirm four appealing properties of our regularizer, which we call LinkGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (1) Any image region can be linked to the latent space, even if the region is pre-selected before training and fixed for all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (2) Two or multiple regions can be independently linked to different latent axes, surprisingly allowing tokenized control of synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (3) Our regularizer can improve the spatial controllability of both 2D and 3D GAN models, barely sacrificing the synthesis performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (4) The models trained with our regularizer are compatible with GAN inversion techniques and maintain editability on real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Project page can be found here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' † indicates equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' This work was done during an internship at Ant Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Introduction Generative adversarial networks (GANs) [10] have been shown to produce photo-realistic and highly diverse images, facilitating a wide range of real world applications [9, 15, 26, 31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The generator in a GAN is formulated to take a randomly sampled latent code as the input and output an image with a feed forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Given a well- learned GAN model, it is generally accepted that a variety of semantics and visual concepts automatically emerge in the latent space [12,16,33,45,51], which naturally support image manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Some recent work also reveals the potential of GANs in local editing by steering the latent code along a plausible trajectory in the latent space [22,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, most studies on the relationship between the latent codes and their corresponding images depend on a posterior discovery, which usually suffers from three major drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (1) Instability: The identification of emerging latent semantics is very sensitive to the samples used for analysis, such that different samples may lead to different results [12,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (2) Inaccuracy: Given the high-dimensional latent space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 512d in the popular StyleGAN family [19, 20]), finding a semantically meaningful subspace can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (3) Inflexibility: Existing manipulation models arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='04604v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='CV] 11 Jan 2023 are usually linear (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', based on vector arithmetic [16,32]), limiting the editing diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' This work offers a new perspective on learning control- lable image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Instead of discovering the semantics from pre-trained GAN models, we introduce an efficient regularizer into the training of GANs, which is able to explicitly link some latent axes with an image region or a semantic category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', sky) in the synthesis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In this way, the selected axes and the remaining axes are related to the in-region pixels and out-region pixels, respectively, with little cross-influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Such a design, termed as LinkGAN, enables a more accurate and more convenient control of the generation, where we can alter the image content within the linked region simply by resampling on the corresponding axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We conduct experiments on various datasets to evaluate the efficacy of our approach and demonstrate its four appealing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (1) It is possible to link an arbitrary image region to the latent axes, no matter the region is pre- selected before training and fixed for all instances, or refers to a semantic category and varies across instances (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (2) Our regularizer is capable of linking multi- ple regions to different sets of latent axes independently and simultaneously, and achieves joint manipulation of these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We even push the limit to tokenized image control (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (3) Our approach lends itself well to both 2D image synthesis models [20] and 3D-aware image synthesis models [4], by sufficiently improving the con- trollability yet barely harming the synthesis performance (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (4) The models trained with our regularizer are compatible with GAN inversion techniques [52] and maintain the editability on real images (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We believe that this work makes a big step towards the spatial controllability of GANs as well as the explicit disentangle- ment of GAN latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Related Work Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Generative adversarial networks (GANs) are composited by a generator and a discriminator, which are trained simultaneously by playing a two-player minimax game [10], have made tremendous progress in generating high quality and diversity images [3, 18–20, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In turn, there are widely used in a variety of tasks, such as representation learning [17, 43], image-to- image translation [6, 15], image segmentation [48], image editing [5,22,32], 3D generation [4,34,42] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Semantic discovery in GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Interpreting the pre-trained GANs [1, 2, 32] has drawn lots of attention recently since it will help us to understand how an image in GANs is rendered, which in turn, enables us to control the generation process of the GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Many recent works [7, 12, 32, 38, 50] have demonstrated the latent space of GANs encodes rich semantic information that can be used to edit the Latent code Synthesis Pixels of interest Explicit link Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Concept diagram of LinkGAN, where some axes of the latent space are explicitly linked to the image pixels of a spatial area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In this way, we can alter the image content within the linked region simply by resampling the latent code on these axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' output images globally or locally by merely shifting the corresponding latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the methods that control the output images globally [5, 12, 16, 21, 29, 32, 33, 36, 38, 39, 45, 47], they either use some off-shell classifiers to find the variation of factors supervised or just find them unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the methods that control the output images locally [1, 2, 7, 22, 37, 41, 43, 50, 51], they either control from the feature maps [2, 37, 43], control from weights of the generator [1], or control from the latent space [7, 22, 41, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, all of these works try to interpret the pre-trained GANs, but few of them apply those findings during GAN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Regularizers for GAN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Many attempts have been made to regularize GANs during training [11, 13, 20, 23, 28, 35, 40, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Some of them try to improve the training stability of GANs by regularizing the gradients of the discriminator [11, 23], the spectral norm of each layer [24], or the singular values of the generator [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Besides, some of them [8,13,20,28,35,40] aim to improve the disentanglement property of GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For example, [28, 40] try to disentangle each component in the latent vectors so that each dimension in the latent codes can only affect one attribute on the output images by adding some regularizers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', Hessian Penalty or Orthogonal Jacobian Regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In StyleGAN2 [20], a path length regularization is added to encourage the learned generator to become much smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' And [8, 35] try to disentangle different semantics in the latent space by borrowing some labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' But none of them tries to relate an arbitrary region to a specific latent part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Method In this section, we first give some background knowledge regarding GANs as well as the conventional manipulation model using GANs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Our method that bridges specified subspaces of the whole latent space with a parti- tion of a synthesized image is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Preliminaries A generative adversarial network (GAN) consists of a generator G(·) that maps latent vectors z ∼ p(z) to fake images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' ˜x = G(z), and a discriminator D(·) that tries to differentiate fake images from real ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' They are trained in an adversarial manner, in the sense that the generator tries to fool the discriminator while the discriminator learns to measure the realness of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The training loss can be formulated as follows: LG = Ep(˜x)[f(1 − D(˜x))], (1) LD = Ep(x)[f(D(x))] − Ep(˜x)[f(1 − D(˜x))], (2) where p(x) and p(˜x) are the distributions of real images and synthesized images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Besides, f is a model- specific function that varies between different GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Conventional manipulation leverages a pre-trained GAN for content editing since many prior arts [30, 32] have shown that the latent space of a GAN is correlated with the property of semantic arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, adding or removing the glass from an image can be fulfilled by performing simple additive or subtractive operations in the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The manipulation model can be formulated as follows: xedit = G(z + αn), (3) where n is the attribute vector in the latent space and α is the step for the manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In other words, when a latent code is moved toward the attribute vector, the corresponding attribute contained in the output image will vary accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Notably, such a manipulation strategy tends to modify global attributes of synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Linking Latents to Pixels With the rapid development of manipulation technique, several works [22, 50] have shown that some subspaces of the latent space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', the w space in StyleGAN [19]) can control local semantics over output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Specifically, traversing a latent code within those subspaces results in a local modification in the synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, there lacks an explicit connection between the local regions and the specified axes of latent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' To this end, we propose a new regularizer explicitly linking the axes to arbitrary partitions of synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Partition of latent codes and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In order to set up the explicit link between some axes of latent space and local regions of an image, we first introduce some notations for the corresponding partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Taking StyleGAN [19] as an example, w ∈ Rdw is the intermediate latent vector of dimension dw derived from the mapping network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Through a generator G(·), an image ˜x ∈ Rdx with dimension dx is produced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', ˜x = G(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Note that here the image in editing is reshaped as a vector for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Namely, the dimension dx is equal to the multiplication between the width and height of images (the color channels are also neglected here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We first divide the latent space into several subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Namely, a latent code w could be divided into K fragments i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', w = [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , wK], where one partition wi consists of multiple channels wi = [wi1, wi2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , wini] and ΣK i=1ni = dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Similarly, an image ˜x could also produce several partitions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', ˜x = [˜x1, ˜x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , ˜xK], where ˜xi ∈ Rmi and ΣK i=1mi = dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Importantly, we further define wc i and ˜xc i are the complements of wi and ˜xi, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', wc i = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , wi−1, wi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , wK], ˜xc i = [˜x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , ˜xi−1, ˜xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , ˜xK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 presents an example (K is equal to 2) where the blue part of the latent code and pixels within the blue bounding box denote the partitions wi and ˜xi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Now, our goal is that the latent fragment wi only controls the pixels in ˜xi and wc i controls the pixels in ˜xc i, namely, building an explicit link illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' To our surprise, we find in practice that a simple regularizer combined with the StyleGAN framework is sufficient to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Formally, we can randomly perturb wi and wc i and then minimize the variations on ˜xc i and ˜xi, respectively, expecting that wi merely controls ˜xi and hardly affects ˜xc i and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' To be specific, we use two vectors p and p to perturb wi and wc i , respectively, where p = [0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , 0, pi, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , 0], p = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , pi−1, 0, pi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' , pK], and each non-zero sub-vector in p and p is sampled from a standard Gaussian distribution N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Thus we can obtain two perturbed images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', ˜x1 = G(w + αp) and ˜x2 = G(w + αp) regarding the original one ˜x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The pixel change in ˜xc i after the perturbation by p can be computed as L1 = ||M1⊙(˜x1−˜x)||2 2 = ||M1⊙(G(w+αp)−G(w))||2 2, (4) where M1 is the binary mask denoting the region out of interest, ||·||2 denotes the ℓ2 norm, and α is the perturbation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We want the pixels to change in the region ˜xc i as minimally as possible after the perturbation by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Similarly, the pixels change in ˜xi after the perturbation by p can be written as L2 = ||M2⊙(˜x2−˜x)||2 2 = ||M2⊙(G(w+αp)−G(w))||2 2, (5) where M2 is the binary mask indicating the chosen pixels of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' These two losses L1 and L2 can be integrated as a regularizer in the StyleGAN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Lreg = λ1L1 + λ2L2, (6) where λ1 and λ2 are the weights to balance these two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Therefore, the total loss to train the generator in StyleGAN can be formulated as L = LG + Lreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (7) Practically, we could apply the new regularization Lreg in a lazy way, in the sense that Lreg is calculated once every several iterations (8 iterations in this paper), greatly improving the training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Additionally, the per- turbed images would be also fed into the discriminator during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Noticeably, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' (6) gives the regularization loss on how to build one explicit link, if we want to build multiple links between image regions and latent fragments, we need to compute Lreg on each link and sum them all, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', Σk j=1Lj reg, where k (1 ≤ k ≤ K) is the number of links we want to build.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Experimental Setup We conduct extensive experiments to evaluate our pro- posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We mainly conduct our experiment on Style- GAN2 [20] and EG3D [4] models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The datasets we use are FFHQ [19], AFHQ [6], LSUN-Church, LSUN-Car [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We also use a segmentation model [49] to select pixels with the same semantic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', all the pixels in the sky on LSUN- Church).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The main metrics we use to qualify our method are Fr´echet Inception Distance (FID) [14] and the masked Mean Squared Error (MSE) [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The experiments are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' First, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2 shows the properties of LinkGAN, which can relate an arbitrary region in the image to the latent fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Second, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='3 gives some applications of our method, such as local control on the 3D generative model, real images, and some comparisons with the baselines regarding the editing precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' At last, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='4 presents an ablation study on the size of the link latent subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Properties of LinkGAN In this section, we mainly demonstrate the effectiveness of the proposed approach by explicitly linking the pixels in any region (both the single region or multi-regions) to a partition of the corresponding latent codes, while seldom deteriorating the quality of synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1 reports FID on different datasets when our regularizer is added, from which we can see our regularizer only has a minor influence on the synthesized quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Empirically, we find that it would more stable if the proposed regularizer is incorporated after the convergence of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Therefore, we start training from a relatively well-trained generator and equipping it with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='1 Linking Latents to Single Region Regarding the partition of latent codes, we could easily choose the first several channels as one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Accord- Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Performance change after introducing our proposed regularizer into 2D and 3D baselines, where the synthesis quality slightly drops but the controllability significantly improves (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3 to 6 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' StyleGAN2 [20] EG3D [4] Dataset FFHQ AFHQ Car Church FFHQ Baseline 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='92 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='82 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='28 LinkGAN (ours) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='11 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='67 ingly, the remaining ones become the complementary code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Therefore, the goal of the proposed regularizer is to enable the explicit control of certain regions of interest through the chosen channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Note that the number of first channels that would be grouped usually depends on the area ratio of the chosen region over the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In the following context, we will show different ways of choosing pixels out of images and building explicit links between the chosen channels and pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Region-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' One general way of grouping pixels is to use a bounding box that could cover a rectangle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3 presents the qualitative results of choosing different regions randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Red bounding boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3 denote the chosen regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In terms of human faces on FFHQ, we randomly select two spatial patches that usually contain either complicated or non-special semantics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', half of faces or just a cube of background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Obviously, after building the explicit link, we could merely change the chosen regions by perturbing the corresponding partition of latent codes, while maintaining the rest regions untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Besides, perturbing the complementary latent codes results in substantial change for regions out of interest, demonstrat- ing that the spatial controlling is well-built by the proposed explicit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Additionally, we also verify the effectiveness of our regularizer on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, the connection between a partition of latent code and two eyes or one ear of animal faces could be also easily set up, causing appealing editing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Moreover, not only a relatively small region but also the larger one could be well linked to several axes of latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Results on LSUN Church suggest that even half of the entire image is also controllable with the aid of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The difference maps further present how well such an explicit link could control a region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Semantic-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Prior experimental results demonstrate the control on a rectangle region that seems to be irrelevant to a certain visual concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Namely, this link is semantic-agnostic since it merely bridges several channels with spatial locations rather than semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Therefore, we further conduct experiments on semantic controlling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' To be specific, by leveraging an off-the-shelf segmentation model [49], we could easily obtain mask annotations that specify various semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Original FFHQ AFHQ Church Input In-Region Out-Region In-Region Out-Region Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Linking latents to single fixed region, which is pre-selected before training and shared by all instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Linked regions are highlighted with red boxes, and the heatmaps reflect the change of pixel values after in-region resampling and out-region resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We find that LinkGAN can robustly link the latents to an arbitrary image region, even semantically meaningless ones (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', the second row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4 presents the semantic control on two datasets, LSUN-Church and LSUN-Car [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In particular, churches and cars are chosen as the semantics that we would like to build a link between latent space to, no matter where the chosen semantics are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Similarly, we manage to connect several channels of latent space with a given semantic such that perturbing the chosen channels will result in the obvious change of semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, the color and shape of a church vary while the sky keeps the same and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Regarding the experiments on cars, the color could be modified no matter what cars face and how many pixels cars occupy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' All these results together with the rectangle region control demonstrate the arbitrary region control enabled by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Y90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='Original Resample Heatmap Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Linking latents to semantic region (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', church, sky, and car) which dynamically varies across instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Our LinkGAN manages to precisely control a particular semantic category simply by resampling on some sparse latent axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Original Resample Heatmap Resample Heatmap Resample Heatmap Original Resample Heatmap Resample Heatmap Resample Heatmap Resample Heatmap Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Linking latents to multiple regions, where the regions highlighted by red boxes are simultaneously linked to some non- overlapping sets of latent axes and can be independently controlled by partially resampling the latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We even achieve tokenized control of the synthesis as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='2 Linking Latents to Multiple Regions After checking the effectiveness of our approach on build- ing one explicit link, a natural question then arises: is it possible to link multiple regions of interest to multiple partitions of latent codes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Joint control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 5 presents the corresponding results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' On top of that, we link three subspaces that contain 64 channels to three image regions i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', eye, nose, and mouth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Even though we could remain to manipulate semantics individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Tokenized control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The bottom one moves forward to a more challenging setting that both latent spaces and images are equally divided into four groups and four corners without any overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' To this end, we could tell that such a regularizer could build a full explicit link between the entire latent space and the whole synthesis in a disentangled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Namely, we can even tokenize an image and assign one subspace to each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Original Original Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Controllability on 3D-aware generative model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', EG3D [4], under the cases of mouth and nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We find that LinkGAN is well compatible with 3D-aware image synthesis and allows controlling both the appearance and the underlying geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Applications of LinkGAN In this part, we show that our proposed method can be used in various applications, such as controlling 3D generative models, real image manipulation, and precise local image editing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Towards 3D-aware generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We implement our regu- larizer on the 3D generative model EG3D [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Surprisingly, our regularizer performs well not only in controlling the RGB images but also in controlling the geometry of the corresponding image, showing the good generalization ability of our regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 6 shows the results of controlling the mouth and nose region by perturbing the first 64 channels of latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Importantly, controlling the linked subspace simultaneously changes the RGB images and their geometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', mouth is opening for both RGB and corresponding 3D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Real image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' After the generator is trained, we can use the property of the trained generator to control real images locally by inversion [20, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 7 shows the editing results on the real image, in which the eyes can be independently controlled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', we can only open one eye yet keep another eye untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In this case, we need to explicitly link two eye regions to two latent subspaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', one subspace controls one eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' And when the generator is well-learned, we can edit the eye region by controlling the corresponding subspace on the inverted latent code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comparison with existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Now we compare our method with some state-of-the-art algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', ReSeFa [51]) regarding local control precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 8, we can observe that our method can reach more precise control on the local regions than ReSeFa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, when modifying eyes, ReSeFa also results in a change of face color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' On the contrary, when editing the specific region, our method has negligible changes in the other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 reports the masked MSE between our method and ReSeFa when controlling those three regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Input Inversion Left Eye Right Eye Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Real image editing achieved by LinkGAN via borrow- ing the GAN inversion technique [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We manage to edit the two eyes of human independently in a very convenient way, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', partially resampling the inverted code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Namely, when editing a specific region, we want the change in this region to be as larger as possible (the higher MSE, the better) and the change in the remaining region as small as possible (the smaller MSE, the better).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We can observe that the MSEs within the edited regions are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, regarding the MSEs out of the edited regions, our method significantly outperforms ReSeFa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Ablation Study on Linking Dimensionality In this part, we conduct an ablation study on how many axes are required to build an explicit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Eyes LinkGAN ReSeFa Original Eyes Nose Mouth Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Qualitative comparison with ReSeFa [51], which posteriorly discovers semantics from a pre-trained model, on the task of local editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' LinkGAN achieves more precise control within the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' See Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 for quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Quantitative comparison with ReSeFa [51] on the task of local editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Pixel-wise mean square error (MSE) within/out of the region of interest (scaled by 1e−3 for better readability) is used as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Lower MSEo and higher MSEi are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Region Eyes Nose Mouth Metrics MSEi MSEo MSEi MSEo MSEi MSEo ReSeFa [51] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='90 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='12 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='02 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='55 LinkGAN (ours) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='21 of faces are chosen as regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3 gives the quantitative results of changing in/out eye regions with the same perturbation strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3, all the training configurations are the same except for the number of axes during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' MSEi and MSEo are computed in and out of the eye region when perturbing on their complementary latent space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Take axes number 8 as an example, and the MSEi is computed within the eye region when perturbing on axes from 8 to 512, while MSEo is computed out of the eye region perturbing on axes from 0 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In such a way, precise control could be obtained since the perturbing on the complementary latent space should barely influence the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Hence, in this situation, both MSEi and MSEo are the smaller, the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Obviously, when occupying the first 64 axes, we can get satisfying results since the sum of them is the smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In practice, we set the number of axes in latent code to 64 in most cases, such as when controlling on eyes, nose, mouth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' And the detailed axes for other datasets can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Ablation study on the linking dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' MSEi measures the effect of unlinked axes on the linked region, while MSEo measures the effect of linked axes on the unlinked region, both of with enjoy a small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' All numbers are scaled by 1e−3 for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' # Linked axes 8 16 32 64 128 256 MSEi 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='45 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='43 MSEo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='86 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='53 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='41 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='20 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='71 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Discussion and Conclusion After linking an arbitrary region to some latent axes with the size of n, any perturbation with randomly sampled n dimension vector on the linked subspace results in the con- tent change only in the linked image region, which can be viewed as a local semantic direction since it only influences the linked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, some of the sampled latent vectors can not generate realistic manipulation, and some of them can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Hence, we need to verify whether the randomly sampled vector can produce a meaningful manipulation posteriorly, just like other unsupervised methods [12,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In this work, LinkGAN is proposed to explicitly link some latent axes to a region of an image or a semantic by utilizing an easy yet powerful regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Extensive experiments demonstrate the powerful ability to precisely control the synthesized image locally from the linked latent subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' References [1] David Bau, Steven Liu, Tongzhou Wang, Jun-Yan Zhu, and Antonio Torralba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Rewriting a deep generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 [2] David Bau, Jun-Yan Zhu, Hendrik Strobelt, Bolei Zhou, Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Tenenbaum, William T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Freeman, and Antonio Torralba.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Alias- free generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Neural Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 [19] Tero Karras, Samuli Laine, and Timo Aila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' A style-based generator architecture for generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Pattern Recog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1, 2, 3, 4 [20] Tero Karras, Samuli Laine, Miika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Analyzing and improving the image quality of StyleGAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Vis.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 [22] Huan Ling, Karsten Kreis, Daiqing Li, Seung Wook Kim, Antonio Torralba, and Sanja Fidler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' EditGAN: High- precision semantic image editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Adv.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2 [24] Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida.' metadata={'source': 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scenes through the ade20k dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 4 [50] Jiapeng Zhu, Ruili Feng, Yujun Shen, Deli Zhao, Zhengjun Zha, Jingren Zhou, and Qifeng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Low-rank subspaces in GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Neural Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1, 2, 3, 4 [51] Jiapeng Zhu, Yujun Shen, Yinghao Xu, Deli Zhao, and Qifeng Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Region-based semantic factorization in GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 1, 2, 7, 8, 11, 12, 13 [52] Jun-Yan Zhu, Philipp Kr¨ahenb¨uhl, Eli Shechtman, and Alexei A Efros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Generative visual manipulation on the natural image manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' In Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Vis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2, 7 Appendix This paper proposes LinkGAN that explicitly links some latent axes to a region of an image or a semantic by utilizing an easy yet powerful regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' The Appendix is orga- nized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We first give the implementation details of our method in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Second, we give an ablation study on whether or not to use the discriminator on the perturbed images in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Third, more comparison results with other methods are given in Appendix C to show the advantages of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Implementation Details We use the official Pytorch implementation of Style- GAN2 [20] and official Pytorch implementation of EG3D [4] to validate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We keep all the param- eters untouched except our newly added regularizer during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For FID, we directly follow the original codebase, and for the masked MSE, we computed it on 10,000 images for each edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the update frequency of our lazy regularization, we calculate once every 8 minibatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For how many axes we use to control the specific region, we list as below: 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For small regions, we use 64 axes, such as the eye, nose, mouth, and ear region on FFHQ or AFHQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the larger region, such as the left region of the human face and the bottom part of the church in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='3 of the main text, we use 128 axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Also, for tokenized control in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='5 of the main text, each part has a size of 128 since we evenly spilt the latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' When the partition size becomes bigger, such as half of the image, we use 256 axes, and for the semantic control (church, sky, and car) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='4 of the main text, we use 256 axes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the loss weight λ1 and λ2, we list as below: 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the latent segment with 64 axes, we set λ1 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='01 and λ2 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the latent segment with 128 axes, we set λ1 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='01 and λ2 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For the latent segment with 256 axes, we set λ1 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='01 and λ2 equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Ablation Study Here we conduct an ablation study, that is, whether or not to use the discriminator on the perturbed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' And we do the study on the eye region of AFHQ [6] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 9 gives some abnormal random perturbed results on the eyes region without involving the discriminator on the perturbed images during training, from which we can discover some unrealistic perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, the eyes of the first cat are distorted, which can not be in the real image, and the eyes of the second cat turn to be dog eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Also, the leopard feature appears when perturbing the dog’s eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Hence, in such cases, we need to involve the discriminator in those perturbed images to hinder the generator from synthesizing those unrealistic perturbed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Original Resample Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Ablation study on AFHQ [6] when not to use discrimi- nator on the perturbed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' More results Besides the method we compare in the main text, here we compare our method with the baseline StyleCLIP [27], which can use text to control the synthesized images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' We also provide the comparison results with ReSeFa [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 10 gives the comparison results on opening eyes, for LinkGAN and ReSeFa [51], we posteriorly discover semantics that can change eye size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For StyleCLIP [27], we use the text “extremely big eyes” for the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 10, StyleCLIP can successfully make the eyes bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' However, the global color of the edited images is easily changed, such as the first column shows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 11 shows the comparison results on the opening mouth, for LinkGAN and ReSeFa [51], we also posteriorly discover semantics that can open the mouth of a face image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For StyleCLIP, we use the text “open mouth” for the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' 11, StyleCLIP also suffers from the global change of the image when editing a specific region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' For instance, the identity of the man changed in the third column, and the background of the man in the six column varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' On the contrary, our LinkGAN achieves much more precise control of the local region thanks to our explicit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' LinkGAN ReSeFa Original StyleCLIP Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Qualitative comparison on eyes with ReSeFa [51] and StyleCLIP [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' LinkGAN achieves more precise control within the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' LinkGAN ReSeFa Original StyleCLIP Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' Qualitative comparison on mouth with ReSeFa [51] and StyleCLIP [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} +page_content=' LinkGAN achieves more precise control within the regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf'} diff --git a/xtE2T4oBgHgl3EQfMAY2/content/2301.03719v1.pdf b/xtE2T4oBgHgl3EQfMAY2/content/2301.03719v1.pdf new 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Not. R. Astron. Soc. 000, 000–000 (0000) +Printed 3 January 2023 +(MN LATEX style file v2.2) +Tidal disruption rate suppression by the event horizon of +spinning black holes +Hao-Tse Huang1,2⋆ and Wenbin Lu1† +1Departments of Astronomy and Theoretical Astrophysics Center, UC Berkeley, Berkeley, CA 94720, USA +2Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong S.A.R., China +3 January 2023 +ABSTRACT +The rate of observable tidal disruption events (TDEs) by the most massive (≳ few × +107M⊙) black holes (BHs) is suppressed due to direct capture of stars by the event +horizon. This suppression effect depends on the shape of the horizon and holds the +promise of probing the spin distribution of dormant BHs at the centers of galaxies. +By extending the frozen-in approximation commonly used in the Newtonian limit, we +propose a general relativistic criterion for the tidal disruption of a star of given interior +structure. The rate suppression factor is then calculated for different BH masses, spins, +and realistic stellar populations. We find that either a high BH spin (≳ 0.5) or a young +stellar population (≲1 Gyr) allows TDEs to be observed from BHs significantly more +massive than 108 M⊙. We call this spin-age degeneracy (SAD). This limits our utility +of the TDE rate to constrain the BH spin distribution, unless additional constraints +on the age of the stellar population or the mass of the disrupted star can be obtained +by modeling the TDE radiation or the stellar spectral energy distribution near the +galactic nuclei. +Key words: Tidal disruption events — black hole — general relativity — transients +1 +INTRODUCTION +A prediction of general relativity is that stars can be directly +swallowed by the most massive black holes (BHs) without +producing an electromagnetic flare (Young et al. 1977; Rees +1988; Kesden 2012; Lu et al. 2017; van Velzen 2018). This +gives a strong, spin-dependent suppression of observable +tidal disruption event (TDE) rate (Kesden 2012; Coughlin +& Nixon 2022b). An important goal of the TDE community +is to use this suppression effect to constrain the spin distri- +bution of a large number of dormant BHs at the nuclei of +galaxies. Given the rapidly growing sample of TDEs enabled +by recent surveys (Holoien et al. 2019; Hung et al. 2020; +van Velzen et al. 2021; Sazonov et al. 2021; Hammerstein +et al. 2022) and future Vera Rubin Observatory (Ivezi´c et al. +2019), it is very promising to accurately measure the TDE +rate as a function of the BH mass, ΓTDE(MBH), provided +that the BH masses can be statistically inferred from galaxy +scaling relations (e.g., the MBH-σ relation, see Kormendy & +Ho 2013) that are well calibrated for MBH ≳ few × 107M⊙. +To approach the goal of constraining the BH spins, in +this work we provide an accurate prediction of the observ- +⋆ haotse813@gmail.com +† wenbinlu@berkeley.edu +able TDE rate function ΓTDE(MBH), for different BH spins +and stellar populations. For a given dimensionless spin and +stellar population, this rate function is decomposed into two +factors, +ΓTDE(MBH) = fTDE × Γlc, +(1) +where Γlc is the rate at which stars are scattered into the +loss cone and fTDE is the fraction of these stars that produce +observable electromagnetic flares (and 1 − fTDE is the frac- +tion of stars that are directly swallowed). Previous studies of +the loss-cone dynamics (Magorrian & Tremaine 1999; Wang +& Merritt 2004; Stone & Metzger 2016; Stone et al. 2020) +show that Γlc is likely only a weak function of the BH mass. +Observations also suggest that the TDE rate depend weakly +on the BH mass for MBH ≲ 107M⊙ for which most loss-cone +events produce bright flares (van Velzen 2018, their Figure +3). +On the other hand, we expect the observable TDE frac- +tion fTDE to drop rapidly at high BH masses as a result +of direct captures — for instance, Kesden (2012) predicted +fTDE(MBH = 107M⊙) ∼ 0.5 independent of spin and that +it drops to ∼ 10−3 (or ∼ 3 × 10−2) for MBH = 108M⊙ and +dimensionless spin parameter j = 0.5 (or 0.9). However, +Kesden (2012) only considered the case of a solar-like star +whereas a realistic stellar population consist of stars of dif- +© 0000 RAS +arXiv:2301.00259v1 [astro-ph.GA] 31 Dec 2022 + +2 +Huang & Lu +ferent masses and ages. Moreover, the criterion for tidal dis- +ruption in Kesden (2012) is based on the maximum tidal +acceleration equaling to the surface gravitational accelera- +tion of the star, but this criterion is not realistic and in fact +disagrees with the results of relativistic hydrodynamic simu- +lations by Ryu et al. (2020c) (see Figure 3 for a comparison). +In the absence of relativistic hydrodynamic simulations +for a large number of inclined orbits for spinning BHs, we +seek for semi-analytic criteria for tidal disruption that in- +volve the interior structure of the star as well as the rel- +ativistic BH spacetime. After briefly introducing the Kerr +spacetime in §2.1, we present our new criteria in §2.2, where +we generalize the “frozen-in” approximation (as adopted in +the Newtonian limit by Lodato et al. 2009; Stone et al. 2013; +Steinberg et al. 2019) to relativistic geodesics by integrat- +ing the tidal acceleration over the orbit, and if a part of +the star (in its interior) can be accelerated to the local es- +cape velocity wrt. the stellar center, then we consider the +fluid element to be tidally stripped from the star. We find +good agreement between this generalized frozen-in approx- +imation and numerical simulations by Ryu et al. (2020c) +(which are for Schwarzschild BHs). This motivates us to ap- +ply the method to the case of spinning BHs and calculate the +observable TDE rate for the highest mass BHs for different +stellar populations in §2.3. +It is important to stress upfront that many aspects of +the electromagnetic emission from TDEs are poorly under- +stood (the origin of optical emission in particular, see e.g., +Piran et al. 2015; Metzger & Stone 2016; Roth et al. 2016; +Dai et al. 2018; Lu & Bonnerot 2020; Bonnerot et al. 2021; +Andalman et al. 2022; Steinberg & Stone 2022). Currently, +there is not a clear mapping between the mass loss from +the star to the properties of the emission. In this paper, we +adopt a mass-loss fraction of 50% from the star as a clear- +cut boundary between observable and dark TDEs, although +our method can be directly applied to other mass-loss frac- +tions. The mass loss fraction is a very strong function of the +orbital pericenter radius — for the cases simulated by Ryu +et al. (2020c), the pericenter radii for mass-loss fraction of +30% (or 70%) only different from that for 50% mass loss by +about ten percent, and this would lead to a small change in +the observable TDE rate as compared to what is presented +in this paper. +The results from our calculations are then presented in +§3, including the size of the loss cones for disruption and +direct capture for stars of different ages and masses (§3.1). +The synthesized TDE rates for the entire stellar population +and our proposed spin-age degeneracy (SAD) are detailed +in §3.2. We discuss the limitations of our calculations in §4 +and summarize our findings in §5. +2 +METHOD +In this section, we describe our criteria for determining the +outcome of a star passing by a BH, and how they can be +used to calculate the disruption rate and direct capture rate +by integrating over the angular momentum and mass distri- +butions of the stellar population. +2.1 +Kerr geodesic and tidal tensor +The spacetime of a rotating BH is given by the Kerr met- +ric, which can be expressed in Boyer-Lindquist coordinates +under geometrized units (G = c = 1) as +ds2 = − +� +1 − 2MBHr +Σ +� +dt2 + Σ +∆dr2 + Σdθ2 ++ +� +r2 + a2 + 2MBHra2 +Σ +sin2 θ +� +sin2 θdφ2 +− 4MBHra sin2 θ +Σ +dtdφ, +(2) +where Σ = r2 + a2 cos2 θ, ∆ = r2 − 2MBHr + a2, and MBH, a +are the mass and spin of the BH (Boyer & Lindquist 1967). +In the following we will also frequently use j = a/MBH, +which is the dimensionless spin of the BH. +For a main-sequence star passing by a BH, its center +of mass follows a time-like geodesic. The radius of the star, +which is of the order R⊙ ≈ 7 × 108 m, is much smaller than +the Schwarzschild radius of the BH with MBH ⩾ 106 M⊙: +rS = 2 rg = 2GMBH +c2 += 2.95 × 109 +� MBH +106 M⊙ +� +m. +(3) +The geodesic equations are (Carter 1968) +∆˙t = +� +r2 + a2 + 2MBHra2 +Σ +sin2 θ +� +E − 2MBHra +Σ +Lz, +(4) +∆ ˙φ = +� +1 − 2MBHr +Σ +� +Lz +sin2 θ + 2MBHra +Σ +E, +(5) +Σ2 ˙θ2 = Q + cos2 θ +� +(E2 − 1)a2 − +1 +sin2 θ L2 +z +� +, +(6) +Σ2 ˙r2 = +� +E(r2 + a2) − aLz +�2 − ∆ +� +Q + (Lz − aE)2 + r2� +, +(7) +where E is the specific energy, Lz is the specific angular mo- +mentum along the black hole spin axis, and Q is the Carter +constant. Note that E, Lz, Q are all constants of motion and +are conserved along the geodesic. Far from the BH (r ≫ rg), +the Carter constant is related to the total specific angular +momentum L by +Q = L2 − L2 +z. +(8) +The above geodesic equations are numerically solved using +the code developed by Rauch & Blandford (1994). Further- +more, we are only interested in the stellar trajectories that +will come close to the BH and potentially produce TDE. As +the initial kinetic energy of the star is negligible compared +to the work done by tidal forces in the BH’s frame as the +star reaches near the pericenter, we simply set E = 1 in all +the calculations. +The tidal disruption can be viewed as the consequence +of the differential motion of fluid elements in the star induced +by the tidal forces of the BH. Due to the smallness of the +star compared to the BH, it is convenient to describe the +process in the local frame of the star, using the tidal tensor. +In the local inertial frame of the center of a free-falling star +(Fermi Normal Coordinates), the motions of fluid elements +will follow the equation of geodesic deviation in the absence +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +3 +of other forces: +d2χi +dτ 2 = −Cijχj, +(9) +where τ is the proper time of the geodesic, χ is the displace- +ment of the fluid element from the stellar center, and Cij +is the tidal tensor. The tidal tensor is a 3 × 3 symmetric +matrix described in Appendix A and see Marck (1983) for +more details. +2.2 +Criteria for tidal disruption +Since we are interested in not the details of each TDE but +the overall rate, we adopt a critical mass loss fraction of 50% +as the threshold for luminous TDEs. In the following the star +is said to be disrupted only if it loses more than 50% of its +mass during the pericenter passage. Our rate calculation is +not sensitive to this choice, since the mass loss fraction is a +very steep function of the pericenter radius of the stellar tra- +jectory (Guillochon & Ramirez-Ruiz 2013; Law-Smith et al. +2020; Ryu et al. 2020a). In fact, our method can be directly +applied to other choices as well (e.g., 30% mass loss). +Whether the star can be disrupted depends on both +the strength and the working time of the BH’s tidal forces. +Without full-scale hydrodynamic simulations, we attempt to +devise the criteria of tidal disruption that capture these two +aspects of tidal forces. Our criteria have two parts. First, +motivated by the frozen-in approximation, we calculate the +work done by tidal forces by integrating the geodesic de- +viation equation to obtain “maximum differential velocity” +∆vmax. As explained below, the value of ∆vmax for a given +stellar geodesic depends on the initial radius r0 from which +we start our integration. Second, the star must fill up its +Roche Lobe (described in Fermi Normal Coordinates) at +radii r < r0 in order for it to lose mass. +In the following, we elaborate on the concept of rela- +tivistic Roche lobe and how the maximum differential ve- +locity ∆vmax is calculated, and based on these, we then con- +struct the criteria for tidal disruption. A comparison of our +criteria to the numerical simulation results by Ryu et al. +(2020a) is then provided as a test of the validity for the +Schwarzschild case. Since our tidal disruption criteria are +based on the time-dependent tidal tensor in the Fermi Nor- +mal frame, the formalism can be directly applied to any +geodesics in the Kerr metric. We stress that our criteria +are only approximate (certainly not perfect) and the va- +lidity needs to be strictly tested against future hydrody- +namic simulations in the Kerr spacetime. However, given +the large computational cost of such numerical simulations, +our method provides the best-effort TDE rate predictions on +the high BH-mass end before such extensive hydrodynamic +simulations are carried out. +2.2.1 +Maximum differential velocity +The frozen-in approximation (Stone et al. 2013) assumes +that a star is unperturbed before reaching the Newtonian +tidal disruption radius rT = R∗(MBH/M∗)1/3 (R∗ and M∗ +being the stellar radius and mass), at which the star is +rapidly torn apart and then the fluid elements start to free- +fall according to the gravitational field of the BH. +Figure 1. +An example of the geodesic and half-mass surface +distorted by tidal forces under the frozen-in approximation. The +top-left panel shows a parabolic geodesic in the Kerr metric of +j = 0.9. In Boyer-Lindquist coordinates and natural units, the +geodesic starts at r = 1000, θ = π/2, φ = 0, with L = 4.62 and +Lz/L = −0.5. Five markers are placed on the geodesic to repre- +sent the different stellar proper times τ (τ = 0 at the pericenter). +Blue circle marker is the point the star center enters r0 = 13; +purple star marker is the point the star center leaves r0 = 13; red +square marker is a point close to the pericenter. We assume that, +as soon as the star enters r0 = 13, the fluid elements at the half- +mass radius start to free-fall under the influence of tidal forces +(eq. 9), causing the sphere to be distorted. Such distortions are +shown in the remaining panels using Fermi Normal Coordinates. +The length unit in Fermi Normal Coordinates is normalized to +the original half-mass radius Rhm. Note that the surfaces shown +in this figure are shown using a very high resolution sampling grid +across the half-mass surface (and triangulation), whereas in our +production-run calculations of ∆v(r0) (see Section 2.2.1), a much +coarser (yet sufficient) grid is used to reduce the computational +cost. +In our relativistic treatment of tidal disruption, we as- +sume the star to be unperturbed before reaching radius r0, +which is not equal to the Newtonian tidal disruption radius +rT. At r = r0, the self-gravity and pressure forces of the star +are overwhelmed by the tidal forces and fluid elements in the +star begin to free-fall. We track the free-fall of the fluid ele- +ments that originate at the half-mass radius Rhm of the star +by solving the equation of geodesic deviation explicitly. The +choice of the half-mass radius is based on our use of 50% +mass loss as the threshold of tidal disruption, and the exact +value of Rhm, which depends on both the age and the mass +of the star, is calculated from a stellar model obtained by +Modules for Experiments in Stellar Astrophysics (MESA, +Paxton et al. 2011, 2013, 2015, 2018, 2019). An illustration +of the free-fall of the fluid elements is provided in Figure 1, +where we show an inclined geodesic in the Kerr metric and +how the sphere of the half-mass radius get distorted with +time under the influence of tidal forces described by eq. (9). +© 0000 RAS, MNRAS 000, 000–000 + +Geodesic +t= - 39.4 +T = - 39.4 +t = - 14.8 +1 +T=0.6 +t= 15.0 +40 +t = 39.9 +-1 +-1 +30 +-1 +0 +0 +20 +X2 +Z +t= - 14.8 +10 +0 +1 +-10 +0 +0 +10 +20 +-10 +30 +X +-1 +y +-20 +40 +0 +-1 +0 +11 +X2 +t= 15.0 +T=0.6 +T= 39.9 +2.5 +5 +-10 +2 +0.0 +0 +0 +0 +以o +-5 +-2 +5 +10 +-2 +-2.5.02.5 +-2.5 +-50 5 +0 +-10 +X2 +2 +12 +124 +Huang & Lu +These fluid elements are uniformly sampled on a sphere +of radius Rhm according to their angular positions (θ, φ) in +the local frame of the star. The number of sampling grid +points is 5×5 and the grid is uniform in the range of cos θ ∈ +(−1, 1), φ ∈ (0, 2π), which we show to be sufficient based +on our convergence test. In order for the star to lose 50% +of its mass, these fluid elements must be able to produce a +sufficiently large differential velocity ∆v so as to break free +from the self-gravity of the star. Based on each of the fluid +elements initially at (θ, φ) at radius Rhm from the stellar +center, we calculate the largest differential velocity by the +time the star exits from the sphere of radius r0 from the BH +and refer to it as ∆v(r0) ≡ max [∆v(θ, φ)]. +The choice of the initial radius coordinate r0 is an issue +in this approach. However, we find that the following method +produces results that are in good agreement with those from +the hydrodynamic simulations by Ryu et al. (2020a). Step +(1): we evaluate ∆v(r0) on a dense grid1of r0 and then +choose the critical r0 that maximizes the function of ∆v(r0). +The maximum differential velocity ∆vmax obtained in Step +(1) characterizes the maximum work that can be done by +the BH’s tidal forces on a given fluid element. Step (2): we +compute the Roche Lobe volume VRL(r0) as a function of r0 +along the geodesic and restrict the choice of r0 in Step (1) +by the requirement of VRL(r0) < V∗ (where V∗ = 4πR3 +∗/3 is +the volume of the unperturbed star). This is a conservative +restriction, which means that the star will lose mass at radii +r < r0. +In Figure 2, we show ∆v(r0) for parabolic, inclined +geodesics of L = 4.62 and |Lz/L| = 0.5 (in natural units) in +the Kerr metric for different BH spins j = 0 and 0.9. The +differential velocity ∆v depends on the initial radius r0 in +the following way. For large enough r0, ∆v decreases with +r0 because of the declining strength of the BH’s tidal forces; +whereas as r0 approaches rp, ∆v vanishes rapidly because +of the decreased time for the tidal forces to do work. For the +example of L = 4.62 in the Schwarzschild metric (j = 0), we +find that differential velocity reaches its maximum ∆vmax +at r0/rp ≈ 1.7rg. Figure 2 also shows the dependence of the +work done by tidal forces on the BH spin. For fixed L, the +prograde (retrograde) orbit in the spinning BH will experi- +ence smaller (larger) tidal forces and has a lower (higher) +value of ∆v. We will return to this point in §3.1. +In the next subsection, we discuss the requirement that +the star must fill up its Roche lobe at radii r < r0 in order +for it to lose mass. +2.2.2 +Relativistic Roche Lobe +To complete our discussion on maximum differential veloc- +ity ∆vmax, we provide a constraint on the initial radius r0. +In the face of the enormous challenge of modeling the hydro- +dynamic effects including the stellar interior structure and +the tidal acceleration history along the geodesic (see Rossi +et al. 2021, for a review), we only seek to place a limit on the +potential range of r0 using the concept of the Roche lobe. +In the studies of binary systems, the Roche lobe is the +largest volume that a star can occupy without losing mass +1 The grid is composed of 20 grid points and equally spaced in +the variable log10 r′ +0 ∈ [−3, 0.3], where r′ +0 = (r0 − rp)/rp. +10−3 +10−2 +10−1 +100 +(r0 − rp) / rp +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Δv +jΔ=Δ0.0 +jΔ=Δ0.9Δ(prograde) +jΔ=Δ0.9Δ(retrograde) +Figure 2. +Maximum differential velocity ∆v as the function +of the tidal radius r0 for three geodesic in the Kerr metric of +different spins j. All geodesics have L = 4.62, which for the +Schwarzschild metric (blue solid line) corresponds to rp = 8.0 rg. +For the geodesic of prograde (retrograde) orbit plotted in orange +dashed (green dotted) line, it has lz = Lz/L = 0.5 (−0.5) and +rp = 8.7 (7.1) rg. The value of ∆v is calculated from an initial +radius of |χ0| = 1 (normalized such that Rhm = 1), in the natural +units G = MBH = c = 1. +to its companion (Paczy´nski 1971). A similar concept can be +applied to the mass loss of the star under the effect of tidal +forces. For a tidal tensor Cij, the motion of a fluid element +in the Fermi Normal frame is governed by the tidal poten- +tial Φtide, the star’s original gravitational potential, as well +as pressure forces. Hydrostatic equilibrium is reached when +the isobars are coincident with the equipotential surfaces +(including both tidal and self-gravity potentials). There is +a critical value of potential Φcrit, above which the equipo- +tential surfaces are no longer closed around the star. The +region enclosed within the equipotential surface correspond- +ing to Φcrit is the largest volume that a star can have before +starting to lose mass, under the assumption of hydrostatic +equilibrium. +We calculate the volume of the Roche Lobe VRL(r) +along the geodesic at different radii. The detailed (but +straightforward) calculations of the shape of the Roche lobe +and its volume VRL are provided in Appendix B. It should +be noted that here we assume that the star is rotating at +angular frequencies that are much below the Keplerian fre- +quency at the stellar surface +� +GM∗/R3∗, and under this +assumption, we ignore the centrifugal forces in the Roche +potential. This is appropriate since most stars that are scat- +tered into the loss cone are expected to be slow rotators. +Since a necessary condition for tidal disruption is mass loss +from the star, the volume of the Roche lobe sets a constraint +on the value of r0 by requiring +VRL(r0) < V∗ = 4πR3 +∗/3, +(10) +where V∗ is the volume of the unperturbed star. This pro- +vides a maximum value of r0, which is referred to as r0,max +hereafter. Beyond this radius, the BH’s tidal forces are too +weak to induce mass loss from the star and the frozen-in +approximation is unlikely to apply. There are non-plunging +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +5 +106 +107 +108 +MBH [M ⊙ ] +101 +102 +rp/rg +M * ⊙=⊙0.3⊙M ⊙ +M * ⊙=⊙1.0⊙M ⊙ +M * ⊙=⊙3.0⊙M ⊙ +Figure 3. The maximum pericenter that is able to produce the +tidal disruption with 50% mass loss for different SMBH and stellar +masses. The SMBHs are non-rotating (j = 0) and the spacetime +is simplified into Schwarzschild metric. The markers are the data +points interpolated from Figure 6 of Ryu et al. (2020c). The solid +lines are the prediction from our criteria and the dashed lines are +the prescription of the tidal disruption used in Kesden (2012). +To ensure a fair comparison, the stellar properties are computed +from MESA and taken at the half age of the main sequence. The +horizontal black line indicates the minimum pericenter 4 rg that +a star can reach without being captured by the SMBH. The gray- +shaded region indicates the pericenters inaccessible by the stars. +geodesics where r0,max does not exist, because VRL(r) is +greater than V∗ everywhere along the geodesic. In those +cases, the star is not tidally disrupted for that geodesic. +2.2.3 +Combination of maximum differential velocity and +Roche lobe +Equipped with the differential velocity as a function of ini- +tial radius ∆v(r0) (already maximized over all fluid elements +at Rhm from the stellar center) as well as the maximum +initial radius r0,max, we can now fully describe our criteria +for tidal disruption. We evaluate ∆v(r0) on a dense grid +of r0 ∈ [rp, r0,max]. The maximum value of ∆v(r0) within +the range r0 ∈ [rp, r0,max] is the largest differential veloc- +ity that the fluid elements originated from Rhm can achieve +and is denoted as ∆vmax. The star is then only classified as +disrupted if ∆vmax is greater than the critical velocity vvir, +where +vvir = +� +GM∗/2 +Rhm +. +(11) +which is the virial velocity for a fluid element at the radius +Rhm. +We test the validity of our criteria of tidal disruption by +computing the maximum pericenter radius that can produce +tidal disruption with 50% mass loss for non-spinning BHs. +Our results are compared with those from the hydrodynamic +simulations by Ryu et al. (2020a,c) and shown in Figure 3. +We interpolate the data points in Figure 6 of Ryu et al. +(2020c) to get the pericenter of the stellar trajectory that +can produce the tidal disruption with 50% mass loss. For all +three stellar masses, 0.3, 1.0, 3.0 M⊙, our criteria match the +result of the hydrodynamic simulations at low BH masses +(∼ 106 M⊙). The value of maximum pericenter from our +criteria is slightly higher for M∗ = 3.0 M⊙ and lower for +M∗ = 0.3 M⊙ than the result of hydrodynamic simulation +at high BH masses (∼ 107.5 M⊙). Such discrepancies are +expected from the approximate nature of our treatment. +In the earlier study of tidal disruption by Kerr BHs, +Kesden (2012) proposed a tidal disruption criterion based +on the comparison between the strength of the maximum +tidal force (given by the maximum eigenvalue of the tidal +tensor) and the surface gravity of the star at the pericenter. +From Figure 3, we see that the critical pericenter radii from +Kesden (2012)’s prescription has a nearly power-law depen- +dence on the BH mass, rp ∝ M −1/3 (close to that expected +from the Newtonian prescription), which fails to reproduce +the hydrodynamic results at high BH masses. In particular, +the hydrodynamic results show a flattening in the critical +pericenter radii at high BH masses, and this flattening is +due to the longer working time of the tidal forces in those +cases, which are not captured by Kesden (2012)’s criterion +based on the comparison of instantaneous forces. Moreover, +Kesden (2012)’s prescription does not make use of the infor- +mation on the stellar interior structure. Stars slightly heavier +than about 1 M⊙ are more difficult to be tidally disrupted +than lower mass stars as the interior structure transits from +convective to radiative envelope without much increase in +the stellar size. Ignoring the interior structure leads to in- +correct critical pericenter radii even for low-mass BHs. +Therefore, despite some small discrepancies, we regard +our criteria as a significant improvement from Kesden (2012) +as we use a more sophisticated, physically motivated treat- +ment on relativistic effects and the stellar structure. +2.3 +Rates of direct captures and observable tidal +disruption events +With the criteria for tidal disruption in hand, we then pro- +ceed to calculate the rates of direct captures and tidal dis- +ruptions based on the orbital angular momentum and mass +distributions of stars that are scattered into the loss cone. +All the stars coming close to the BH pass through the +surface of a sphere of radius rinit ≫ rg centered at the +BH. Following Kesden (2012), we assume the “full loss-cone” +regime in this work (as also adopted by Coughlin & Nixon +2022b), meaning that the velocity distribution on the surface +of the rinit sphere is taken to be uniform. Our formalism can +be directly applied to any angular momentum distribution +for the stars scattered into the loss cone. We leave a detailed +exploration of other angular momentum distributions (e.g., +allowing a certain fraction of galaxies to be in the “empty +loss-cone” regime, see Stone & Metzger 2016) to a future +work. +Let n and v stand for the number density and velocity +of stars on the surface of the sphere at radius rinit. The +differential rate of the stars entering the sphere in terms of +specific angular momentum is given by +∂2Γ +∂L∂lz = πnL +v +, +(12) +where L is the total specific angular momentum of the star +© 0000 RAS, MNRAS 000, 000–000 + +6 +Huang & Lu +and lz = Lz/L ∈ [−1, 1] is the fractional angular momentum +projected in the direction of the BH spin axis. The deriva- +tion of the above expression is provided in Appendix C. Note +that to fully specify a geodesic, the initial radius rinit and +polar angle θinit (wrt. the BH spin axis) are needed in addi- +tion to L and lz. Since tidal interactions are only important +near the BH, the results do not depend sensitively on the +choice of rinit, as long as it is sufficiently large. Based on this, +we fix rinit = 1000 rg. We also find that, for rinit ≫ rg, the +strength of the BH’s tidal forces at small radii are rather +insensitive of the initial polar angle θinit, with maximum +fractional difference in the eigenvalues of the tidal tensor +typically less than 10% (only in the most extreme rare cases +of j = 0.99 and nearly plunging geodesics, the maximum dif- +ference reaches to about 20%). To save computational cost, +we fix θinit = π/2 for all geodesics. The fact that the tidal +forces at small radii are insensitive to θinit also allows us to +integrate the differential rate over θinit to obtain eq. (12). +For a given lz (specifying the inclination angle of the or- +bit), there are two important values of the total angular mo- +mentum, Lcapt(lz) and LTD(lz), which are the critical values +of the total angular momentum for direct capture and tidal +disruption, respectively. The value of Lcapt only depends on +the constants of motion and can be calculated numerically +(see also Coughlin & Nixon 2022b). On the other hand, LTD +also depends on the stellar properties and only exists if the +BH’s tidal forces are strong enough to produce tidal disrup- +tion. We further define the loss cone angular momentum Llc +as the maximum of Lcapt and LTD, if the latter exists, i.e. +Llc = max (Lcapt, LTD) . +With the above definitions, the TDE rate ΓTDE of a stellar +population of fixed stellar mass and age is given by +ΓTDE = +� 1 +−1 +� Llc +Lcapt +πnL +v +dLdlz +=πn +2v +� 1 +−1 +� +L2 +lc(lz) − L2 +capt(lz) +� +dlz. +(13) +Similarly, the capture rate can be expressed as +Γcapt = +� 1 +−1 +� Lcapt +0 +πnL +v +dLdlz +=πn +2v +� 1 +−1 +L2 +capt(lz) dlz. +(14) +The sum of ΓTDE and Γcapt is the rate at which the stars +enter the loss cone +Γlc = ΓTDE + Γcapt. +(15) +To calculate Γlc, ΓTDE, Γcapt and thus LTD, we sample the +geodesic on the grid of L, lz. For j = 0.0, the grid number +is 8000 in the range of L ∈ [4 + 10−5, 14.5] and uniform in +log scale. For j ̸= 0.0, the grid number is 1000 × 100 in the +range of L ∈ [Lcapt(lz = 1.0) + 10−5, 14.5] and lz ∈ (−1, 1). +The grid is uniform in log scale of L and in linear scale of +lz. +The above “monochromatic” (for single M∗ and tage) +TDE and capture rates depend on the mass MBH and spin +j of the BH, as well as the mass M∗ and age tage of the star. +The observed TDE rates from the galactic nuclei are then +obtained by averaging ΓTDE(MBH, j, M∗, tage) over a given +stellar population of different tage, M∗. +The stellar populations in extragalactic nuclei, espe- +cially the stars within a few parsecs from the BH, are un- +certain due to the lack of observational constraints. In the +following, we adopt a single stellar population of a given age +following the Kroupa (2001) initial mass function (IMF). +Realistically, the stars that are scattered into the loss cones +in a given galactic nucleus were formed at different epochs +throughout the history of the galaxy, and different galax- +ies may have different stellar populations near the centers +depending on their evolutionary history. Our results for a +single stellar population of different ages can be statistically +combined to mimic any realistic stellar populations. +We use MESA to model the evolution of solar- +metallicity stars on a sufficiently wide mass grid2 and record +the structures of all stars below the main-sequence turn-over +mass Mmax(tage) at a given age tage. The stellar population +at this age can be approximately described as a truncated +Kroupa IMF within the mass range Mmin ⩽ M∗ ⩽ Mmax, +where Mmin = 0.085 M⊙ is the minimum stellar mass3 con- +sidered in this work. It is further assumed that the stellar +populations of different masses have the same initial veloc- +ity v and only differ in their number densities n. With these +assumptions, the stellar population-averaged TDE rate and +the direct-capture rate are given by +⟨Γk⟩ = +� Mmax(tage) +Mmin +Γk(MBH, j, M∗, tage) dN +dM∗ dM∗ +� Mmax(tage) +Mmin +dN +dM∗ dM∗ +, +(16) +where the subscript k = TDE (for observable TDEs) or +capt (for direct captures), and dN/dM∗ is the properly nor- +malized Kroupa IMF. Note that the averaged TDE and +capture rates, ⟨ΓTDE⟩ and ⟨Γcapt⟩, are both functions of +MBH, j, tage. The total population-averaged rate of loss- +cone scatterings is +⟨Γlc⟩ = ⟨ΓTDE⟩ + ⟨Γcapt⟩ . +(17) +We further define the observable TDE fraction fTDE as +the fraction of stars that are scattered into the loss cone +fTDE = ΓTDE +Γlc , +(18) +which does not depend on the stellar number density n and +velocity v as they are canceled in the expression. Our goal is +to calculate the observable TDE fraction fTDE as a function +of MBH, j, M∗, tage (i.e., for different BH and stellar prop- +erties). Finally, the stellar population-averaged observable +TDE fraction ⟨fTDE⟩ can similarly be defined based on the +averaged rates +⟨fTDE⟩ = ⟨ΓTDE⟩ +⟨Γlc⟩ , +(19) +which can be directly compared with observations provided +that we know the total loss-cone scattering rate ⟨Γlc⟩ (which +depends weakly on BH masses, see below). +2 The stellar mass M∗ is sampled uniformly in the log scale in +the range of M∗ ∈ [10−1, 100.71] with 13 grid points. +3 TDEs by stars with even lower masses, mostly brown dwarfs, +are fainter (due to smaller energy budget) and faster fading (due +to shorter fallback timescale), so they are not expected to domi- +nate the observed rate in current surveys. +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +7 +Figure 4. +The ratio ∆vmax/vvir (see Section 2.2 for their definitions) in the L, lz parameter space for different BH masses +log10(MBH/M⊙) = 6.5, 7.5 and spins j = 0.0, 0.9. The calculations are for a 1.0 M⊙ main-sequence star of age 1.0 Gyr. The angu- +lar momentum L is expressed in natural units G = MBH = c = 1. The red line in each panel indicates the location of LTD, for which +∆vmax = vvir. The black region at the bottom indicates the geodesics directly captured by the BH. Between the red line and black +region, the star can produce an observable TDE. The gray region indicates where the star does not fill up its Roche lobe and hence there +will be no mass loss. +.0 +.5 +.9 .99 +j +0.92 +0.94 +0.96 +0.98 +1.00 +Γlc(j) / Γlc(j = 0.0) +Young et al. 1977 +log10(MBH/M ⊙ ) = 6.0 +log10(MBH/M ⊙ ) = 7.0 +log10(MBH/M ⊙ ) = 8.0 +log10(MBH/M ⊙ ) = 9.0 +Figure 5. +The spin-dependent loss-cone scattering rate Γlc(j) +normalized to the value for a non-spinning BH Γlc(j = 0.0), for a +1.0 M⊙, 1.0 Gyr main-sequence star. The red curve is the analyt- +ical fit given by eq. B2 in Young et al. (1977). +3 +RESULTS +In this section we present the TDE rate calculation based +on the method in Section 2. We first calculate the TDE rate +fraction fTDE of a single star of given initial mass and age in +§3.1, and then go on to show the averaged TDE rate fraction +⟨fTDE⟩ for a stellar population of a given age in §3.2. +3.1 +Test case of a single star +In this subsection, we consider the TDE fraction fTDE as +a function of BH mass and spin for a fixed 1 M⊙, 1 Gyr-old +main-sequence star. +3.1.1 +TDE and direct-capture cross-sections +There are generally three distinct regions in the parameter +space L, lz for the geodesics of the star. (1) For sufficiently +large L, the star stays far from the BH and is neither tidally +disrupted nor captured; (2) For sufficiently small L, the star +directly plunges into the BH and there is no observable TDE; +(3) In between these two regions, the star is tidally disrupted +by the tidal force without being captured and produces an +observable TDE. In the following, we use the criteria de- +veloped in Section 2.2, which is based on the ratio between +the maximum differential velocity ∆vmax (given by the work +done by tidal forces) and the virial velocity vvir (eq. 11) at +the half-mass radius of the star, to examine how these re- +gions are influenced by BH mass MBH and spin j. +Figure 4 shows the ratio ∆vmax/vvir in the parameter +space of L, lz for two different BH masses log10(MBH/M⊙) = +6.5, 7.5 and two different spins j = 0.0, 0.9. In each panel, +the capture region in the parameter space L, lz is colored in +black. For each inclination angle as specified by lz, the upper +bound of the capture region is Lcapt, which independent of +the stellar properties, and when expressed in the natural +units, independent of BH mass MBH. +For L > Lcapt(lz), we calculate the maximum differen- +tial velocity ∆vmax based on the method described in Sec- +tion 2.2. The value of ∆vmax contains information on the +stellar structure and the relativistic tidal forces. The star is +only considered to be tidally disrupted when ∆vmax exceeds +the virial velocity vvir at the star’s half-mass radius. The +critical angular momentum LTD for which ∆vmax = vvir is +marked by a red solid line. Between LTD and Lcapt, the star +experiences sufficiently strong tidal forces that lead to an +observable TDE. For very large angular momenta L, tidal +forces are so weak that mass loss from the star is not possible +and this region is shaded in gray. +The presence of the BH spin leads to asymmetry in pro- +grade (lz > 0) and retrograde orbits (lz < 0). For a fixed +total angular momentum L, the retrograde orbits experience +stronger tidal forces and have larger values of ∆vmax than +the prograde orbits. This effect is reflected in the value of +LTD, which is larger for retrograde orbits and smaller for +prograde orbits. The asymmetry of LTD(lz) between pro- +grade and retrograde orbits are more important for large +BH masses, where LTD can be very close to Lcapt, meaning +that TDEs can only occur close to the capture region. The +strongest tidal forces experienced by a given star, however, +occur not in retrograde orbits but in prograde orbits. This is +because the BH spin lowers the value of Lcapt for prograde +© 0000 RAS, MNRAS 000, 000–000 + +log10(MBH/Mo) = 6.5,j= 0.0 +log10(MBH/M) = 7.5,j= 0.0 +log10(MBH/Mo) = 6.5,j= 0.9 +log10(MBH/Mo) = 7.5,j= 0.9 +2 +8 +8 +8 +8 +6 +-9 +6 +6 +0 +4 +4 +4. +4 +-1 +0 +1 +-1 +0 +1 +-1 +0 +1 +-1 +0 +1 +Iz8 +Huang & Lu +10−1 +100 +M * [M ⊙ ] +101 +102 +rp / rg +tage = 0.1⊙Gyr +10−1 +100 +M * [M ⊙ ] +101 +102 +rp / rg +tage = 1.0⊙Gyr +Figure 6. The maximum pericenter radii for TDEs as a function of stellar mass and BH mass, for two different stellar ages tage = 0.1 Gyr +(left panel) and 1 Gyr (right panel). The BH is non-spinning (j = 0). In each panel, different curves, from top to bottom, represent the +increasing BH masses, starting from MBH = 106 M⊙ with a step size ∆ log10(MBH/M⊙) = 0.2. The cases of MBH = 106, 107, 108 M⊙ +are highlighted with dark blue lines. The horizon black line at rp = 4 rg indicates the minimum pericenter below which the star would +be captured by the BH. +orbits, and the star can reach closer to the BH without being +captured. +3.1.2 +Weak spin-dependence of the loss-cone cross-section +For a given inclination lz, the BH spin changes the critical +angular momenta for capture, Lcapt, and for TDE, LTD. In +this subsection, we show that when averaged over all inclina- +tion angles (assuming that stars at large distances are not +aware of the BH’s spin direction), the total loss-cone rate +Γlc, the rate at which stars are scattered into the loss cone, +is nearly independent of the BH’s spin for all BH masses +relevant for observable TDEs (MBH < 109M⊙). +Such a weak dependence is expected for low BH masses +for which LTD ≫ Lcapt or Γcapt ≪ ΓTDE, because most +TDEs occur far from the BH’s horizon where the spin effects +are minor. For high BH masses, the functional form of Γlc(j) +is not obvious. In Figure 5, we show the ratio Γlc(j)/Γlc(j = +0.0) in the spin range of 0.0 ⩽ j ⩽ 0.99 for different BH +masses MBH for a 1 M⊙, 1 Gyr main-sequence star. We find +that this ratio is nearly constant over the entire range of +j considered here for all relevant BH masses up to 109M⊙. +For the extreme case of MBH = 109 M⊙ and j = 0.99, the +loss-cone rate is only slightly smaller than that for j = 0, +with a difference no more than 10%. +The weak dependence of the inclination-averaged loss- +cone cross-section on the BH spin is physically due to the +fact that increased values of LTD and Lcapt in the retrograde +orbits compensate for their decreased values in the prograde +orbits in the full loss-cone case4 considered in this work. A +similar conclusion was obtained by Young et al. (1977) and +4 We caution that the weak spin-dependence is not necessarily +true in the empty loss-cone limit. +Kesden (2012), who found that the capture rate is nearly +independent of the black hole spin j if stars come from an +isotropic velocity distribution. In Figure 5, we also show, by +a red curve, the analytical fit of Γcapt(j)/Γcapt(j = 0.0) for +the isotropic stellar flux given by Young et al. (1977). The +analytical fit agrees with our data points of Γlc(j)/Γlc(j = +0.0) at MBH = 109 M⊙, because at such high BH masses, +a 1 M⊙ main-sequence star cannot be disrupted outside the +horizon and hence Γlc = Γcapt. +3.1.3 +Effects of stellar interior structure +We now isolate the effects of stellar mass and age on the +TDE rate by considering the maximum pericenter radii be- +low which a star can be tidally disrupted by non-spinning +BHs of different masses. The results are shown in Figure 6. +In the hydrodynamic simulations by Ryu et al. (2020a), +it was found that the maximum pericenter for a complete +tidal disruption cannot be described by a simple function of +M∗ but instead varies near an average value (see the right +panel of their Figure 3). This complex behavior is related +to the change in stellar interior structure with M∗. The ef- +fect of stellar interior structure is included in our criteria +for tidal disruption based on the work done by tidal forces +on the fluid elements at the half-mass radius Rhm of the +star. Similar to what Ryu et al. (2020a) found, the maxi- +mum pericenter radius for tidal disruption also cannot be +easily described by a simple function of M∗ (for a given +stellar age). One particular feature in Figure 6 is the slight +decrease of the maximum pericenter when M∗ goes above +about 1 M⊙. This is the consequence of the stellar density +structure transitioning from that of a convective envelope +(for M∗ ≲ 1M⊙) to a radiative envelope (for M∗ ≳ 1M⊙) +(Kippenhahn et al. 2013). +The interior structure of a star is also affected by its +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +9 +age in a number of ways. The main effects are: (1) at a +given age, there is a main-sequence turn-over mass Mmax +and we ignore contributions to the TDE rate by post-main- +sequence stars; (2) low mass stars may take up to ∼ 1 Gyr +to contract to the main-sequence. The pre-main-sequence +(PMS) contraction over time makes the star more compact +and hence harder to be tidally disrupted as the star ages. To +our knowledge, tidal disruption of PMS stars have not been +carefully considered in the literature before. We find that +young PMS stars can be disrupted by very massive BHs. +For instance, a 0.2 M⊙, 0.1 Gyr-old PMS star can be tidally +disrupted by non-spinning BHs up to MBH ≃ 5 × 107M⊙ +(see Table D1 in Appendix D). +3.1.4 +TDE rate fraction and the maximum BH mass +In +this +subsection, +we +discuss +the +TDE +fraction +fTDE(MBH, j) += +ΓTDE/Γlc as a function of BH mass +and spin, for (single) stars of different masses and ages. The +results are shown in Figure 7. +For all stellar masses and ages, the universal trend is +that fTDE drops rapidly at high BH masses, as the loss-cone +scattering rates are dominated by direct captures instead +of observable TDEs. The BH spin plays an important role +in creating a region of large tidal forces for prograde orbits +in the parameter space of L, lz that is inaccessible for non- +spinning BHs. This makes it possible to, at least in principle, +probe the BH spin distribution using the measured TDE rate +at different BH masses. +For instance, the plummet of fTDE with the BH mass in- +dicates the existence of a maximum BH mass MBH,max that +is able to produce the observable TDE for fixed j, M∗, tage. +We define MBH,max(j, M∗, tage) to be the critical BH mass +at which fTDE = 10−3 (the TDE rate at even higher BH +masses are extremely small). The value of MBH,max for +different j, M∗, tage are shown in Figure 8 and listed in +Table D1 in Appendix D. We find that a solar-like star +(M∗ = 1 M⊙, tage = 1.0 Gyr) can be tidally disrupted by +BHs up to a maximum mass that depends on the spin: +MBH,max/(108M⊙) ≃ 1.2 (for j = 0), 1.6 (j = 0.5), 3 +(j = 0.9), 6 (j = 0.99). +Unfortunately, MBH,max(j, M∗, tage) also depends on +the stellar mass and age in a way that is degenerate with +the effects of the BH spin. Roughly speaking, more mas- +sive stars in a younger stellar population can be disrupted +by more massive BHs. For instance, a high-mass young star +with M∗ = 3.7M⊙ and tage = 0.1 Gyr can be tidally dis- +rupted by BHs up to MBH,max/(108M⊙) ≃ 3 (for j = 0) and +8 (j = 0.9). This means that in order to strongly constrain +the BH spin distribution, one must independently constrain +the mass and age distributions of the stellar population. We +will return to the degeneracy in 3.2.3 when considering the +entire stellar population. +3.2 +Averaging over the stellar population +After examining the TDE rate by stars of fixed mass and +age, we now proceed to integrate the stellar populations of +different masses. We consider the simplest case where the +stars surrounding the BH are formed at the same time fol- +lowing the Kroupa IMF. +106 +107 +108 +109 +MBH [M +] +10 +3 +10 +2 +10 +1 +100 +fTDE +M * = 0.2 M +106 +107 +108 +109 +MBH [M +] +10 +3 +10 +2 +10 +1 +100 +fTDE +M * = 1.0 M +106 +107 +108 +109 +MBH [M +] +10 +3 +10 +2 +10 +1 +100 +fTDE +M * = 3.7 M +j = 0.0 +j = 0.5 +j = 0.9 +j = 0.99 +tage = 0.1 Gyr +tage = 1.0 Gyr +tage = 5.0 Gyr +Figure 7. The TDE rate fraction, fTDE (the fraction of loss-cone +scatterings that produce observable TDEs, eq. 18) as a function +of BH mass and spin, for different single-star masses (different +panels) and ages (different line styles). The line colors indicate +different BH spins j. From the middle panel (for M⊙ = 1 M⊙), we +see that the stellar age plays a minor role for the tidal disruption +of solar-like stars (only the BH spin is important). In the bottom +panel (for M∗ = 3.7 M⊙), only the results at tage = 0.1 Gyr are +shown since the star evolves off the main-sequence long before +tage = 1 Gyr. +3.2.1 +TDE rate dependence on stellar population age +The population-averaged TDE fraction ⟨fTDE⟩, obtained +based on eq. (19), is a function of the BH mass and spin +as well as the stellar age tage. This is shown in the left pan- +els of Figure 9. +Overall, ⟨fTDE⟩ for the stellar population follows a sim- +ilar trend as in Figure 7 for single stars. The pre-main- +sequence contraction of low-mass stars causes ⟨fTDE⟩ to de- +crease at early times (before 1 Gyr). At a fixed age tage, the +TDE fraction drops rapidly at high BH masses due to the +dominance of direct captures. For higher spins, TDEs can be +observed up to higher BH masses. We define a population- +© 0000 RAS, MNRAS 000, 000–000 + +10 +Huang & Lu +Figure 8. The maximum BH mass, defined by fTDE(MBH,max) = 10−3, for observable TDEs for different stellar masses M∗ and BH +spins j. Different panels show the results for different stellar ages tage. The underlying color plot is smoothed using third-order spline +interpolation for better visualization. The horizontal axis is logarithmic in log(1 − j). +averaged maximum BH mass ⟨MBH,max⟩ by +⟨fTDE⟩ (MBH = ⟨MBH,max⟩) = 10−3, +(20) +which means that at higher BH masses MBH > ⟨MBH,max⟩, +less than 10−3 of the stars that are scattered into the loss- +cone would give rise to observable TDEs. The values of +⟨MBH,max⟩ for different BH spins j and stellar population +ages tage are plotted in Figure 10 and listed in Table D2 in +Appendix D. +To convert ⟨fTDE⟩ into the per-galaxy TDE rate, we +adopt +⟨ΓTDE⟩ = ⟨fTDE⟩ ˙N, +(21) +where ˙N = ⟨Γlc⟩ has the following power-law form +˙N(MBH) = ⟨Γlc⟩ = 10−4 +� MBH +106 M⊙ +�−0.3 +yr−1. +(22) +It should be noted that our theoretical understanding of +the stellar dynamics near galactic nuclei and hence the +rate of loss-cone scatterings is rather limited (with a num- +ber of major uncertainties, see Stone et al. 2020, for a re- +cent review). However, one aspect of the functional form of +˙N(MBH) that is widely agreed upon is the relatively weak +dependence on the BH mass. For instance, Wang & Merritt +(2004) found +˙N ∝ M −0.25 +BH +by combining the M-σ relation +with two-body relaxation in galactic nuclei with spherically +symmetric, isotropic stellar distribution function. Stone & +Metzger (2016) found ˙N ∝ M −0.4 +BH +by applying the loss-cone +theory to a large galaxy sample. The reason for a gradu- +ally decreasing loss-cone scattering rate with the BH mass +is that the 2-body relaxation timescale near the sphere of +influence is longer for galactic nuclei hosting bigger BHs, +trel ∝ MBHr/σ ∝ M 2 +BH/σ3, where r ∼ GMBH/σ2 is the ra- +dius of the BH’s sphere of influence and σ is the velocity dis- +persion near r. Making use of the M ∝ σ4.4 correlation (Ko- +rmendy & Ho 2013), we obtain trel ∝ M 1.3 +BH, so the loss-cone +scattering rate roughly scales as Γlc ∝ MBH/trel ∝ M −0.3 +BH . +We defer to a future work the detailed calculation of the +loss-cone scattering rate, which depends on the distribution +function of stars and its variation among different galax- +ies. In the current work, the main point is that the drop +in the TDE fraction ⟨fTDE⟩ above MBH ∼ few × 107 M⊙ +(due to direction captures) is much steeper than the func- +tional dependence of the loss-cone scattering rate ˙N(MBH). +Therefore, the sharp drop of the observed TDE rate function +⟨ΓTDE⟩ (MBH) (see Figure 3 of van Velzen 2018) is largely +due to the general relativistic effects of the event horizon. +Our eq. (22) is a rough representation of the weak depen- +dence of ˙N(MBH), and our conclusions are only weakly af- +fected by our choice here, although we caution that the nor- +malization value of 10−4 yr−1 in eq. (22) should not be taken +too seriously. +3.2.2 +Demographics in the masses of the disrupted stars +We then study the contributions to the TDE rate fraction +fTDE from different stellar mass ranges. The results are +shown in Figure 11, where we divide the entire stellar popu- +lation at a given age into three ranges roughly in logarithmic +bins: 0.085 < M∗/M⊙ < 0.3 (mostly M-dwarfs, 57.5% of +stars in the Kroupa IMF), 0.3 < M∗/M⊙ < 1 (mostly K/G- +types, 32.5%), and M∗/M⊙ > 1 (F-type and above, 10%). +Figure 11 shows the normalized fractional contributions to +⟨fTDE⟩ by these three stellar mass bins. In Figure D1 in Ap- +pendix D, we show a different version of the decomposition +into three stellar mass bins, without normalizing the sum of +the contributions to unity. +For low BH masses MBH ≲ 107M⊙, the observable TDE +rate is dominated by the stars in the lowest two mass bins +and their contributions are comparable to each other (as +dictated by the Kroupa IMF). For high BH masses MBH ≳ +107M⊙, the stellar demographics changes depending on the +BH spin and the age of the stellar population. At the highest +BH mass end, TDEs are always dominated by the highest +mass stars that are still on the main-sequence (for a given +age). For instance, at an age of tage = 0.1 Gyr, the highest +stellar mass bin dominates the TDE rate at MBH ≳ 108M⊙ +for both spins j = 0 and 0.9. At older ages tage ≳ 1 Gyr, +stars more massive than about 2.5(tage/Gyr)−0.4M⊙ have +evolved off the main-sequence and hence the contribution +to the observable TDE rate by the highest stellar mass bin +decreases rapidly with stellar age. +© 0000 RAS, MNRAS 000, 000–000 + +tage = 0.1 Gyr +tage = 1.0 Gyr +tage = 10.0Gyr +9.25 +3.0 - +9 +.00 +3.0 - +3.0 - +9.00 +8.75 +8.75 +8.50 +8.75. +8.50 +1.0- +1.0 - +1.0 - +8.50 +M +8.25 +8.00 +8 +8 +0.3 - +0.3 +.00 +0.3 +7.75 +7.50 +7.50 +Q +0.1 - +0.1 +0.1 +7.25 +0 0.5 +0.9 +0.99 +0 0.5 +0.9 +0.99 +0 0.5 +0.9 +0.99 +i +j +jTDEs by spinning BHs +11 +106 +107 +108 +109 +MBH [M ⊙ ] +10−3 +10−2 +10−1 +100 +⟨fTDE⟩ +tage = 0⟩1 Gyr +j = 0.0 +j = 0.5 +j = 0.9 +j = 0.99 +j = 0.0, +tage = 10.0 Gyr +106 +107 +108 +109 +MBH [M ⊙ ] +10−7 +10−6 +10−5 +10−4 + ΓTDE⟩ [gal−1yr−1] +tage = 0.1⟩Gyr +∝ M−0.3 +106 +107 +108 +109 +MBH [M ⊙ ] +10−3 +10−2 +10−1 +100 +⟨fTDE⟩ +tage = 1.0 Gyr +j = 0.0 +j = 0.5 +j = 0.9 +j = 0.99 +j = 0.0, +tage = 10.0 Gyr +106 +107 +108 +109 +MBH [M ⊙ ] +10−7 +10−6 +10−5 +10−4 + ΓTDE⟩ [gal−1yr−1] +tage = 1.0⟩Gyr +∝ M−0.3 +106 +107 +108 +109 +MBH [M ⊙ ] +10−3 +10−2 +10−1 +100 +⟨fTDE⟩ +tage = 10.0 Gyr +j = 0.0 +j = 0.5 +j = 0.9 +j = 0.99 +j = 0.0, +tage = 10.0 Gyr +106 +107 +108 +109 +MBH [M ⊙ ] +10−7 +10−6 +10−5 +10−4 + ΓTDE⟩ [gal−1yr−1] +tage = 10.0⟩Gyr +∝ M−0.3 +Figure 9. +The Kroupa-population-averaged TDE rate. Left Panels: ⟨fTDE⟩ = ⟨ΓTDE⟩ / ⟨Γlc⟩, the fraction of the TDE rate over the +rate of loss-cone scatterings. From top to bottom, the panels show the result of different stellar population ages tage = 0.1, 1, 10 Gyr. In +each panel, the curve for j = 0.0, tage = 10.0 Gyr is shown in dark-gray color to guide the eye. Right Panels: Per-galaxy TDE rates given +by ⟨ΓTDE⟩ = ⟨fTDE⟩ ˙N for a power-law loss-cone scattering rate ˙N = ⟨Γlc⟩ = 10−4 � +MBH/106 M⊙ +�−0.3 gal−1yr−1. The power-law itself +is shown as the black dashed line. +3.2.3 +Spin-Age Degeneracy (SAD) +As can be clearly seen from Figure 9, 10 and Table D2 +in Appendix D, there is a degeneracy between the age of +the stellar population and the BH spin, because BHs with +MBH ≳ 108M⊙ can produce observable (by “observable” we +mean ⟨fTDE⟩ > 10−3) TDEs provided that the stellar popu- +lation is young tage ≲ 1 Gyr or the spin is high a ≳ 0.5 (or a +combination of these two). A possible example of such TDEs +is ASASSN-15lh, where the inferred mass of the BH at the +nucleus of the host galaxy is MBH ≳ 2 × 108 M⊙ (Leloudas +et al. 2016). At such high BH masses, TDEs are necessarily +limited to the region close to the horizon. High spins make +it possible for low-mass stars in prograde orbits to reach +closer to the BH and experience stronger tidal forces. Mean- +while, a younger age means that short-lived high-mass stars, +which are easier to tidally disrupt, can give rise to observable +TDEs. Hereafter, we call this spin-age degeneracy (SAD). +To further explore the competition between the spin and +the age of the stellar population, we show the population- +averaged TDE fraction ⟨fTDE⟩ for two selected BH masses, +MBH = 107, 107.7 M⊙, in Figure 12 and in Table D3 in Ap- +pendix D. We see that for MBH = 107 M⊙ (or lower BH +© 0000 RAS, MNRAS 000, 000–000 + +12 +Huang & Lu +Figure +10. +The population-averaged maximum BH mass +� +MBH,max +� +(defined by eq. 20) that can produce observable TDEs +for different stellar population ages tage and BH spins j. The un- +derlying color plot is smoothed using third-order spline interpo- +lation for better visualization. The horizontal axis is logarithmic +in log(1 − j). +masses), the effects of spin and stellar age are both mild. The +main change in ⟨fTDE⟩ (MBH = 107 M⊙) occurs between +tage = 0.1 Gyr and 1.0 Gyr, mainly due to the contraction +of low-mass pre-main-sequence stars. For MBH = 107 M⊙, +TDE fraction at tage ≳ 1 Gyr does not depend on the age +of the stellar population and only slightly increases with j. +However, for MBH = 107.7 M⊙ (or higher BH masses), the +effects of j and tage are both prominent. For instance, a +young stellar population at tage = 0.1 Gyr produces a much +larger TDE fraction than an old population at tage = 10 Gyr +by a factor of 3 to 5 (depending on the spin). A high BH +spin of j = 0.9 also produces a larger TDE fraction than +the case of zero spin by a factor of 2 to 5 (depending on the +stellar age). +Therefore, we emphasize the importance of including +the effects of stellar population ages when using the TDE +rates to constrain the BH spin distribution. +4 +DISCUSSION +In this section, we discuss the limitations and caveats in this +work and how future works can improve upon our calcula- +tions. +(1) Our relativistic criteria for tidal disruption are dif- +ferent from the previous studies that only consider the max- +imum strength of the tidal forces at the pericenter (Kes- +den 2012; Coughlin & Nixon 2022b,a). By integrating the +geodesic deviation equation (eq. 9), our criteria explicitly +calculate the work done by the tidal forces during the peri- +center passage under the framework of the frozen-in approx- +imation. The consideration of the work done by the tidal +forces, as opposed to only the maximum strength, is impor- +tant in the tidal disruption condition (see Figure 3). +In Ryu et al. (2020b), it is argued that the physical tidal +disruption radius can be modeled by equating the tidal force +at the pericenter to the stellar gravity scaled by a fixed con- +stant (see their eq. 12 and Figure 10). A similar treatment is +106 +107 +108 +MBH [M ⊙ ] +10−2 +10−1 +100 +⟨fTDE, mass bin⟩/⟨fTDE⟩ +tage = 0⟩1⊙G r +j = 0⟩0 +j = 0⟩9 +M ⟨ /M ⊙ < 0⟩3 +0⟩3 < M ⟨ /M ⊙ < 1⟩0 +1⟩0 < M ⟨ /M ⊙ +106 +107 +108 +MBH [M ⊙ ] +10−2 +10−1 +100 +⟨fTDE, mass bin⟩/⟨fTDE⟩ +tage = 1⟩0⊙G r +106 +107 +108 +MBH [M ⊙ ] +10−2 +10−1 +100 +⟨fTDE, mass bin⟩/⟨fTDE⟩ +tage = 10⟩0⊙G r +Figure 11. The fractional contributions to the observable TDE +fraction ⟨fTDE⟩ by stars in the three mass bins, M∗/M⊙ < 0.3 +(solid), 0.3 < M∗/M⊙ < 1.0 (dashed), 1.0 < M∗/M⊙ (dotted +lines). The blue and red lines represent BH spins of j = 0.0, 0.9, +respectively. The data are only plotted if +� +fTDE,mass bin +� +> 10−5 +and ⟨fTDE⟩ > 10−3. +also proposed by Coughlin & Nixon (2022a). Even though +(Ryu et al. 2020b) has demonstrated the applicability of +this argument for low BH masses (106 M⊙), it is unlikely to +hold at higher BH masses, where the TDE is increasingly +relativistic. If the disruption criteria via the maximum tidal +forces were to hold, we would expect the hydrodynamic sim- +ulation results of Ryu et al. (2020c) to trace out the contours +of maximum tidal forces5 in Figure 3. However, even with +5 Consequently, the suppression of the TDE rate at high BH +masses would be much steeper (see Figure 6 of Coughlin & Nixon +2022b) compared to our results. +© 0000 RAS, MNRAS 000, 000–000 + +101 +8.60 +log10((MBH, max)/M ) +8.40 +O +2 +8 +.40 +100- +8 +8.20 +09 +8.00 +10-1, +0 +0.5 +0.9 +0.99 +jTDEs by spinning BHs +13 +10−6 +10−5 +⟨ΓTDE⟩ [gal−1yr−1] +0.00 +0.25 +0.50 +0.75 +0.99 +j +10−2 +10−1 +⟨fTDE⟩ +tage = 0.1 Gyr +tage = 1.0 Gyr +tage = 5.0 Gyr +tage = 10.0 Gyr +log10(MBH/M ⊙ ⊙ = 7.0 +log10(MBH/M ⊙ ⊙ = 7.7 +Figure +12. +Population-averaged TDE fraction ⟨fTDE⟩ for +different ages tage and spins j, at two selected BH masses, +log10(MBH/M⊙) += +7.0 +(triangles) +and +7.7 +(circles). +The +right vertical axis uses the loss-cone scattering rate +˙N += +10−4 � +MBH/106 M⊙ +�−0.3 gal−1yr−1 evaluated at the geometric- +mean BH mass, log10(MBH/M⊙) = 7.35, to convert ⟨fTDE⟩ into +the observable TDE rate ⟨ΓTDE⟩. The data used in the figure are +listed in Table D3. +the explicit form of the relativistic tidal force, the contours +of maximum tidal forces (dashed lines) fail to reproduce the +results of hydrodynamic simulations. This shows that there +is no single scaling factor to predict the disruption of a given +star across all BH masses; instead, the work done by tidal +forces must be taken into consideration. +We caution that our criteria, based on the combination +of the work done by tidal forces and the relativistic Roche +lobe condition, are only approximate and need to be further +verified by future hydrodynamic simulations in Kerr space- +time. Nevertheless, this is the first attempt to incorporate +full general relativity and stellar density structure to predict +the TDE rate. This allows us to make direct comparison to +the observed TDE rate from on-going and future surveys. +(2) Our model cannot predict the loss-cone scattering +rate in a galaxy, which involves the distribution function of +stars, their mutual interactions, as well as interactions with +other massive objects in galactic nuclei. Instead, we assume +a loss-cone scattering rate ˙N, which can in principle be ob- +tained from the loss-cone theory (Merritt 2013), and then +calculate the fraction of these stars that produce observable +TDEs fTDE. The independent combination of the loss-cone +theory with our relativistic tidal disruption criteria, how- +ever, has some important caveats. +The first caveat is that the loss-cone theory must be +fully general relativistic, which is non-trivial6. The second +caveat is the nature of the loss cone. Our assumption of +an isotropic velocity distribution of stars can be effectively +viewed as the full loss-cone regime. This assumption, how- +ever, is highly idealized, as the relaxation timescale for stars +near high-mass BHs is much longer. Taking stars near the +sphere of influence r ∼ GMBH/σ2 as an example (where +σ is the velocity dispersion near r), the angular momen- +tum diffusion timescale for highly eccentric orbits with peri- +center radii rp ≪ r is roughly given by tJ ∼ (rp/r)trel, +where trel is the 2-body relaxation timescale near r. The +orbital period is P(r) ∼ r/σ, so we obtain the following +scaling tJ/P ∝ rptrelσ5/M 2 +BH. For high-mass BHs, we take +rp ∝ rg ∝ MBH and trel ∝ M 1.3 +BH obtained based on the M-σ +correlation of M ∝ σ4.4 (see §3.2.1), and then the ratio be- +tween the two timescales scales as tJ/P ∝ M 1.4 +BH. Stars are +in the empty (or full) loss-cone regime when tJ ≫ P (or +tJ ≪ P). We see that it is likely that the stars in the nu- +clei of very massive BHs are in the empty loss-cone regime, +as argued by Merritt (2013). If this is the case, our full-loss- +cone assumption then leads to an underestimate of the TDE +fraction fTDE, meaning that our predicted TDE rate should +be considered as lower limits when compared with observa- +tions. The investigation of the effects of the more realistic +loss cone near spinning BHs is deferred to a future work. +(3) Throughout this work, we take 50% mass loss as +a representative criterion for an observable TDE. This as- +sumption is partly due to our poor understanding of the +(optical and X-ray) emission mechanisms of TDEs. More +realistically, partial TDEs with smaller mass loss fractions +may be observable from nearby galaxies. In the future when +better understanding of the TDE emission (especially in the +optical band) is available, one might revise our strict cut of +50% mass loss to other values. For instance, for the case of +30% mass loss, our tidal disruption criteria would then need +to be modified to consider the fluid elements at a radius of +R30 (for exterior mass of 0.3M∗) and the maximum differen- +tial velocity ∆vmax due to tidal forces needs to be compared +with the virial velocity of vvir = +� +GM∗(1 − 30%)/R30. Be- +cause the fractional mass loss from the star is a very steep +function of the pericenter radius (Guillochon & Ramirez- +Ruiz 2013; Ryu et al. 2020a), we expect that our results will +only be weakly affected by this aspect of uncertainty. We +leave to a future work to explore in detail other fractions of +mass loss from the star. +(4) We have assumed, for simplicity, a single stellar pop- +ulation as given by the Kroupa IMF. There is evidence that +the IMF near our own Galactic Center is more top-heavy (Lu +et al. 2013). Such top-heavy IMFs will increase the TDE rate +from the most massive BHs as compared to our predictions. +Recent observations showed that the post-starburst galax- +ies, despite their rarity, are over-represented in the detected +TDE sample (French et al. 2020; Hammerstein et al. 2021). +Moreover, Bortolas (2022) studied the dynamics of nuclear +6 In the empty loss-cone limit and for sufficiently high BH masses, +the spin of the BH makes the stellar distribution function non- +spherical and anisotropic, since retrograde orbits have higher di- +rect capture cross-sections which depends on the orbital inclina- +tion wrt. the BH spin axis. This makes the general relativistic +loss-cone theory non-trivial (on top of the large uncertainties in +the velocity and density distributions of stars). +© 0000 RAS, MNRAS 000, 000–000 + +14 +Huang & Lu +star clusters with IMFs with various degrees of top-heaviness +and found that the loss-cone scattering rate is strongly en- +hanced due to mass segregation in the early evolution of +the cluster at tage ≲ 1 Gyr as compared to an old cluster +at tage = 10 Gyr. These previous works, together with our +findings, demonstrate that the influence of the stellar pop- +ulation age on the observed TDE rate cannot be ignored +— the stellar population age, if unconstrained, will severely +compromise the utility of TDEs as a probe of BH spin dis- +tribution (see 3.2.3). Self-consistent calculations of the TDE +rate must consider the current stellar mass function and the +dynamics in galactic nuclei in the framework of the relativis- +tic loss-cone theory for spinning BHs. +(5) Finally, we have assume that a star that is tidally +disrupted outside the horizon would produce an electromag- +netically bright and detectable signal. An important point +is that, for the extreme cases near the highest mass BHs, +about half of the tidally stripped debris would plunge into +the BH and the other half becomes unbound. The question +is whether the bound (but plunging) debris would produce +bright emission before entering the event horizon. Recently, +Ryu et al. (2022) carried out hydrodynamic simulation of a +TDE with stellar pericenter distance of rp = 4.02 rg for a +non-spinning BH. They found that, although the bound de- +bris are largely in plunging orbits (and certainly do not form +a rotationally supported accretion disk), internal shocks +form as the material fall towards the BH due to apsidal +precession and that the radiative efficiency (in terms of rest +mass) is of the order a few percent. Therefore, it is possible +that even extremely relativistic TDEs produce detectable +emission. +5 +SUMMARY +In this paper, we quantify the suppression of the observable +TDE rate due to direct captures of stars by the event horizon +of spinning BHs, improving upon the work of Kesden (2012). +We first generalize the commonly adopted frozen-in ap- +proximation from the Newtonian limit to the general rela- +tivistic case, including the effects of stellar interior structure. +This is achieved by integrating the equation of motion ac- +cording to the tidal tensor in the comoving frame of the +center of mass of the star on a Kerr geodesic. Our integra- +tion starts when the star first enter the sphere of radius +r0 from the BH and ends when it exits from the sphere. +By uniformly sampling the fluid elements on the sphere at +half-mass radius Rhm from the stellar center, we obtain the +maximum velocity ∆v(r0) achieved by all these fluid ele- +ments in the comoving frame of the star’s center of mass. +The choice of half-mass radius is motivated by our consid- +eration of 50% mass loss from the star as the threshold for +bright TDEs (partial TDEs with a much smaller fractional +mass loss would be fainter and more difficult to detect). +Then, by varying the initial radius r0, we obtain the max- +imum velocity ∆vmax = max{all r0}[∆v(r0)], which repre- +sents the maximum possible work done by tidal forces on +any of the fluid elements at the star’s half-mass radius. We +further restrict the initial radius r0 by requiring that an un- +perturbed star fills up the relativistic Roche lobe at all radii +r < r0, because otherwise there will be no mass loss from the +star. Finally, if the maximum differential velocity between +the fluid element and the star’s center of mass exceeds the +virial velocity at half-mass radius vvir = +� +GM∗/(2Rhm), +we infer that the star will lost more than 50% of its mass +and that there will be a bright, detectable TDE. The above +tidal disruption criteria are in good agreement with the re- +sults from the relativistic hydrodynamic simulations carried +out by Ryu et al. (2020c) (see Figure 3 for a comparison). +The next step is to consider the angular momentum dis- +tribution of stars that are scattered into the loss-cone. We +consider the full loss-cone case and calculate the ratio be- +tween the rates of TDEs and direct captures, ΓTDE/Γcapt, +for a given star (the ratio is directly given by the cross- +sections of TDEs and direct captures). The observable TDE +fraction is then defined as fTDE = ΓTDE/(ΓTDE + Γcapt), +where the denominator is the total rate of loss-cone scatter- +ings Γlc = ΓTDE + Γcapt. We then consider a stellar pop- +ulation as given by the Kroupa IMF at different ages. By +integrating over the stellar mass distribution at a given age +of the population, we calculate the population-averaged ob- +servable TDE fraction ⟨fTDE⟩ = ⟨ΓTDE⟩ / ⟨Γlc⟩, which de- +pends on the BH mass MBH and dimensionless spin j, and +the age of the stellar population tage. +The population-averaged TDE rate per galaxy is then +given by ⟨ΓTDE⟩ = ⟨fTDE⟩ (MBH, j, tage) × ⟨Γlc⟩, where ⟨Γlc⟩ +is the population-averaged rate of loss-cone scatterings. To +make a direct comparison with observational measurements +of the TDE rate as a function of the BH mass (provided +that MBH can be inferred from e.g., the M-σ correlation), +we further need the loss-cone scattering rate ⟨Γlc⟩. Despite +uncertainties in the distribution function of stars near galac- +tic nuclei, it is theoretically expected that ⟨Γlc⟩ only de- +pends weakly on the BH mass across different galaxies (e.g., +Wang & Merritt 2004; Merritt 2013; Stone & Metzger 2016). +Thus, the observed TDE rate is mainly sensitive to the TDE +fraction ⟨fTDE⟩ (MBH, j, tage), and this makes it possible to +achieve an important goal of the TDE community — to con- +strain the spin distribution of dormant BHs. +The results in this work make the first step towards this +goal. However, we find that a serious hurdle to overcome is +the spin-age degeneracy (SAD), which means that either a +young stellar population (tage ≲ 1 Gyr) or a high BH spin +(a ≳ 0.5), or a combination of these two factors, can extend +the mass function of TDE-hosting BHs significantly above +108 M⊙. A possible example of such TDEs is ASASSN-15lh, +where the inferred mass of the hosting BH is MBH ≳ 2 × +108 M⊙ (Leloudas et al. 2016). +To break this degeneracy with complimentary informa- +tion, we suggest the following strategies: +• Make use of the electromagnetic signals from the TDE +to independently constrain the mass of the disrupted star +(e.g., Mockler et al. 2022). However, this requires a sig- +nificant improvement in our understanding of the multi- +wavelength emission mechanisms of TDEs. +• Systematically search for signatures (in the X-ray and +radio bands) of relativistic jets from TDEs hosted by the +most massive BHs. According to the Blandford & Znajek +(1977) mechanism, relativistic jets are most likely associated +with high BH spins. +• Use high spatial resolution imaging or spectroscopy to +obtain the spectral energy distribution (SED) or spectrum +of the stars near the TDE-hosting galactic nuclei, after the +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +15 +TDE has faded away. However, the TDE emission may over- +shine the stellar emission for many decades in the UV bands +(van Velzen et al. 2019). +ACKNOWLEDGMENTS +HTH would like to thank the financial support from Depart- +ment of Physics, The Chinese University of Hong Kong for +this research. +DATA AVAILABILITY +The data underlying this article will be shared on reasonable +request to the corresponding author. +REFERENCES +Andalman Z. L., Liska M. T. P., Tchekhovskoy A., Coughlin +E. R., Stone N., 2022, MNRAS, 510, 1627 +Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433 +Bonnerot C., Lu W., Hopkins P. 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S., 2019, ApJ, 878, 82 +van Velzen S., et al., 2021, ApJ, 908, 4 +APPENDIX A: TIDAL TENSOR +Here we provide the full expression of tidal tensor in Kerr +spacetime: +C11 = +� +1 − 3ST(r2 − a2 cos2 θ) +KΣ2 +cos2 Ψ +� +I1 ++ 6ar cos θ ST +KΣ2 cos2 ΨI2, +(A1) +C12 = +� +−ar cos θ(S + T)I1 + (a2 cos2 θS − r2T)I2 +� +× 3 +√ +ST +KΣ2 cos Ψ, +(A2) +C13 = +� +(a2 cos2 θ − r2)I1 + 2ar cos θI2 +� +3 ST +KΣ2 cos Ψ sin Ψ, +(A3) +C22 = +� +1 + 3r2T 2 − a2 cos2 θS2 +KΣ2 +� +I1 − 6ar cos θ ST +KΣ2 I2, +(A4) +C23 = +� +−ar cos θ(S + T)I1 + (a2 cos2 θS − r2T)I2 +� +× 3 +√ +ST +KΣ2 sin Ψ, +(A5) +C33 = +� +1 − 3ST(r2 − a2 cos2 θ) +KΣ2 +sin2 Ψ +� +I1 ++ 6ar cos θ ST +KΣ2 sin2 ΨI2, +(A6) +where +S = K + r2, +(A7) +T = K − a2 cos2 θ, +(A8) +I1 = MBHr +Σ3 +(r2 − 3a2 cos2 θ), +(A9) +© 0000 RAS, MNRAS 000, 000–000 + +16 +Huang & Lu +I2 = MBHa cos θ +Σ3 +(3r2 − a2 cos2 θ), +(A10) +and K is related to the Carter constant Q by +K = Q + (Lz − aE)2 . +(A11) +The expression of C21, C31, C32 follows directly from the +symmetry of the tidal tensor. The angle Ψ in the above ex- +pressions is the rotation angle that ensures the basis of the +coordinates is parallel transported. The value of Ψ changes +along the geodesic according to +dΨ +dτ = +√ +K +Σ +�(r2 + a2)E − aLz +K + r2 ++ aLz − aE sin2 θ +K − a2 cos2 θ +� +. +(A12) +We refer the readers to Marck (1983) for a complete deriva- +tion behind those expressions. +APPENDIX B: RELATIVISTIC ROCHE LOBE +Consider a star subjected to the forces from a tidal tensor +Cij. The tidal force F i at a given position can be written in +Fermi Normal Coordinates as +F i = −Cijχj, +(B1) +where χ is the displacement 3-vector from the star center. +The symmetric tidal tensor allows the definition of the tidal +potential +Φtide = 1 +2Cijχiχj, +(B2) +such that +⃗F = −∇Φtide. +(B3) +Furthermore, the symmetry of the tidal tensor ensures the +existence of a rotated coordinate ˜χ in which the tidal tensor +is diagonal: +˜Cij = +� +� +ξ1 +0 +0 +0 +ξ2 +0 +0 +0 +ξ3 +� +� , +(B4) +Φtide =1 +2 +� +i=1,2,3 +ξi(˜χi)2, +(B5) +where ξi are the eigenvalues of the tidal tensor. Let ξ1 be +the smallest eigenvalue among ξi and ξ1 is guaranteed to be +negative because the tidal tensor is traceless. +The total gravitational potential experienced by the +star Φtotal is the sum of Φtide and its own self-gravity Φstar: +Φtotal = Φtide + Φstar. +(B6) +The tidal forces will deform the star and Φstar will in general +deviate from the potential of a perfect sphere. Nonetheless, +we assume that prior to reaching the initial radius r0, the +deformation of the star is small and its potential can be well- +approximated as spherical. In the natural units G = MBH = +c = 1, +Φstar = −M∗/MBH +|˜χ| +. +(B7) +Unlike the binary system, Φtotal here does not contain the +potential of the centrifugal force as we assume the star to +be non-rotating. +The inclusion of Φtide introduces a critical value of po- +tential Φcrit, above which the equipotential surface is no +longer closed within the inner Lagrangian point. The equipo- +tential surface of Φcrit thus marks the maximum extent that +a star in hydrostatic equilibrium can reach before starting to +lose mass. We refer to the equipotential surface Φcrit as the +“Roche lobe”, although it differs from the classical Roche +lobe in a circular binary system where the donor star is +assumed to be in synchronous rotation, because here we as- +sume the star to be non-rotating (and hence the potential +Φtot does not include the centrifugal term). +The value of Φcrit is determined by the saddle point in +Φtotal, which can be obtained by solving ∇Φtotal = 0 and is +given by +Φcrit = −3 +2(−ξ1)1/3(M∗/MBH)2/3. +(B8) +The volume of the Roche lobe VRL can be written as +VRL = +� π +0 +� 2π +0 +˜r3(˜θ, ˜φ) +3 +d˜φd˜θ, +(B9) +where ˜r, ˜θ, ˜φ are the spherical coordinates of the frame ˜χ: +˜χ1 =˜r sin ˜θ cos ˜φ, +(B10) +˜χ2 =˜r sin ˜θ sin ˜φ, +(B11) +˜χ3 =˜r cos ˜θ. +(B12) +The value of ˜r for a given set of ˜θ, ˜φ is determined by +Φcrit =1 +2 +� +i=1,2,3 +ξi(˜χi)2 − M∗/MBH +|˜χ| +=1 +2 ˜r2 � +ξ1 sin2 ˜θ cos2 ˜φ + ξ2 sin2 ˜θ sin2 ˜φ + ξ3 cos2 ˜θ +� +− M∗/MBH +˜r +. +(B13) +An additional property of the Roche lobe is gained by +observing that the above equation (Eq. B13) can be rewrit- +ten in a scaled radius ˜r′ = ˜r(M∗/MBH)−1/3 and has the +form +− 3 +2(−ξ1)1/3 = +1 +2(˜r′)2 � +ξ1 sin2 ˜θ cos2 ˜φ + ξ2 sin2 ˜θ sin2 ˜φ + ξ3 cos2 ˜θ +� +− 1 +˜r′ . +(B14) +The value of ˜r′ is therefore independent of the stellar mass +and completely given by the eigen values ξi and the angles +˜θ, ˜φ. This implies that the volume of the Roche Lobe VRL is +proportional to M∗. +APPENDIX C: DIFFERENTIAL RATE +Consider the stars being shot from the surface of a sphere +with radius rinit towards the SMBH located at the origin. +The stellar number density n is uniform on the sphere and +the velocity of the stars v is fixed. The direction of stellar +velocity is assumed to be isotropically distributed. The ini- +tial motion of a star on the sphere can be described by a set +of angles θinit, φinit, θv, φv and is illustrated in Figure C1: +θinit and φinit describe the initial position of the star in the +Boyer-Lindquist coordinates of the Kerr metric (eq. 2). At +© 0000 RAS, MNRAS 000, 000–000 + +TDEs by spinning BHs +17 +the large distance rinit ≫ rg, θinit, φinit can be viewed as the +polar and azimuthal angles of the spherical coordinates. In +the local frame (ˆr, ˆθ, ˆφ) of a given star at its initial posi- +tion, θv and φv are the angles that the velocity of the star +makes with the inward radial direction −ˆr and ˆθ in the plane +perpendicular to ˆr, respectively. In terms of θinit, θv, φv, the +differential rate of the stars can be expressed as +∂3Γ +∂θinit∂θv∂φv = 1 +2r2 +0nv sin θinit sin θv cos θv. +(C1) +Note that we only consider the stars entering the sphere, +with θv ∈ [0, π/2]. +The total specific angular momentum L and the frac- +tional angular momentum in the direction of the black hole +spin axis lz = Lz/L are related to the above angles by +L =r0v sin θv, +(C2) +lz = − sin θinit sin θv. +(C3) +Using the change of variables, the differential rate can be +expressed in terms of L, lz, θinit as +∂3Γ +∂θinit∂L∂lz = n +v +L +� +1 − (lz/ sin θinit)2 . +(C4) +For a given lz, the range of θinit is restricted to the range +θinit ∈ [π/2 − θl, π/2 + θl] , +(C5) +where cos θl = |lz|. By marginalizing the differential rate +over L and lz, we get the differential rate of stars as +∂2Γ +∂L∂lz = πnL +v +. +(C6) +The above expression coincidentally does not depend on lz. +APPENDIX D: ADDITIONAL TABLES AND +FIGURES +In this Appendix, we include Table D1 for the maximum BH +mass for individual stars of given a mass and age and for +different BH spins, Table D2 for the maximum BH mass for +a Kroupa stellar population at a given age and for different +BH spins, and Table D3 for the population-averaged TDE +fraction at different BH masses, spins, and stellar ages. We +also include Figure D1 for the TDE fraction decomposed +into contributions by three stellar mass bins. +Figure C1. The initial movement of the star described by the +angles θinit, φinit, θv, φv. The SMBH is located at the origin with +the spin axis in z direction. The brown vector indicates the initial +position of the star ⃗rinit and the red vector indicates the initial +velocity of the star ⃗v, which are described by black, local coordi- +nates ˆx, ˆy, ˆz, and blue, spherical coordinates ˆr, ˆθ, ˆφ, respectively. +Notice that the velocity of the star (red vector) is pointing inward +as we only consider the star that can approach the vicinity of the +SMBH. +© 0000 RAS, MNRAS 000, 000–000 + +